SQOPT

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Introduction

TOMVIEW /SQOPT (hereafter referred to as SQOPT ) is a program for solving the large-scale linear or quadratic programming problem, which is assumed to be stated in following form:

${\displaystyle {\begin{array}{cccc}\min \limits _{x}&q(x)\\s/t&\leq \left({\begin{array}{c}x\\Ax\end{array}}\right)\leq u,\\\end{array}}}$

where l and u are constant lower and upper bounds, and A is a sparse matrix and q(x) is a linear or quadratic function objective function that may be specified in a variety of ways, depending upon the particular problem being solved. The optional parameter maximize may be used to specify a problem in which q is maximized instead of minimized.

Upper and lower bounds are specified for all variables and constraints. This form allows full generality in specifying various types of constraint. In particular, the jth constraint may be defined as an equality by setting lj = uj . If certain bounds are not present, the associated elements of l or u may be set to special values that are treated as ${\displaystyle -\infty }$ or ${\displaystyle +\infty }$.

The possible forms for the function q(x) are summarized in #Table: Choices for the objective function q(x).. The most general form for q(x) is ${\displaystyle q(x)=f+\sum _{j=1}^{n}c_{j}x_{j}+{\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}x_{i}H_{ij}x_{j}=f+c^{T}x+{\frac {1}{2}}x^{T}Hx,}$

where f is a constant, c is a constant n vector and H is a constant symmetric n × n matrix with elements ${\displaystyle \{H_{ij}\}}$. In this form, q is a quadratic function of x and Problem LCQP is known as a quadratic program (QP). SQOPT is suitable for all convex quadratic programs. The defining feature of a convex QP is that the matrix H must be positive semidefinite -i.e., it must satisfy xTHx >= 0 for all x. If not, q(x) is nonconvex and SQOPT will terminate with the error indicator inform = 4.

Table: Choices for the objective function q(x).

Problem type Objective function q Hessian matrix
Quadratic Programming (QP) ${\displaystyle f+c^{T}x+{\frac {1}{2}}x^{T}Hx}$ Symmetric positive semidefinite
Linear Programming (LP) ${\displaystyle f+c^{T}x}$ ${\displaystyle H=0}$
Feasible Point (FP) Not Applicable ${\displaystyle f=0}$, ${\displaystyle c=0}$, ${\displaystyle H=0}$

If H = 0, then q(x) = f + cTx and the problem is known as a linear program (LP).

If H = 0, f = 0, and c = 0, there is no objective function and the problem is a feasible point problem (FP), which is equivalent to finding a point that satisfies the constraints on x. In the situation where no feasible point exists, several options are available for finding a point that minimizes the constraint violations (see the optional parameter Elastic option).

SQOPT is suitable for large LPs and QPs in which the matrix A is sparse -i.e., when there are sufficiently many zero elements in A to justify storing them implicitly.

SQOPT exploits structure or sparsity in H by requiring H to be defined implicitly in 'a subroutine that computes the product Hx for any given vector x. In many cases, the product Hx can be computed very efficiently for any given x-e.g., H may be a sparse matrix, or a sum of matrices of rank-one.

Brief description of the method

Here we briefly describe some features of the algorithm used in SQOPT and introduce some terminology used in the description of the subroutine and its arguments. For further details, see #Algorithmic Details.

Formulation of the problem

The upper and lower bounds on the m components of Ax are said to define the general constraints of the problem. Internally SQOPT converts the general constraints to equalities by introducing a set of slack variables s, where ${\displaystyle s=(s_{1},s_{2},\dots ,s_{m})^{T}}$. For example, the linear constraint ${\displaystyle 5\leq 2x_{1}+3x_{2}\leq +\infty }$ is replaced by ${\displaystyle 2x_{1}+3x_{2}-s_{1}=0}$ together with the bounded slack ${\displaystyle 5\leq s_{1}\leq +\infty }$. Problem LCQP can therefore be rewritten in the following equivalent form

${\displaystyle \min _{x,s}\quad q(x)\quad {\text{subject to}}\quad Ax-s=0,\quad l\leq \left({\begin{array}{c}x\\s\end{array}}\right)\leq u.}$

Since the slack variables s are subject to the same upper and lower bounds as the components of Ax, they allow us to think of the bounds on Ax and x as simply bounds on the combined vector (x, s). (In order to indicate their special role in problem QP, the original variables x are sometimes known as "column variables", and the slack variables s are known as "row variables")

Each LP or QP is solved using an active-set method. This is an iterative procedure with two phases: a phase 1 (sometimes called the feasibility phase ), in which the sum of infeasibilities is minimized to find a feasible point; and a phase 2 (or optimality phase ), in which q is minimized by constructing a sequence of iterations that lies within the feasible region.

Phase 1 involves solving a linear program of the form

${\displaystyle \min {x,v,w}\ \ \ \sum _{j=1}^{n+m}v_{j}+w_{j}\quad {\text{subject to}}\quad Ax-s=0,\quad \quad l\leq \left({\begin{array}{c}x\\s\end{array}}\right)-v+w\leq v,\quad v\geq 0,w\geq 0}$

which is equivalent to minimizing the sum of the constraint violations. If the constraints are feasible (i.e., at least one feasible point exists), eventually a point will be found at which both v and w are zero. The associated value of (x, s) satisfies the original constraints and is used as the starting point for the phase 2 iterations for minimizing q.

If the constraints are infeasible (i.e., v = 0 or w = 0 at the end of phase 1), no solution exists for Problem LCQP and the user has the option of either terminating or continuing in so-called elastic mode (see the discussion of the optional parameter Elastic option). In elastic mode, a "relaxed" or "perturbed" problem is solved in which q(x) is minimized while allowing some of the bounds to become "elastic"-i.e., to change from their specified values. Variables subject to elastic bounds are known as elastic variables. An elastic variable is free to violate one or both of its original upper or lower bounds.

To make the relaxed problem meaningful, SQOPT minimizes q while (in some sense) finding the "smallest" violation of the elastic variables. In the situation where all the variables are elastic, the relaxed problem has the form

${\displaystyle {\text{Phase2}}(\gamma )}$

${\displaystyle \min _{x,v,w}q(x)+\gamma \sum _{j=1}^{n+m}(v_{j}+w_{j})}$

${\displaystyle {\text{subject to}}Ax-s=0,\quad l\leq \left({\begin{array}{c}x\\s\end{array}}\right)-v+w\leq v,\quad v\geq 0,w\geq 0}$

where ${\displaystyle \gamma }$ is a nonnegative parameter known as the elastic weight, and ${\displaystyle q(x)+\gamma \sum _{j}(v_{j}+w_{j})}$ is called the composite objective. In the more general situation where only a subset of the bounds are elastic, the vs and ws for the non-elastic bounds are fixed at zero.

The elastic weight can be chosen to make the composite objective behave like either the original objective q(x) or the sum of infeasibilities. If γ= 0, SQOPT will attempt to minimize q subject to the (true) upper and lower bounds on the nonelastic variables (and declare the problem infeasible if the nonelastic variables cannot be made feasible). At the other extreme, choosing γ sufficiently large, will have the effect of minimizing the sum of the violations of the elastic variables subject to the original constraints on the non-elastic variables. Choosing a large value of the elastic weight is useful for defining a "least-infeasible" point for an infeasible problem.

In phase 1 and elastic mode, all calculations involving v and w are done implicitly in the sense that an elastic variable xj is allowed to violate its lower bound (say) and an explicit value of v can be recovered as vj = lj - xj .

The main iteration

A constraint is said to be active or binding at x if the associated component of either x or Ax is equal to one of its upper or lower bounds. Since an active constraint in Ax has its associated slack variable at a bound, we can neatly describe the status of both simple and general upper and lower bounds in terms of the status of the variables (x, s). A variable is said to be nonbasic if it is temporarily fixed at its upper or lower bound. It follows that regarding a general constraint as being active is equivalent to thinking of its associated slack as being nonbasic.

At each iteration of an active-set method, the constraints Ax - s = 0 are (conceptually) partitioned into the form BxB + SxS + NxN = 0, where xN comprises the nonbasic components of (x, s) and the basis matrix B is square and nonsingular. The elements of xB and xS are called the basic and superbasic variables respectively; with xN they are a permutation of the elements of x and s. At a QP solution, the basic and superbasic variables will lie somewhere between their bounds, while the nonbasic variables will be equal to one of their upper or lower bounds. At each iteration, xS is regarded as a set of independent variables that are free to move in any desired direction, namely one that will improve the value of the QP objective (or sum of infeasibilities). The basis variables are then adjusted in order to ensure that (x, s) continues to satisfy Ax - s = 0. The number of superbasic variables (nS say) therefore indicates the number of degrees of freedom remaining after the constraints have been satisfied. In broad terms, nS is a measure of how nonlinear the problem is. In particular, nS will always be zero for FP and LP problems.

If it appears that no improvement can be made with the current definition of B, S and N , a nonbasic variable is selected to be added to S, and the process is repeated with the value of nS increased by one. At all stages, if a basic or superbasic variables encounters one of its bounds, the variable is made nonbasic and the value of nS is decreased by one.

Associated with each of the m equality constraints Ax - s = 0 is a dual variable πi . Similarly, each variable in (x, s) has an associated reduced gradient dj (also known as a reduced cost ). The reduced gradients for the variables x are the quantities g - ATπ, where g is the gradient of the QP objective; and the reduced gradients for the slacks s are the dual variables π. The QP subproblem is optimal if dj >= 0 for all nonbasic variables at their lower bounds, dj <= 0 for all nonbasic variables at their upper bounds, and dj = 0 for all superbasic variables. In practice, an approximate QP solution is found by slightly relaxing these conditions on dj (see the Optimality tolerance described in #Description of the optional parameters).

The process of computing and comparing reduced gradients is known as pricing (a term first introduced in the context of the simplex method for linear programming). To "price" a nonbasic variable xj means that the reduced gradient dj associated with the relevant active upper or lower bound on xj is computed via the formula dj = gj - ajTπ, where aj is the jth column of ( A - I ). (The variable selected by the price, and its corresponding value of dj (i.e., its reduced gradient) are printed in the columns marked +SBS and dj in the Print file output.) If A has significantly more columns than rows (i.e., ${\displaystyle n\gg m}$), pricing can be computationally expensive. In this case, a strategy known as partial pricing can be used to compute and compare only a subset of the dj 's.

The SPECS file

The performance of SQOPT is controlled by a number of parameters or "options". These are normally set in Prob.SOL.optPar. Each option has a default value that should be appropriate for most problems. (The defaults are given in the next section.) For special situations it is possible to specify non-standard values for some or all of the options, using data in the following general form:

Begin  SQOPT  options
Iterations limit 		500
Feasibility  tolerance	1.0e-7
Scale  all variables
End SQOPT  options

We shall call such data a SPECS file, since it specifies various options. The file starts with the keyword Begin and ends with End. The file is in free format. Each line specifies a single option, using one or more items as follows:

1. A keyword (required for all options).
2. A phrase (one or more words) that qualifies the keyword (only for some options).
3. A number that specifies an integer or real value (only for some options). Such numbers may be up to 16 contiguous characters in Fortran 77's I, F, E or D formats, terminated by a space.

The items may be entered in upper or lower case or a mixture of both. Some of the keywords have synonyms, and certain abbreviations are allowed, as long as there is no ambiguity. Blank lines and comments may be used to improve readability. A comment begins with an asterisk (*), which may appear anywhere on a line. All subsequent characters on the line are ignored.

It may be useful to include a comment on the first (Begin) line of the file. This line is echoed to the SUMMARY file.

Most of the options described in the next section should be left at their default values for any given model. If experimentation is necessary, we recommend changing just one option at a time.

SPECS File Checklist and Defaults

The following example SPECS file shows all valid keywords and their default values. The keywords are grouped according to the function they perform.

Some of the default values depend on E, the relative precision of the machine being used. The values given here correspond to double-precision arithmetic on most current machines (E ˜ 2.22 × 10-16 ). Similar values would apply to any machine having about 15 decimal digits of precision.

BEGIN checklist of SPECS file parameters and their default values
* Printing
Print level			1 	* 1-line iteration log
Print file 			15 	*
Summary file 			6 	* typically the screen
Print frequency 		1 	* iterations log on PRINT file
Summary frequency 		1 	* iterations log on SUMMARY file
Solution 			Yes 	* on the PRINT file
* Suppress options listing 		* default: options are listed

* Convergence Tolerances
Feasibility tolerance 	1.0e-6 	* for satisfying the simple bounds
Optimality tolerance 		1.0e-6 	* target value for reduced gradients

* Scaling
Scale option 			2	* All constraints and variables
Scale tolerance 		0.9 	*
* Scale Print 				* default: scales are not printed

* Other Tolerances
Crash tolerance 		0.1 	*
LU factor tolerance 		10.0 	* limits size of multipliers in L
LU update tolerance 		10.0	* the same during updates
LU singularity tolerance 	2.0e-6 	*
Pivot tolerance 		3.7e-11 * ?^(2/3)

* LP/QP problems
Crash option 			0 	* all slack initial basis
Elastic weight 		1.0 	* used only during elastic mode
Iterations limit 		10000 	* or m if that is more
Minimize				* (opposite of Maximize)
Partial price 			1 	* 10 for large LPs
Superbasics limit 		500 	* or n1+1 if that is less
Reduced Hessian dimension 	500 	* or Superbasics limit
Unbounded step size 		1.0e+18 *

* Infeasible problems
Elastic weight 		100 	* used only during elastic mode
Elastic mode 			1 	* use elastic mode when infeasible
Elastic objective 		2 	* infinite weight on the elastics

* Frequencies
Check frequency 		60 	* test row residuals ||Ax - s||
Expand frequency 		10000 	* for anti-cycling procedure
Factorization frequency 	100 	*
Save frequency 		100 	* save basis map

Description of the optional parameters

The following is an alphabetical list of the options that may appear in the SPECS file, and a description of their effect.

 Check frequency k Default = 60

Every kth iteration after the most recent basis factorization, a numerical test is made to see if the current solution x satisfies the general constraints. The constraints are of the form Ax - s = 0, where s is the set of slack variables. To perform the numerical test, the residual vector r = s - Ax is computed. If the largest component of r is judged to be too large, the current basis is refactorized and the basic variables are recomputed to satisfy the general constraints more accurately.

Check frequency 1 is useful for debugging purposes, but otherwise this option should not be needed.

 Crash option i Default = 0 Crash tolerance r Default = 0.1

Except on restarts, a CRASH procedure is used to select an initial basis from certain rows and columns of the constraint matrix ( A - I ). The Crash option i determines which rows and columns of A are eligible initially, and how many times CRASH is called. Columns of -I are used to pad the basis where necessary.

i Meaning
0 The initial basis contains only slack variables: B = I .
1 CRASH is called once, looking for a triangular basis in all rows and columns of the matrix A.
2 CRASH is called once, looking for a triangular basis in linear rows.
3 CRASH is called twice. The two calls treat linear equalities and linear inequalities separately.

If i >= 1, certain slacks on inequality rows are selected for the basis first. (If i >= 2, numerical values are used to exclude slacks that are close to a bound.) CRASH then makes several passes through the columns of A, searching for a basis matrix that is essentially triangular. A column is assigned to "pivot" on a particular row if the column contains a suitably large element in a row that has not yet been assigned. (The pivot elements ultimately form the diagonals of the triangular basis.) For remaining unassigned rows, slack variables are inserted to complete the basis.

The Crash tolerance r allows the starting procedure CRASH to ignore certain "small" nonzeros in each column of A. If amax is the largest element in column j, other nonzeros aij in the column are ignored if |aij| = amax × r. (To be meaningful, r should be in the range 0 <= r < 1.)

When r > 0.0, the basis obtained by CRASH may not be strictly triangular, but it is likely to be nonsingular and almost triangular. The intention is to obtain a starting basis containing more columns of A and fewer (arbitrary) slacks. A feasible solution may be reached sooner on some problems.

For example, suppose the first m columns of A form the matrix shown under LU factor tolerance; i.e., a tridiagonal matrix with entries -1, 4, -1. To help CRASH choose all m columns for the initial basis, we would specify Crash tolerance r for some value of r > 1/4.

 Elastic moden i Default = 1

This parameter determines if (and when) elastic mode is to be started. Three elastic modes are available as follows:

i Meaning
0 Elastic mode is never invoked. SQOPT will terminate as soon as infeasibility is detected. There may be other points with significantly smaller sums of infeasibilities.
1 Elastic mode is invoked only if the constraints are found to be infeasible (the default). If the constraints are infeasible, continue in elastic mode with the composite objective determined by the values of Elastic objective and Elastic weight.
2 The iterations start and remain in elastic mode. This option allows you to minimize the composite objective function directly without first performing phase-1 iterations.

The success of this option will depend critically on your choice of Elastic weight. If Elastic weight is sufficiently large and the constraints are feasible, the minimizer of the composite objective and the solution of the original problem are identical. However, if the Elastic weight is not sufficiently large, the minimizer of the composite function may be infeasible, even though a feasible point for the constraints may exist.

 Elastic objective i Default = 2

This option determines the form of the composite objective. Three types of composite objectives are available.

i Meaning
0 Include only the true objective q(x) in the composite objective. This option sets ? = 0 in the composite objective and will allow SQOPT to ignore the elastic bounds and find a solution that minimizes q subject to the nonelastic constraints. This option is useful if there are some "soft" constraints that you would like to ignore if the constraints are infeasible.
1 Use a composite objective defined with ? determined by the value of Elastic weight. This value is intended to be used in conjunction with Elasticmode = 2.
2 Include only the elastic variables in the composite objective. The elastics are weighted by ? = 1. This choice minimizes the violations of the elastic variable at the expense of possibly increasing the true objective. This option can be used to find a point that minimizes the sum of the violations of a subset of constraints determined by the parameter helast.
 Elastic weight r Default = 1.0

This keyword defines the value of γ in the composite objective.

• At each iteration of elastic mode, the composite objective is defined to be

${\displaystyle \min \ \ \sigma q(x)+r({\hbox{sum of infeasibilities}}),}$

where σ = 1 for Minimize, σ = -1 for Maximize, and q is the current objective value.

• Note that the effect of r is not disabled once a feasible iterate is obtained.
 Expand frequency i Default = 10000

This option is part of the EXPAND anti-cycling procedure designed to make progress even on highly degenerate problems.

The strategy is to force a positive step at every iteration, at the expense of violating the bounds on the variables by a small amount. Suppose that the Feasibility tolerance is δ. Over a period of i iterations, the tolerance actually used by SQOPT increases from 0.5δ to δ (in steps of 0.5δ/i).

Increasing i helps reduce the number of slightly infeasible nonbasic basic variables (most of which are eliminated during a resetting procedure). However, it also diminishes the freedom to choose a large pivot element (see Pivot tolerance).

 Factorization Frequency k Default = 100 (LP) or 50 (QP)

At most k basis changes will occur between factorizations of the basis matrix.

• With linear programs, the basis factors are usually updated every iteration. The default k is reasonable for typical problems. Higher values up to k = 100 (say) may be more efficient on problems that are extremely sparse and well scaled.
• When the objective function is quadratic, fewer basis updates will occur as an optimum is approached. The number of iterations between basis factorizations will therefore increase. During these iterations a test is made regularly (according to the Check frequency) to ensure that the general constraints are satisfied. If necessary the basis will be refactorized before the limit of k updates is reached.
 Feasibility tolerance t Default = 1.0E-6

A feasible problem is one in which all variables satisfy their upper and lower bounds to within the absolute tolerance t. (This includes slack variables. Hence, the general constraints are also satisfied to within t.)

• SQOPT attempts to find a feasible point for the non-elastic constraints before optimizing the objective. If the sum of the infeasibilities of these constraints cannot be reduced to zero, the problem is declared INFEASIBLE. If sInf is quite small, it may be appropriate to raise t by a factor of 10 or 100. Otherwise, some error in the data should be suspected.
• Note: if sInf is not small and you have not asked SQOPT to minimize the violations of the elastic variables (i.e., you have not specified Elasticobjective = 2, there may be other points that have a significantly smaller sum of infeasibilities. SQOPT will not attempt to find the solution that minimizes the sum unless Elasticobjective = 2.
• If scale is used, feasibility is defined in terms of the scaled problem (since it is then more likely to be meaningful).
 Infinite Bound Size r Default = 1.0E + 20

If r > 0, r defines the "infinite" bound BigBnd in the definition of the problem constraints. Any upper bound greater than or equal to BigBnd will be regarded as plus infinity (and similarly for a lower bound less than or equal to -BigBnd). If r <= 0, the default value is used.

 Iterations Limit k Default = 3 * m

This is the maximum number of iterations of the simplex method or the QP reduced-gradient algorithm allowed.

• Itns is an alternative keyword.
• k = 0 is valid. Both feasibility and optimality are checked.
 Log Frequency k Default = 1

see Print Frequency

 LU factor tolerance r1 Default = 100.0 LU update tolerance r2 Default = 10.0

These tolerances affect the stability and sparsity of the basis factorization B = LU during refactorization and updating, respectively. They must satisfy r1 , r2 >= 1.0. The matrix L is a product of matrices of the form

${\displaystyle \left({\begin{array}{cc}1\\\mu &\ 1\end{array}}\right),}$

where the multipliers ${\displaystyle \mu }$ satisfy ${\displaystyle |\mu |\leq r_{i}}$. Smaller values of ${\displaystyle r_{i}}$ favor stability, while larger values favor sparsity. The default values usually strike a good compromise.

• For large and relatively dense problems, r1 = 10.0 or 5.0 (say) may give a useful improvement in stability without impairing sparsity to a serious degree.
• For certain very regular structures (e.g., band matrices) it may be necessary to reduce r1 and/or r2 in order to achieve stability. For example, if the columns of A include a submatrix of the form

${\displaystyle \left({\begin{array}{rrrrrr}4&-1\\-1&4&-1\\&-1&4&-1\\&&\ddots &\ddots &\ddots \\&&&-1&4&-1\\&&&&-1&4\end{array}}\right),}$

one should set both r1 and r2 to values in the range 1.0 <= ri < 4.0.

 LU singularity tolerance r3 Default = ε2/3

This tolerance should satisfy ${\displaystyle r_{3}\leq \epsilon ^{1/4}\approx 10^{-4}}$. It helps guard against ill-conditioned basis matrices. When the LU factors of B are computed directly (not updated), the diagonal elements of U are tested as follows: if ${\displaystyle |U_{jj}|\leq r_{3}}$ or ${\displaystyle |U_{jj}|, the j-th column of the basis is replaced by the corresponding slack variable. (Replacements are rare because the LU updating method is stable. They are most likely to occur during the first factorization.)

 LU density tolerance r1 Default = 0.6 LU singularity tolerance r2 Default = ${\displaystyle \epsilon ^{2/3}\approx 10^{-11}}$

The density tolerance r1 is used during LU factorization of the basis matrix. Columns of L and rows of U are formed one at a time, and the remaining rows and columns of the basis are altered appropriately. At any stage, if the density of the remaining matrix exceeds r1, the Markowitz strategy for choosing pivots is altered to reduce the time spent searching for each remaining pivot. Raising the density tolerance towards 1.0 may give slightly sparser LU factors, with a slight increase in factorization time.

The singularity tolerance r2 helps guard against ill-conditioned basis matrices. When the basis is refactorized, the diagonal elements of U are tested as follows: if $\displaystyle |U_{jj}| \le r_2$ or $\displaystyle |U_{jj}| < r_2 \max_i |U_{ij}|$ , the jth column of the basis is replaced by the corresponding slack variable. (This is most likely to occur after a restart, or at the start of a major iteration.)

 Minimize Default Maximize

This specifies the required direction of optimization. It applies to both linear and quadratic terms in the objective.

 Optimality tolerance t Default = 1.0e-6

This is used to judge the size of the reduced gradients dj = gj - πTaj , where gj is the jth component of the gradient, aj is the associated column of the constraint matrix tmatA-I, and π is the set of dual variables.

• By construction, the reduced gradients for basic variables are always zero. The problem will be declared optimal if the reduced gradients for nonbasic variables at their lower or upper bounds satisfy

$\displaystyle d_j / ||{ \pi }|| \ge -t \quad \text{or} \quad d_j / ||{ \pi }|| \le t$

respectively, and if $\displaystyle \mod{d_j} / ||{ \pi }|| \le t$ for superbasic variables.

• In the above tests, p is a measure of the size of the dual variables. It is included to make the tests independent of a scale factor on the objective function.
• The quantity p actually used is defined by

$\displaystyle ||{\pi}|| = \max\{\sigma / \sqrt{m}, 1\}, \text{where} \sigma = \sum_{i=1}^m \mod{\pi_i},$

so that only large scale factors are allowed for.

• If the objective is scaled down to be very small, the optimality test reduces to comparing dj against 0.01t.
 Partial Price i Default = 10 (LP) or 1 (QP)

This parameter is recommended for large problems that have significantly more variables than constraints. It reduces the work required for each "pricing" operation (when a nonbasic variable is selected to become superbasic).

• When i = 1, all columns of the constraint matrix ( A - I ) are searched.
• Otherwise, A and I are partitioned to give i roughly equal segments Aj , Ij (j = 1 to i). If the previous pricing search was successful on Aj , Ij , the next search begins on the segments Aj+1 , Ij+1 . (All subscripts here are modulo i.)
• If a reduced gradient is found that is larger than some dynamic tolerance, the variable with the largest such reduced gradient (of appropriate sign) is selected to become superbasic. If nothing is found, the search continues on the next segments Aj+2 , Ij+2 , and so on.
• Partial price t (or t/2 or t/3) may be appropriate for time-stage models having t time periods.
 Pivot Tolerance r Default = $\displaystyle \epsilon^{2/3}$ /tt>

Broadly speaking, the pivot tolerance is used to prevent columns entering the basis if they would cause the basis to become almost singular.

• When x changes to x + αp for some search direction p, a "ratio test" is used to determine which component of x reaches an upper or lower bound first. The corresponding element of p is called the pivot element.
• For linear problems, elements of p are ignored (and therefore cannot be pivot elements) if they are smaller than the pivot tolerance r.
• It is common for two or more variables to reach a bound at essentially the same time. In such cases, the Feasibility tolerance (say t) provides some freedom to maximize the pivot element and thereby improve numerical stability. Excessively small values of t should therefore not be specified.
• To a lesser extent, the Expand frequency (say f ) also provides some freedom to maximize the pivot element. Excessively large values of f should therefore not be specified.
 Print frequency k Default = 1

One line of the iteration log will be printed every kth iteration. A value such as k = 10 is suggested for those interested only in the final solution.

 Print level k Default = 1

This controls the amount of printing produced by SQOPT as follows.

 0 No output except error messages. If you want to suppress all output, set Printfile = 0. >= 1 The set of selected options (including workspace limits), problem statistics, summary of the scaling procedure, information about the initial basis resulting from a CRASH or a BASIS file. A single line of output each iteration (controlled by Print frequency), and the exit condition with a summary of the final solution. >= 10 Basis factorization statistics.
 Save frequency k Default = 100

If a NEW BASIS file has been specified, a basis map describing the current solution will be saved on the appropriate file every kth iteration. A BACKUP BASIS file will also be saved if specified.

 Scale option i Default = 2 (LP) or 1 (QP) Scale tolerance r Default = 0.9 Scale Print

Three scale options are available as follows:

i Meaning
0 No scaling. This is recommended if it is known that x and the constraint matrix never have very large elements (say, larger than 1000).
1 The constraints and variables are scaled by an iterative procedure that attempts to make the matrix coef- ficients as close as possible to 1.0. This will sometimes improve the performance of the solution procedures.
2 The constraints and variables are scaled by the iterative procedure. Also, a certain additional scaling is performed that may be helpful if the right-hand side b or the solution x is large. This takes into account columns of ( A - I ) that are fixed or have positive lower bounds or negative upper bounds.

Scale tolerance affects how many passes might be needed through the constraint matrix. On each pass, the scaling procedure computes the ratio of the largest and smallest nonzero coefficients in each column:

$\displaystyle rho_j = \max_{i} |a_{ij}| / \min_i |a_{ij}| \qquad (a_{ij}\ne 0).$

If maxj pj is less than r times its previous value, another scaling pass is performed to adjust the row and column scales. Raising r from 0.9 to 0.99 (say) usually increases the number of scaling passes through A. At most 10 passes are made.

Scale Print causes the row-scales r(i) and column-scales c(j) to be printed. The scaled matrix coefficients are $\displaystyle \bar{ij} = a_{ij} c(j) / r(i)$ , and the scaled bounds on the variables and slacks are $\displaystyle \bar{j} = l_j / c(j),\ \bar{j} = u_j / c(j)$ where $\displaystyle c(j) \equiv r(j-n)$ if $\displaystyle j>n$ .

 Solution Yes Solution No Solution If Optimal, Infeasible, or Unbounded
 Summary file f Default = 6 Summary frequency k Default = 100/tt>

If f > 0, a brief log will be output to file f , including one line of information every kth iteration. In an interactive environment, it is useful to direct this output to the terminal, to allow a run to be monitored on-line. (If something looks wrong, the run can be manually terminated.) Further details are given in #The Summary File.

 Superbasics limit i Default = min{500, n1 + 1} Summary frequency k Default = 100

This places a limit on the storage allocated for superbasic variables. Ideally, i should be set slightly larger than the "number of degrees of freedom" expected at an optimal solution.

For linear programs, an optimum is normally a basic solution with no degrees of freedom. (The number of variables lying strictly between their bounds is no more than m, the number of general constraints.) The default value of i is therefore 1.

For quadratic problems, the number of degrees of freedom is often called the "number of independent variables".

• Normally, i need not be greater than ncolH + 1, where ncolH is the number of leading nonzero columns of H .
• For many problems, i may be considerably smaller than ncolH. This will save storage if ncolH is very large.
• This parameter also sets the Reduced Hessian dimension, unless the latter is specified explicitly (and conversely).
 Unbounded Step Size amax Default = 1.0E + 18

This parameter is intended to detect unboundedness in a quadratic problem. (It may or may not achieve that purpose!) During a line search, q is evaluated at points of the form x + αp, where x and p are fixed and α varies. if a exceeds αmax, iterations are terminated with the exit message problem is unbounded.

Note that unboundedness in x is best avoided by placing finite upper and lower bounds on the variables.

File Output

The following information is output to the PRINT file during the solution of each problem referred to in the SPECS file.

• A listing of the relevant part of the SPECS file.
• A listing of the parameters that were or could have been set in the SPECS file.
• An estimate of the amount of working storage needed, compared to how much is available.
• Some statistics about the problem.
• The amount of storage available for the LU factorization of the basis matrix.
• A summary of the scaling procedure, if Scale was specified.
• Notes about the initial basis resulting from a CRASH procedure or a BASIS file.
• The iteration log.
• Basis factorization statistics.
• The EXIT condition and some statistics about the solution obtained.
• The printed solution, if requested.

The last four items are described in the following sections. Further brief output may be directed to the SUMMARY file, as discussed in #The Summary File.

The iteration Log

If Printlevel > 0, one line of information is output to the PRINT file every kth iteration, where k is the specified Print frequency (default k = 1). A heading is printed before the first such line following a basis factorization. The heading contains the items described below. In this description, a PRICE operation is defined to be the process by which one or more nonbasic variables are selected to become superbasic (in addition to those already in the superbasic set). The variable selected will be denoted by jq. If the problem is purely linear, variable jq will usually become basic immediately (unless it should happen to reach its opposite bound and return to the nonbasic set).

If Partial price is in effect, variable jq is selected from App or Ipp , the ppth segments of the constraint matrix (A - I ).

Label Description
Itn The current iteration number.
pp The Partial Price indicator. The variable selected by the last PRICE operation came from the ppth partition of A and -I . pp is set to zero when the basis is refactored.
dj This is dj, the reduced cost (or reduced gradient) of the variable jq selected by PRICE at the start of the present iteration. Algebraically, dj is $\displaystyle d_j=g_j - \pi^T a_ j$ , where gj is the gradient of the current objective function, π is the vector of dual variables, and aj is the jth column of the constraint matrix (A - I ).

Note that dj is the norm of the reduced-gradient vector at the start of the iteration, just after the PRICE operation.

+SBS The variable jq selected by PRICE to be added to the superbasic set.
-SBS The variable chosen to leave the set of superbasics. It has become basic if the entry under -B is nonzero; otherwise it has become nonbasic.
-BS The variable removed from the basis (if any) to become nonbasic.
-B The variable removed from the basis (if any) to swap with a slack variable made superbasic by the latest PRICE. The swap is done to ensure that there are no superbasic slacks.
Step The step length a taken along the current search direction p. The variables x have just been changed to x + αp. If a variable is made superbasic during the current iteration (i.e., +SBS is positive), Step will be the step to the nearest bound. During phase 2, the step can be greater than one only if the reduced Hessian is not positive definite.
Pivot If column aq replaces the rth column of the basis B, Pivot is the rth element of a vector y satisfying By = aq . Wherever possible, Step is chosen to avoid extremely small values of Pivot (since they cause the basis to be nearly singular). In rare cases, it may be necessary to increase the Pivot tolerance to exclude very small elements of y from consideration during the computation of Step.
L The number of nonzeros representing the basis factor L. Immediately after a basis factorization B = LU , this is lenL, the number of subdiagonal elements in the columns of a lower triangular matrix. Further nonzeros are added to L when various columns of B are later replaced. (Thus, L increases monotonically.)
U The number of nonzeros in the basis factor U . Immediately after a basis factorization, this is lenU, the number of diagonal and superdiagonal elements in the rows of an upper-triangular matrix. As columns of B are replaced, the matrix U is maintained explicitly (in sparse form). The value of U may fluctuate up or down; in general it will tend to increase.
ncp The number of compressions required to recover storage in the data structure for U . This includes the number of compressions needed during the previous basis factorization. Normally ncp should increase very slowly. If not, the amount of integer and real workspace available to SQOPT should be increased by a significant amount. As a suggestion, the work arrays iw(*) and rw(*) should be extended by L + U elements.
nInf The number of infeasibilities before the present iteration. This number will not increase unless the iterations are in elastic mode.
Sinf,Objective If nInf > 0, this is sInf, the sum of infeasibilities before the present iteration. (It will usually decrease at each nonzero Step, but if nInf decreases by 2 or more, sInf may occasionally increase. However, in elastic mode, it will decrease monotonically.)

Otherwise, it is the value of the current objective function after the present iteration. Note: If Elasticmode = 2, the heading is Composite Obj.

The following items are printed if the problem is a QP or if the superbasic set is non-empty (i.e., if the current solution is nonbasic).
Norm rg This quantity is rg, the norm of the reduced-gradient vector at the start of the iteration. (It is the Euclidean norm of the vector with elements dj for variables j in the superbasic set. During phase 2 this norm will be approximately zero after a unit step.
nS The current number of superbasic variables.
Cond Hz An estimate of the condition number of the reduced Hessian. It is the square of the ratio of the largest and smallest diagonals of the upper triangular matrix R. This constitutes a lower bound on the condition number of the reduced Hessian RTR.

To guard against high values of cond Hz, attention should be given to the scaling of the variables and the constraints.

Basis Factorization Statistics

When PrintLevel = 20 and Printfile > 0, the following lines of intermediate printout (< 120 characters) are produced on the unit number specified by Print file whenever the matrix B or BS = (B S )T is factorized. Gaussian elimination is used to compute an LU factorization of B or BS , where PLPT is a lower triangular matrix and PUQis an upper triangular matrix for some permutation matrices P and Q. This factorization is stabilized in the manner described under LU factor tolerance in #Description of the optional parameters.

Label Description
Factorize The number of factorizations since the start of the run.
Demand A code giving the reason for the present factorization.
Code Meaning
0 First LU factorization.
1 Number of updates reached the value of the optional parameter Factorization Frequency.
2 Excessive non-zeros in updated factors.
7 Not enough storage to update factors.
10 Row residuals too large (see the description for Check Frequency).
11 Ill-conditioning has caused inconsistent results.
Iteration The current iteration number.
Infeas The number of infeasibilities at the start of the previous iteration.
Objective If Infeas > 0, this is the sum of infeasibilities at the start of the previous iteration.

If Infeas = 0, this is the value of the objective function after the previous iteration.

Nonlinear The number of nonlinear variables in the current basis B. (not printed if BS is factorized). Linear The number of linear variables in B. (not printed if BS is factorized).
Slacks The number of slack variables in B. (not printed if BS is factorized).
Elems The number of nonzero matrix elements in B. (not printed if BS is factorized).
Density The percentage nonzero density of B, 100 × Elems/(m × m), where m is the number of rows in the problem (m = Linear + Slacks).
Comprssns The number of times the data structure holding the partially factored matrix needed to be compressed, to recover unused storage. Ideally this number should be zero. If it is more than 3 or 4, the amount of workspace available to SQOPT should be increased for efficiency.
Merit The average Markowitz merit count for the elements chosen to be the diagonals of P U Q. Each merit count is defined to be (c - 1)(r - 1) where c and r are the number of nonzeros in the column and row containing the element at the time it is selected to be the next diagonal. Merit is the average of m such quantities. It gives an indication of how much work was required to preserve sparsity during the factorization.
lenL The number of nonzeros in L. On most machines, each nonzero is represented by one eight-byte REAL and two two-byte integer data types.
lenU The number of nonzeros in U . The storage required for each nonzero is the same as for the nonzeros of L.
Increase The percentage increase in the number of nonzeros in L and U relative to the number of nonzeros in B; i.e., 100 × (lenL + lenU - Elems)/Elems.
m is the number of rows in the problem. Note that m = Ut + Lt + bp.
Ut is the number of triangular rows of B at the top of U .
d1 is the number of columns remaining when the density of the basis matrix being factorized reached 0.3.
Lmax The maximum subdiagonal element in the columns of L. This will be no larger than the LU factor tolerance.
Bmax The maximum nonzero element in B.
Umax The maximum nonzero element in U , excluding elements of B that remain in U unaltered. (For example, if a slack variable is in the basis, the corresponding row of B will become a row of U without alteration. Elements in such rows will not contribute to Umax. If the basis is strictly triangular, none of the elements of B will contribute, and Umax will be zero.)

Ideally, Umax should not be substantially larger than Bmax. If it is several orders of magnitude larger, it may be advisable to reduce the LU factor tolerance to some value nearer 1.0. (The default value is 10.0.)

Umax is not printed if BS is factorized.
Umin The smallest diagonal element of P U Q in absolute magnitude.
Growth The ratio Umax/Bmax, which should not be too large (see above).

As long as Lmax is not large (say 10.0 or less), the ratio max{Bmax, Umax} / Umin gives an estimate of the condition number of B. If this number is extremely large, the basis is nearly singular and some numerical difficulties could conceivably occur. (However, an effort is made to avoid near-singularity by using slacks to replace columns of B that would have made Umin extremely small. Messages are issued to this effect, and the modified basis is refactored.)

Lt is the number of triangular columns of B at the beginning of L.
bp is the size of the "bump" or block to be factorized nontrivially after the triangular rows and columns have been removed.
d2 is the number of columns remaining when the density of the basis matrix being factorized reached 0.6.

Crash statistics

When PrintLevel = 20 and Printfile > 0, the following CRASH statistics (< 120 characters) are produced on the unit number specified by Print file whenever Start = 'Cold', see #Description of the optional parameters. They refer to the number of columns selected by the CRASH procedure during each of several passes through A, whilst searching for a triangular basis matrix.

Label Description
Slacks is the number of slacks selected initially.
Free cols is the number of free columns in the basis.
Preferred is the number of "preferred" columns in the basis (i.e., hs(j) = 3 for some j = n).
Unit is the number of unit columns in the basis.
Double is the number of double columns in the basis.
Triangle is the number of triangular columns in the basis.

EXIT conditions

For each problem in the SPECS file, a message of the form EXIT -- message is printed to summarize the final result. Here we describe each message and suggest possible courses of action.

A number is associated with each message below. It is the final value assigned to the integer variable inform.

The following messages arise when the SPECS file is found to contain no further problems.

-2.  EXIT -- input error.

Otherwise, the SPECS file may be empty, or cards containing the keywords Skip or Endrun may imply that all problems should be ignored (see #the SPECS File).

-1.  ENDRUN

This message is printed at the end of a run if SQOPT terminates of its own accord. Otherwise, the operating system will have intervened for one of many possible reasons (excess time, missing file, arithmetic error in the user routine, etc.).

The following messages arise when optimization terminates gracefully.

0. 	EXIT -- optimal solution  found

The final point seems to be a unique solution of LCQP. This means that x is feasible (it satisfies the constraints to the accuracy requested by the Feasibility tolerance), the reduced gradient is negligible, the reduced costs are optimal, and R is nonsingular.

1. 	EXIT -- the  problem  is infeasible

Feasibility is measured with respect to the upper and lower bounds on the variables. The message tells us that among all the points satisfying the general constraints Ax - s = 0, there is apparently no point that satisfies the bounds on x and s. Violations as small as the Feasibility tolerance are ignored, but at least one component of x or s violates a bound by more than the tolerance.

Note: Although the objective function is the sum of infeasibilities (when nInf > 0), this sum will usually not have been minimized when SQOPT recognizes the situation and exits. There may exist other points that have a significantly lower sum of infeasibilities.

2. 	EXIT -- the  problem  is unbounded	(or badly  scaled)

For linear problems, unboundedness is detected by the simplex method when a nonbasic variable can apparently be increased or decreased by an arbitrary amount without causing a basic variable to violate a bound. A message prior to the EXIT message will give the index of the nonbasic variable. Consider adding an upper or lower bound to the variable. Also, examine the constraints that have nonzeros in the associated column, to see if they have been formulated as intended.

Very rarely, the scaling of the problem could be so poor that numerical error will give an erroneous indication of unboundedness. Consider using the Scale option.

3. 	EXIT -- iteration limit exceeded

The Iterations limit was exceeded before the required solution could be found. Check the iteration log to be sure that progress was being made. If so, restart the run using a basis file that was saved (or should have been saved!) at the end of the run.

4. 	EXIT -- QP Hessian appears to be indefinite

The problem appears to be nonconvex and cannot be solved using this version of SQOPT. The matrix H cannot be positive semidefinite, i.e., there must exist a vector y such that yT H y < 0.

5. 	EXIT -- the  superbasics limit is too  small:	nnn

The problem appears to be more nonlinear than anticipated. The current set of basic and superbasic variables have been optimized as much as possible and a PRICE operation is necessary to continue, but there are already nnn superbasics (and no room for any more).

In general, raise the Superbasics limit s by a reasonable amount, bearing in mind the storage needed for the reduced Hessian. (The Hessian dimension h will also increase to s unless specified otherwise, and the associated storage will be about 1 s2 words.) In extreme cases you may have to set h < s to conserve storage, but beware that the rate of convergence will probably fall off severely.

6. 	EXIT -- weak solution  found

The final point is a weak minimizer. (The objective value is a global optimum, but it may be achieved by an infinite set of points x.)

This exit will occur when (i) the problem is feasible, (ii) the reduced gradient is negligible, (iii) the Lagrange multipliers are optimal, and (iv) the reduced Hessian is singular or there are some very small multipliers. This exit cannot occur if H is positive definite (i.e., q(x) is strictly convex).

10.    EXIT -- cannot  satisfy the  general  constraints

An LU factorization of the basis has just been obtained and used to recompute the basic variables xB , given the present values of the superbasic and nonbasic variables. A single step of "iterative refinement" has also been applied to increase the accuracy of xB . However, a row check has revealed that the resulting solution does not satisfy the current constraints Ax - s = 0 sufficiently well.

This probably means that the current basis is very ill-conditioned. Request the Scale option if there are any linear constraints and variables.

For certain highly structured basis matrices (notably those with band structure), a systematic growth may occur in the factor U . Consult the description of Umax, Umin and Growth in [[Basis Factorization Statistics, and set the LU factor tolerance to 2.0 (or possibly even smaller, but not less than 1.0).

If the following exits occur during the first basis factorization, the basic variables xB will have certain default values that may not be particularly meaningful, and the dual vector π will be zero.

20.    EXIT -- not  enough integer/real storage  for the  basis  factors

An estimate of the additional storage required is given in messages preceding the exit.

21.    EXIT -- error in basis  package

A preceding message will describe the error in more detail. One such message says that the current basis has more than one element in row i and column j.

22.    EXIT -- singular basis  after nnn factorization attempts

This exit is highly unlikely to occur. The first factorization attempt will have found the basis to be structurally or numerically singular. (Some diagonals of the triangular matrix PUQ were respectively zero or smaller than a certain tolerance.) The associated variables are replaced by slacks and the modified basis is refactorized. The ensuing singularity must mean that the problem is badly scaled, or the LU factor tolerance is too high.

32.    EXIT -- system error.    Wrong no.    of  basic  variables:   nnn

This exit should never happen. If it does, something is seriously awry in the SQOPT source code.

The following messages arise if additional storage is needed to allow optimization to begin. The problem is abandoned.

42. 	EXIT -- not  enough 8-character storage  to  start  solving the  problem
43. 	EXIT -- not  enough integer storage  to  start  solving the  problem
44. 	EXIT -- not  enough real storage  to  start  solving the  problem

Solution Output

At the end of a run, the final solution will be output to the PRINT file. Some header information appears first to identify the problem and the final state of the optimization procedure. A ROWS section and a COLUMNS section then follow, giving one line of information for each row and column. The format used is similar to that seen in commercial systems, though there is no rigid industry standard.

The ROWS section

The general constraints take the form $\displaystyle l \le Ax \le u$ . The ith constraint is therefore of the form

$\displaystyle \alpha \le a^T x \le \beta.$

Internally, the constraints take the form Ax - s = 0, where s is the set of slack variables (which happen to satisfy the bounds $\displaystyle l \le s \le u$ ). For the ith constraint it is the slack variable si that is directly available, and it is sometimes convenient to refer to its state. To reduce clutter, a "·" is printed for any numerical value that is exactly zero.

Label Description
Number The value n + i. This is the internal number used to refer to the ith slack in the iteration log.
Row The name of the ith row.
State The state of the ith row relative to the bounds α and β. The various states possible are as follows.
 LL The row is at its lower limit, α. UL The row is at its upper limit, β. EQ The lower and upper limit are the same, α= β. BS The constraint is not binding. si is basic. SBS The constraint is not binding. si is superbasic.

A key is sometimes printed before the State to give some additional information about the state of the slack variable.

 A Alternative optimum possible. The slack is nonbasic, but its reduced gradient is essentially zero. This means that if the slack were allowed to start moving away from its bound, there would be no change in the value of the objective function. The values of the basic and superbasic variables might change, giving a genuine alternative solution. However, if there are any degenerate variables (labelled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case, the values of dual variables might also change. D Degenerate. The slack is basic or superbasic, but it is equal to (or very close to) one of its bounds. I Infeasible. The slack is basic or superbasic and it is currently violating one of its bounds by more than the Feasibility tolerance. N Not precisely optimal. The slack is nonbasic or superbasic. If the Optimality tolerance were tightened by a factor of 10 (e.g., if it were reduced from 10-5 to 10-6 ), the solution would not be declared optimal because the reduced gradient for the slack would not be considered negligible. (If a loose tolerance has been used, or if the run was terminated before optimality, this key might be helpful in deciding whether or not to restart the run.) Note: If Scale is specified, the tests for assigning the A, D, I, N keys are made on the scaled problem, since the keys are then more likely to be correct. Activity The row value; i.e., the value of aTx. Slack activity The amount by which the row differs from its nearest bound. (For free rows, it is taken to be minus the Activity.) Lower limit α the lower bound on the row. Upper limit β the upper bound on the row. Dual activity The value of the dual variable pi , often called the shadow price (or simplex multiplier) for the ith constraint. The full vector p always satisfies BTp = gB , where B is the current basis matrix and gB contains the associated gradients for the current objective function. I The constraint number, i.

The COLUMNS section

Here we talk about the "column variables" x. For convenience we let the jth component of x be the variable xj and assume that it satisfies the bounds a = xj = β. A "·" is printed for any numerical value that is exactly zero.

LL xj is nonbasic at its lower limit, α. UL xj is nonbasic at its upper limit, β. EQ xj is nonbasic and fixed at the value α = β. FR xj is nonbasic and currently zero, even though it is free to take any value. (Its bounds are α = -8, β = +8. Such variables are normally basic.) BS xj is basic. SBS xj is superbasic.
Label Description
Number The column number, j. This is the internal number used to refer to xj in the iteration log.
Column The name of xj .
State The state of xj relative to the bounds α and β. The various states possible are as follows.

A key is sometimes printed before the State to give some additional information about the state of xj . The possible keys are A, D, I and N. They have the same meaning as described above (for the ROWS section of the solution), but the words "the slack" should be replaced by "xj ". |-valign="top" |Activity||The value of the variable xj . |-valign="top" |Obj Gradient||gj , the jth component of the linear and quadratic objective function q(x) + cTx. (We define gj = 0 if the current solution is infeasible.) |-valign="top" |Lower limit||α, the lower bound on xj . |-valign="top" |Upper limit β||the upper bound on xj . |-valign="top" |Reduced gradnt||The reduced gradient $\displaystyle d_j = g_j - \pi^T a_j$ , where aj is the jth column of the constraint matrix (or the jth column of the Jacobian at the start of the final major iteration). |-valign="top" |M+J||The value m + j. |}

An example of the printed solution is given in #File Output. Infinite Upper and Lower limits are output as the word None. Other real values are output with format f16.5. The maximum record length is 111 characters, including the first (carriage-control) character.

Note : If two problems are the same except that one minimizes q(x) and the other maximizes -q(x), their solutions will be the same but the signs of the dual variables πi and the reduced gradients dj will be reversed.

The SUMMARY file

If Summary file f is specified certain brief information will be output to file f . For batch jobs a disk file should be used, to retain a concise log of each run if desired. (A SUMMARY file is more easily perused than the associated PRINT file).

A SUMMARY file (like the PRINT file) is not rewound after a problem has been processed. It can therefore accumulate a log for every problem in the SPECS file, if each specifies the same file. The maximum record length is 72 characters, including a carriage-control character in column 1.

The following information is included:

1. The Begin card from the SPECS file.
2. The status of the solution after each basis factorization (whether feasible; the objective value; the number of function calls so far).
3. The same information every kth iteration, where k is the specified Summary frequency (default k = 100).
4. Warnings and error messages.
5. The exit condition and a summary of the final solution.

Item 4 is preceded by a blank line, but item 5 is not.

All items are illustrated below, where we give the SUMMARY file for the first problem in the example program (Summary frequency = 1).

==============================
S Q O P T  5.3       (Oct  97)
==============================

Begin sqmain (Example program for sqopt)

Scale option  2,    Partial price   1

----------------------------------------------------------------
Itn      0: Phase 1A -- making the linear equality rows feasible

Itn       dj    Step nInf    SumInf       Objective
0  0.0E+00 0.0E+00    1 8.868E+01  0.00000000E+00
1  0.0E+00 3.3E+01    0 0.000E+00  0.00000000E+00
Itn      1: Feasible linear equality rows
Itn      1: Phase 1B -- making all linear rows feasible

Itn       dj    Step nInf    SumInf       Objective Norm rg   nS
1  0.0E+00 0.0E+00    2 5.317E+01  0.00000000E+00 3.4E+00    3
2  0.0E+00 0.0E+00    2 5.317E+01  0.00000000E+00 4.6E-01    2
3  0.0E+00 4.7E+02    1 2.896E+01  0.00000000E+00 4.8E-02    1
4  0.0E+00 9.2E+02    1 2.681E+01  0.00000000E+00 0.0E+00    0

This is problem  sqmain.   ncolH =   5

5  6.4E-02 6.5E+03    0 0.000E+00 -1.46750000E+06 0.0E+00    0
Itn      5: Feasible linear rows
6 -4.1E+03 2.1E-01    0 0.000E+00 -1.78368567E+06 0.0E+00    0
7  1.4E+03 1.0E+00    0 0.000E+00 -1.98453602E+06 1.4E-12    1
8 -6.3E+02 9.8E-01    0 0.000E+00 -2.04366386E+06 1.3E+01    1
9  0.0E+00 1.0E+00    0 0.000E+00 -2.04366504E+06 1.1E-12    1
EXIT -- optimal solution found

Problem name                 sqdat1..
No. of iterations                   9   Objective value     -2.0436650381E+06
No. of Hessian products             8   Linear objective     0.0000000000E+00
No. of superbasics                  1   No. of basic nonlinears             2
No. of degenerate steps             1   Percentage                      11.11
Norm of xs  (scaled)          3.5E+03   Norm of pi  (scaled)          8.9E+03
Norm of xs                    1.6E+03   Norm of pi                    1.1E+04
Max Prim inf(scaled)        0 0.0E+00   Max Dual inf(scaled)        6 2.0E-12
Max Primal infeas           0 0.0E+00   Max Dual infeas             3 9.6E-13

Solution printed on file   9

Algorithmic Details

SQOPT is based on an inertia-controlling method that maintains a Cholesky factorization of the reduced Hessian (see below). The method follows Gill and Murray. Here we briefly summarize the main features of the method. Where possible, explicit reference is made to items listed in the printed output, and to the names of the relevant optional parameters.

Overview

SQOPT's method has a feasibility phase (also known as phase 1 ), in which a feasible point is found by minimizing the sum of infeasibilities, and an optimality phase (or phase 2 ), in which the quadratic objective is minimized within the feasible region. The computations in both phases are performed by the same subroutines, with the change of phase being characterized by the objective changing from the sum of infeasibilities (the printed quantity sInf) to the quadratic objective (the printed quantity Objective).

In general, an iterative process is required to solve a quadratic program. Given an iterate (x, s) in both the original variables x and the slack variables s, a new iterate $\displaystyle (\bar{x}, \bar{s})$ is defined by

$\displaystyle \left(\begin{array}{c} \bar{x}\\ \bar{s} \end{array}\right) = \left(\begin{array}{c} x\\ s \end{array}\right) + \alpha p,$

where the step length a is a non-negative scalar, and p is called the search direction. (For simplicity, we shall consider a typical iteration and avoid reference to the index of the iteration.). Once an iterate is feasible (i.e., satisfies the constraints), all subsequent iterates remain feasible.

Definition of the working set

At each iterate (x, s), a working set of constraints is defined to be a linearly independent subset of the constraints that are satisfied "exactly" (to within the value of the Feasibility tolerance). The working set is the current prediction of the constraints that hold with equality at a solution of the LP or QP. Let mw denote the number of constraints in the working set (including bounds), and let W denote the associated mw × (n + m) working-set matrix consisting of the mw gradients of the working-set constraints.

The search direction is defined so that constraints in the working set remain unaltered for any value of the step length. It follows that p must satisfy the identity W p = 0. This characterization allows p to be computed using any $\displaystyle n\times n\Z$ full-rank matrix Z that spans the null space of W . (Thus, nZ = n - mw and W Z = 0.) The null-space matrix Z is defined from a sparse LU factorization of part of W ). The direction p will satisfy W p = 0 if p = ZpZ for any nZ-vector pZ .

The working set contains the constraints Ax - s = 0 and a subset of the upper and lower bounds on the variables (x, s). Since the gradient of a bound constraint $\displaystyle x_j \ge l_j$ or $\displaystyle x_j \le u_j$ is a vector of all zeros except for $\displaystyle \pm1$ in position j, it follows that the working-set matrix contains the rows of ( A - I ) and the unit rows associated with the upper and lower bounds in the working set.

The working-set matrix W can be represented in terms of a certain column partition of the matrix ( A - I ). As in #Brief description of the method we partition the constraints Ax - s = 0 so that

BxB + SxS + NxN = 0,

where B is a square non-singular basis and xB , xS and xN are the basic, superbasic and nonbasic variables respectively. The nonbasic variables are equal to their upper or lower bounds at (x, s), and the superbasic variables are independent variables that are chosen to improve the value of the current objective. The number of superbasic variables is nS , which is printed as the quantity nS. Given values of xN and xS , the basic variables xB are adjusted so that (x, s) satisfies BxB + SxS + NxN = 0.

If P is a permutation such that (A - I) P = (B S N), then the working-set matrix W satisfies

$\displaystyle WP = \left(\begin{array}{ccc} B & S & N \\ 0 & 0 & I\N \end{array}\right),$

where IN is the identity matrix with the same number of columns as N .

The null-space matrix Z is defined from a sparse LU factorization of part of W . In particular, Z is maintained in "reduced-gradient" form, using the package LUSOL to maintain sparse LU factors of the basis matrix B that alters as the working set W changes. Given the permutation P , the null-space basis is given by

$\displaystyle Z = P \left(\begin{array}{c} -B^{-1} S \\ I \\ 0 \end{array}\right).$

This matrix is used only as an operator, i.e., it is never computed explicitly. Products of the form Zv and ZTg are obtained by solving with B or BT. This choice of Z implies that nZ , the number of "degrees of freedom" at (x, s), is the same as nS , the number of superbasic variables.

Let gZ and HZ denote the reduced gradient and reduced Hessian:

gZ = ZTg and HZ = ZTHZ,

where g is the objective gradient at (x, s). Roughly speaking, gZ and HZ describe the first and second derivatives of an nS-dimensional unconstrained problem for the calculation of pZ . (The quantity Cond Hz printed in the summary-file output is a condition estimator of HZ.)

At each iteration, an upper-triangular factor R is available such that HZ = RTR. Normally, R is computed from

RTR = ZTHZ at the start of phase 2 and is then updated as the QP working set changes. For efficiency the dimension of R should not be excessive (say, nS <= 1000). This is guaranteed if the number of nonlinear variables is "moderate".

If the QP contains linear variables, H is positive semi-definite and R may be singular with at least one zero diagonal. However, an inertia-controlling active-set strategy is used to ensure that only the last diagonal of R can be zero.

If the initial R is singular, enough variables are fixed at their current value to give a nonsingular R. This is equivalent to including temporary bound constraints in the working set. Thereafter, R can become singular only when a constraint is deleted from the working set (in which case no further constraints are deleted until R becomes nonsingular).

The main iteration

If the reduced gradient is zero, (x, s) is a constrained stationary point on the working set. During phase 1, the reduced gradient will usually be zero only at a vertex (although it may be zero elsewhere in the presence of constraint dependencies). During phase 2, a zero reduced gradient implies that x minimizes the quadratic objective when the constraints in the working set are treated as equalities. At a constrained stationary point, Lagrange multipliers $\displaystyle \lambda$ are defined from the equations $\displaystyle W^T\lambda = g(x)$ . A Lagrange multiplier λj corresponding to an inequality constraint in the working set is said to be optimal if $\displaystyle \lambda_j \le \sigma$ when the associated constraint is at its upper bound, or if $\displaystyle \lambda_j \ge - \sigma$ when the associated constraint is at its lower bound, where σ depends on the Optimality tolerance. If a multiplier is non-optimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by continuing the minimization with the corresponding constraint excluded from the working set (this step is sometimes referred to as "deleting" a constraint from the working set). If optimal multipliers occur during the feasibility phase but the sum of infeasibilities is not zero, there is no feasible point.

The special form of the working set allows the multiplier vector λ, the solution of WTλ = g, to be written in terms of the vector

$\displaystyle d = \left(\begin{array}{c} g \\ 0 \end{array}\right) - \left(\begin{array}{c} A - I \end{array}\right)^T\pi = \left(\begin{array}{c} g - A^T\pi\\ \pi \end{array}\right),$

where π satisfies the equations BTp = gB , and gB denotes the basic components of g. The components of π are the Lagrange multipliers &lambdaj associated with the equality constraints Ax - s = 0. The vector dN of nonbasic components of d consists of the Lagrange multipliers δj associated with the upper and lower bound constraints in the working set. The vector dS of superbasic components of d is the reduced gradient gZ . The vector dB of basic components of d is zero, by construction. (The Euclidean norm of dS , and the final values of dS , g and π are the quantities norm rg, Reduced Gradnt, Obj Gradient and Dual Activity in the PRINT file output.)

If the reduced gradient is not zero, Lagrange multipliers need not be computed and the search direction is given by p = ZZpZ , where pZ is defined below. The step length is chosen to maintain feasibility with respect to the satisfied constraints.

There are two possible choices for pZ , depending on whether or not HZ is singular. If HZ is nonsingular, R is nonsingular and pZ is computed from the equations,

$\displaystyle R^TRp_Z = -g_Z,$

where gZ is the reduced gradient at x. In this case, (x, s) + p is the minimizer of the objective function subject to the working-set constraints being treated as equalities. If (x, s) + p is feasible, α is defined to be one. In this case, the reduced gradient at $\displaystyle (\bar{x}, \bar{s})$ will be zero, and Lagrange multipliers are computed at the next iteration. Otherwise, α is set to αN , the step to the boundary of the "nearest" constraint along p. This constraint is added to the working set at the next iteration.

If HZ is singular, then R must also be singular, and an inertia-controlling strategy is used to ensure that only the last diagonal element of R is zero. In this case, pZ satisfies

$\displaystyle p_Z^T H_Z p_Z = 0 \quad \text{and} \quad g_Z^T p_Z \le 0,$

which allows the objective function to be reduced by any step of the form (x, s) + αp, a > 0. The vector p = ZpZ is a direction of unbounded descent for the QP in the sense that the QP objective is linear and decreases without bound along p. If no finite step of the form (x, s) + αp (α > 0) reaches a constraint not in the working set, the QP is unbounded and SQOPT terminates at (x, s) and declares the problem to be unbounded. Otherwise, α is defined as the maximum feasible step along p and a constraint active at (x, s) + αp is added to the working set for the next iteration.

Miscellaneous

If the basis matrix is not chosen carefully, the condition of the null-space matrix Z could be arbitrarily high. To guard against this, SQOPT implements a "basis repair" feature in the following way. LUSOL is used to compute the rectangular factorization

$\displaystyle \left(\begin{array}{cc} B & S \end{array}\right)^T = LU,$

returning just the permutation P that makes PLPT unit lower triangular. The pivot tolerance is set to require $\displaystyle |PLP^T|_{ij} \le 2$ , and the permutation is used to define P . It can be shown that $\displaystyle ||{Z}||$ is likely to be little more than 1. Hence, Z should be well-conditioned regardless of the condition of W .

This feature is applied at the beginning of the optimality phase if a potential B-S ordering is known.

The EXPAND procedure is used to reduce the possibility of cycling at a point where the active constraints are nearly linearly dependent. Although there is no absolute guarantee that cycling will not occur, the probability of cycling is extremely small. The main feature of expand is that the feasibility tolerance is increased at every iteration, perhaps at the expense of violating the bounds on (x, s) by a simple amount.

Suppose that the value of Feasibility tolerance is δ. Over a period of K iterations (where K is the value of the optional parameter Expand frequency, the feasibility tolerance used in SQOPT (i.e., the working feasibility tolerance) increases from $\displaystyle \frac{1}{2}\delta$ to $\displaystyle \delta$ in steps of $\displaystyle \frac{1}{2}\delta/K$ .

At certain stages, the following "resetting procedure" is used to remove small constraint infeasibilities. First, all nonbasic variables are moved exactly onto their bounds. A count is kept of the number of non-trivial adjustments made. If the count is nonzero, the basic variables are recomputed. Finally, the working feasibility tolerance is reinitialized to $\displaystyle \frac{1}{2}\delta$ .

If a problem requires more than K iterations, the resetting procedure is invoked and a new cycle of iterations is started. (The decision to resume phase 1 or phase 2 is based on comparing any constraint infeasibilities with δ.)

The resetting procedure is also invoked if when SQOPT reaches an apparently optimal, infeasible, or unbounded solution, unless this situation has already occurred twice. If any non-trivial adjustments are made, iterations are continued.

The EXPAND procedure not only allows a positive step to be taken at every iteration, but also provides a potential choice of constraint to be added to the working set. All constraints at a distance $\displaystyle \alpha (\alpha \le \alpha_N)$ along p from the current point are then viewed as acceptable candidates for inclusion in the working set. The constraint whose normal makes the biggest angle with the search direction is added to the working set. This strategy helps keep the basis matrix B well-conditioned.