QPOPT

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This page is part of the SOL Manual. See SOL.

Contents

Introduction

TOMLAB /QPOPT (hereafter referred to as QPOPT) is based on an inertia-controlling method that maintains a Cholesky factorization of the reduced Hessian (see below). Here we briefly summarize the main features of the method.

Overview

QPOPT's method has a feasibility phase (finding a feasible point by minimizing the sum of infeasibilities) and an optimality phase (minimizing the quadratic objective function within the feasible region). The computations in both phases are performed by the same subroutines, but with different objective functions. The feasibility phase does not perform the standard simplex method; i.e., it does not necessarily find a vertex (with n constraints active), except in the LP case if mL <= n. Once an iterate is feasible, all subsequent iterates remain feasible. Once a vertex is reached, all subsequent iterates are at a vertex.

QPOPT is designed to be efficient when applied to a sequence of related problems-for example, within a sequential quadratic programming method for nonlinearly constrained optimization (e.g., the NPOPT package). In particular, the user may specify an initial working set (the indices of the constraints believed to be satisfied exactly at the solution); see the discussion of Warm Start.

In general, an iterative process is required to solve a quadratic program. Each new iterate \bar{x} is defined by


\bar{x}	= x + \alpha p,

where the step length α 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 iteration index.)

The working set

At each point x, a working set of constraints is defined to be a linearly independent subset of the constraints that are satisfied "exactly" (to within the Feasibility tolerance). The working set is the current prediction of the constraints that hold with equality at a solution of LCQP. Let mw denote the number of constraints in the working set (including bounds), and let W denote the associated mw × n matrix of constraint gradients.

The definition of the search direction ensures that constraints in the working set remain unaltered for any value of the step length. Thus,

Wp = 0.

In order to compute p, a TQ factorization of W is used:


W Q = \left( {\begin{array}{cc}
0 & T  \\
\end{array} } \right),

where T is a nonsingular mw × mw upper-triangular matrix, and Q is an n × n nonsingular matrix constructed from a product of orthogonal transformations. If the columns of Q are partitioned so that


Q = \left( {\begin{array}{cc}
Z & Y  \\
\end{array} } \right),

where Y is n × mw and Z is n × nZ (where nZ = n - mw ), then the columns of Z form a basis for the null space of W . Let nR be an integer such that 0 <= nR <= nZ , and let ZR denote a matrix whose nR columns are a subset of the columns of Z . (The integer nR is the quantity "Zr" in the printed output from qpopt). In many cases, ZR will include all the columns of Z . The direction p will satisfy if

p = ZRpR,

where pR is any nR -vector.

The reduced Hessian

Let gQ and HQ denote the transformed gradient and transformed Hessian :

gQ = QTg(x) and HQ = QTHQ.

The first nR elements of the vector gQ will be denoted by gR , and the first nR rows and columns of the matrix HQ will be denoted by HR . The quantities gR and HR are known as the reduced gradient and reduced Hessian of q(x), respectively. Roughly speaking, gR and HR describe the first and second derivatives of an unconstrained problem for the calculation of pR .

At each iteration, a triangular factorization of HR is available. If HR is positive definite, HR = RTR, where R is the upper-triangular Cholesky factor of HR . If HR is not positive definite, HR = RTDR, where D = diag(1, 1, . . . , 1, w), with w <= 0.

In QPOPT, the computation is arranged so that the reduced-gradient vector is a multiple of eR , a vector of all zeros except in the last (nRth) position. This allows pR to be computed from a single back-substitution,


R p\R = \gamma e\R,

where γ is a scalar whose definition depends on whether the reduced Hessian is positive definite at x. In the positive-definite case, x + p is the minimizer of the objective function subject to the working-set constraints being treated as equalities. If HR is not positive definite, pR satisfies

p\R^T_R H_R p_R < 0 and g^T_R p_R \le 0,

allowing the objective function to be reduced by any step of the form x + αp, α > 0.

Optimality conditions

If the reduced gradient is zero, x is a constrained stationary point in the subspace defined by Z . During the feasibility phase, the reduced gradient will usually be zero only at a vertex (although it may be zero elsewhere in the presence of constraint dependencies). During the optimality phase, 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 λ are defined from the equations

WTλ = g(x).

A Lagrange multiplier λ corresponding to an inequality constraint in the working set is said to be optimal if \lambda_j \le \sigma when the associated constraint is at its upper bound , or if \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 deleting the corresponding constraint from the working set (see #The Summary File).

If optimal multipliers occur during the feasibility phase but the sum of infeasibilities is not zero, there is no feasible point. The user can request QPOPT to continue until the sum of infeasibilities is minimized (see the discussion of Min sum). At such a point, the Lagrange multiplier λj corresponding to an inequality constraint in the working set will be such that -(1+\sigma) \le \lambda_j \le \sigma when the associated constraint is at its upper bound , and -\sigma \le \lambda_j
\le 1+\sigma when the associated constraint is at its lower bound . Lagrange multipliers for equality constraints will satisfy


|\lambda_j| \le 1 + \sigma

If the reduced gradient is not zero, Lagrange multipliers need not be computed and the search direction p is given by ZR pR. The step length is chosen to maintain feasibility with respect to the satisfied constraints. If HR is positive definite and x + p is feasible, α is defined to be one. In this case, the reduced gradient at \bar{x} will be zero, and Lagrange multipliers are computed. Otherwise, α is set to αM , the step to the "nearest" constraint (with index Jadd; see #The Summay File). This constraint is added to the working set at the next iteration.

If the reduced Hessian HR is not positive definite and aM does not exist (i.e., no positive step aM reaches the boundary of a constraint not in the working set), then QPOPT terminates at x and declares the problem to be unbounded.

Further Details of the Method

The following sections are not essential knowledge for normal users. They give background on the active-set strategy and the anti-cycling procedure.

Treatment of simple upper and lower bounds

Bound constraints \ell \le x \le u are treated specially by qpopt. The presence of a bound constraint in the working set has the effect of fixing the corresponding component of the search direction to zero. Thus, the associated variable is fixed , and specification of the working set induces a partition of x into fixed and free variables. For some permutation P, the working-set matrix satisfies


W P = \left( {\begin{array}{cc}
F & N  \\
& I_N
\end{array} } \right),

where ( F N ) is part of the matrix A, and IN corresponds to some of the bounds. The matrices F and N contain the free and fixed columns of the general constraints in the working set. A T Q factorization F QF = ( 0 TF ) of the smaller matrix F provides the required T and Q as follows:


Q = P \left( {\begin{array}{cc}
Q_F &  \\
& I_N
\end{array} } \right),


T = \left( {\begin{array}{cc}
T_F & N  \\
& I_N
\end{array} } \right),

The matrix QF is implemented as a dense orthogonal matrix. Each change in the working set leads to a simple change to F : if the status of a general constraint changes, a row of F is altered; if a bound constraint enters or leaves the working set, a column of F changes. The matrices TF , QF and R are held explicitly; together with the vectors QTg, and QTc. Products of plane rotations are used to update QF and TF as the working set changes. The triangular factor R associated with the reduced Hessian is updated only during the optimality phase.

The initial working set

For a cold start, the initial working set includes equality constraints and others that are close to being satisfied at the starting point. ("Close" is defined under Crash tolerance.) For a warm start, the initial working is specified by the user (and possibly revised to improve the condition of W ).

At the start of the optimality phase, QPOPT must ensure that the initial reduced Hessian HR is positive-definite. It does so by including a suitably large number of constraints (real or artificial) in the initial working set. (When W contains n constraints, HR has no rows and columns. Such a matrix is positive definite by definition.)

Let HZ denote the first nZ rows and columns of HQ = QTHQ at the beginning of the optimality phase. A partial Cholesky factorization with interchanges is used to find an upper-triangular matrix R that is the factor of the largest positive-definite leading submatrix of HZ. The use of interchanges tends to maximize the dimension of R. (The condition of R may be controlled by setting the Rank Tolerance.) Let ZR denote the columns of Z corresponding to R, and let Z be partitioned as Z = ZR ZA . A working set for which ZR defines the null space can be obtained by including the rows of ZAT as "artificial constraints" (with bounds equal to the current value of ZATx). Minimization of the objective function then proceeds within the subspace defined by ZR , as described in #Introduction.

The artificially augmented working set is given by


\bar{W} = \left( {\begin{array}{cc}
Z_A^T  \\
W
\end{array} } \right),

so that p will satisfy W p = 0 and ZATp = 0. By definition of the T Q factors of W , we have


\bar{W}Q = \left( {\begin{array}{cc}
Z_A^T  \\
W
\end{array} } \right),

Q = \left( {\begin{array}{cc}
Z_A^T  \\
W
\end{array} } \right)

\left( {\begin{array}{ccc}
Z_R & Z_A & Y  \\
\end{array} } \right)
= 
\left( {\begin{array}{cc}
0 & \bar{T}
\end{array} } \right),

where


\bar{T} = \left( {\begin{array}{cc}
I & 0 \\
0 & T
\end{array} } \right).

Hence the T Q factors of \bar{W} are available trivially.

The matrix ZA is not kept fixed, since its role is purely to define an appropriate null space; the T Q factorization can therefore be updated in the normal fashion as the iterations proceed. No work is required to "delete" the artificial constraints associated with ZA when ZRTg = 0, since this simply involves repartitioning Q. The "artificial" multiplier vector associated with the rows of ZAT is equal to ZATg, and the multipliers corresponding to the rows of the "true" working set are the multipliers that would be obtained if the artificial constraints were not present. If an artificial constraint is "deleted" from the working set, an A appears alongside the entry in the Jdel column of the printed output (see #The Summary File). The multiplier may have either sign.

The number of columns in ZAand ZR, the Euclidean norm of ZRTg, and the condition estimator of R appear in the printed output as Art, Zr, Norm gZ and Cond Rz (see #The Summary File).

Under some circumstances, a different type of artificial constraint is used when solving a linear program. Although the algorithm of qpopt does not usually perform simplex steps (in the traditional sense), there is one exception: a linear program with fewer general constraints than variables (i.e., mL <= n). (Use of the simplex method in this situation leads to savings in storage.) At the starting point, the "natural" working set (the set of constraints exactly or nearly satisfied at the starting point) is augmented with a suitable number of "temporary" bounds, each of which has the effect of temporarily fixing a variable at its current value. In subsequent iterations, a temporary bound is treated similarly to normal constraints until it is deleted from the working set, in which case it is never added again. If a temporary bound is "deleted" from the working set, an F (for "Fixed") appears alongside the entry in the Jdel column of the printed output (see #The Summary File). Again, the multiplier may have either sign.

The anti-cycling procedure

The EXPAND procedure is used to reduce the possibility of cycling at a point where the active constraints are nearly linearly dependent. The main feature of EXPAND is that the feasibility tolerance is increased slightly at the start of every iteration. This allows a positive step to be taken every iteration, perhaps at the expense of violating the constraints slightly.

Suppose that the Feasibility tolerance is δ. Over a period of K iterations (where K is defined by the Expand frequency), the feasibility tolerance actually used by QPOPT-the working feasibility tolerance-increases from 0.5δ to δ (in steps of 0.5δ/K).

At certain stages the following "resetting procedure" is used to remove constraint infeasibilities. First, all variables whose upper or lower bounds are in the working set are moved exactly onto their bounds. A count is kept of the number of nontrivial adjustments made. If the count is positive, iterative refinement is used to give variables that satisfy the working set to (essentially) machine precision. Finally, the working feasibility tolerance is reinitialized to 0.5δ.

If a problem requires more than K iterations, the resetting procedure is invoked and a new cycle of iterations is started with K incremented by 10. (The decision to resume the feasibility phase or optimality phase is based on comparing any constraint infeasibilities with δ.)

The resetting procedure is also invoked when QPOPT reaches an apparently optimal, infeasible or unbounded solution, unless this situation has already occurred twice. If any nontrivial 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 constraints to be added to the working set. Let aM denote the maximum step at which x + αMp does not violate any constraint by more than its feasibility tolerance. All constraints at distance α (α <= αM ) along p from the current point are then viewed as acceptable candidates for inclusion in the working set. The constraint whose normal makes the largest angle with the search direction is added to the working set. This strategy helps keep the working-set matrix W well-conditioned.

The Options File

Observe that options are normally set in Prob.SOL.optPar.

Several choices in QPOPT's algorithm logic may be defined by various optional parameters (more briefly known as options or parameters).

In order to reduce the number of subroutine parameters for qpopt, the options have default values that are appropriate for most problems. Options need be specified only if their values should be different from the default.

Format of option strings

Each optional parameter is defined by an option string of up to 72 characters, containing one or more items separated by spaces or equal signs (=). Alphabetic characters may be in upper or lower case. An example option string is Print level = 5. In general, an option string contains the following items:

  1. A keyword such as Print.
  2. A phrase such as level that qualifies the keyword. (Typically 0, 1 or 2 words.)
  3. A number that specifies either an integer or a real value (only for some options). Such numbers may be up to 16 contiguous characters in Fortran 77's F, E or D formats, terminated by a space.

Blank strings and comments may be used to improve readability. A comment begins with an asterisk (*) and all subsequent characters are ignored. Synonyms are recognized for some of the keywords, and abbreviations may be used if there is no ambiguity.

The following are examples of valid option strings for QPOPT:

NOLIST
COLD START
Warm start
Problem type  = LP
Problem type  = Quadratic Program        * Same as QP or QP2
Problem Type    QP4
Min sum         Yes
Feasibility Phase iteration limit   100
Feasibility tolerance            1.0e-8  * for IEEE double precision
Crash tolerance                   0.002
Defaults
* This string will be ignored.             So will a blank line.

Description of the optional parameters

Permissible options are defined below in alphabetical order. For each option, we give the keyword, any essential qualifiers, the default value, and the definition. The minimum abbreviation of each keyword and qualifier is underlined. If no characters of a qualifier are underlined, the qualifier may be omitted. The letters i and r denote integer and real values required for certain options. The letter a denotes a character string value. The number u represents unit roundoff for floating-point arithmetic (typically about 10-16).

Check frequencyiDefault = 50

Every ith iteration, a numerical test is made to see if the current solution x satisfies the constraints in the working set. If the largest residual of the constraints in the working set is judged to be too large, the working-set matrix is refactorized and the variables are recomputed to satisfy the constraints more accurately.

Cold startDefault = Coldstart
Warm start

This option specifies how the initial working set is chosen. With a cold start, QPOPT chooses the initial working set based on the values of the variables and constraints at the initial point. Broadly speaking, the first working set will include all equality constraints and also any bounds or inequality constraints that are "nearly" satisfied (to within the Crash tolerance).

With a warm start, the user must provide a valid definition of every element of the array istate. The specification of istate will be overridden if necessary, so that a poor choice of the working set will not cause a fatal error. A warm start will be advantageous if a good estimate of the initial working set is available-for example, when qpopt is called repeatedly to solve related problems.

Crash tolerancerDefault = 0.01

This value is used for cold starts when QPOPT selects an initial working set. Bounds and inequality constraints are selected if they are satisfied to within r. More precisely, a constraint of the form aTx = l will be included in the initial working set if | a_j^T x - l | \le r(1 +
|l|). If r < 0 or r > 1, the default value is used.

Defaults

This is a special option to reset all options to their default values.

Expand frequencyiDefault = 5

This defines the initial value of an integer K that is used in an anti-cycling procedure designed to guarantee progress even on highly degenerate problems. See #The anti-cycling procedure.

If i = 9999999, no anti-cycling procedure is invoked.

Feasibility tolerancerDefault = \sqrt{u}

This defines the maximum acceptable absolute violation in each constraint at a "feasible" point. For example, if the variables and the coefficients in the general constraints are of order unity, and the latter are correct to about 6 decimal digits, it would be appropriate to specify r as 10-6. If r < u, the default value is used.

Before optimizing the objective function, QPOPT must find a feasible point for the constraints. If the sum of infeasibilities cannot be reduced to zero and Min sum = Yes is requested, QPOPT will find the minimum value of the sum. Let sinf be the corresponding sum of infeasibilities. If sinf is quite small, it may be appropriate to raise r by a factor of 10 or 100. Otherwise, some error in the data should be suspected.

Feasibility Phase Iteration Limiti1Default = max(50, 5(n + mL))
Optimality Phase Iteration Limiti2Default = Default = max(50, 5(n + mL ))

The scalars i1 and i2 specify the maximum number of iterations allowed in the feasibility and optimality phases. Optimality Phase iteration limit is equivalent to Iteration limit. Setting i1 = 0 and PrintLevel > 0 means that the workspace needed will be computed and printed, but no iterations will be performed.

Hessian rowsiDefault = 0 or n

This specifies m, the number of rows in the Hessian matrix H or its trapezoidal factor G (as used by the default subroutine qpHess).

For problem type FP or LP, the default value is m = 0.

For problems QP1 or QP2, the first m rows and columns of H are obtained from H, and the remainder are assumed to be zero. For problems QP3 or QP4, the factor G is assumed to have m rows and n columns. They are obtained from the associated rows of H.

If a nonstandard subroutine qpHess is provided, it may access the problem type and m via the lines

integer              lqptyp, mHess 
common	    /sol1qp/ lqptyp, mHess

For example, Problem type FP, LP or QP4 sets lqptyp = 1, 2 or 6 respectively, and Hessian rows 20 sets mHess = 20.

Infinite Bound sizerDefault = 1020

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.

Infinite Step sizerDefault = max(bigbnd, 1020 )

If r > 0, r specifies the magnitude of the change in variables that will be considered a step to an unbounded solution. (Note that an unbounded solution can occur only when the Hessian is not positive definite.) If the change in x during an iteration would exceed the value of Infinite Step, the objective function is considered to be unbounded below in the feasible region. If r <= 0, the default value is used.

Iteration limitiDefault = max(50, 5(n + mL ))
Iters
Itns

This is equivalent to Optimality Phase iteration limit. See Feasibility Phase.

List

If Nolist was previously specified, List restores output to the Print file whenever an optional parameter is reset.

Maximum degrees of freedomiDefault = n

This places a limit on the storage allocated for the triangular factor R of the reduced Hessian HR . Ideally, i should be set slightly larger than the value of nR expected at the solution. (See #The working set and #The reduced Hessian.) It need not be larger than mN + 1, where mN is the number of variables that appear nonlinearly in the quadratic objective function. For many problems it can be much smaller than mN .

For quadratic problems, a minimizer may lie on any number of constraints, so that nR may vary between 1 and n. The default value of i is therefore the number of variables n. If Hessian rows m is specified, the default value of i is the same number, m.

Min sumaDefault = No

This option comes into effect if the constraints cannot be satisfied. If Min sum = No, QPOPT terminates as soon as it is evident that no feasible point exists. The final point will generally not be the point at which the sum of infeasibilities is minimized. If Min sum = Yes, QPOPT will continue until either the sum of infeasibilities is minimized or the iteration limit is reached, whichever occurs first.

Nolist

This suppresses output to the Print file whenever an optional parameter is reset.

Optimality tolerancerDefault = \sqrt{u}

This affects the tolerance used to determine if the Lagrange multipliers associated with the bounds and general constraints have the right "sign" for the solution to be judged optimal. Increasing r tends to cause earlier termination. For example, if r = 1.0e - 4, the final objective value will probably agree with the true optimum to about 4 digits.

Print leveliDefault = 10

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

i
0No output.
1The final solution only, sent to the Print file.
5One line of output for each iteration (no printout of the final solution).
>= 10The final solution and one line of output for each iteration (Print file only).
>= 20At each iteration, the Lagrange multipliers, the variables x, the constraint values Ax and the constraint status (Print file only).
>= 30At each iteration, the diagonal elements of the upper-triangular matrix T associated with the T Q factorization of the working set, and the diagonal elements of the upper-triangular matrix R (Print file only).
Problem typeaDefault = QP2

This option specifies the type of objective function to be minimized during the optimality phase. The following are the six values of a and the dimensions of the arrays that must be specified to define the objective function:

FP		H and cvec not accessed;
LP		H not accessed, cvec(n) required;
QP1H(ldH,*) 	symmetric, cvec not referenced;
P2H(ldH,*) 	symmetric, cvec(n);
QP3H(ldH,*) 	upper-trapezoidal, cvec not referenced;
QP4H(ldH,*) 	upper-trapezoidal, cvec(n);

Linear program is equivalent to LP. Quadratic program and QP are equivalent to the default option QP2. For the QP options, the default subroutine qpHess requires array H(ldH,*) as shown. If a non-standard qpHess is provided, H(*,*) may be used in any convenient way.

RankrDefault = 100u

This parameter enables the user to control the condition number of the triangular factor R (see #Introduction). If ρi denotes the function \rho_i = \max{\{|R_{11}|, |R_{22}|, \dots, |R_{ii}|\}}, the dimension of R is defined to be smallest index i such that |R_{i+1,i+1}| \le \sqrt{r}|\rho_{i+1}|. If r <= 0, the default value is used.

Summary fileiDefault = 6

This specifies the unit number for the Summary file (see #The Summary File).

If i > 0 and PrintLevel > 0, a brief log in 80-column format is output to unit i. On many systems, the default value refers to the screen. Summary file = 0 suppresses output, including error messages .

Warm start

See Cold start.

Optional parameter checklist and default values

For easy reference, the following list shows all valid options and their default values. The quantity u represents floating-point precision (≈1.1 × 10-16 in IEEE double-precision arithmetic).

Check frequency50*
Cold start*
Crash tolerance.01*
Expand frequency5*
Feasibility tolerance1.1e-8*\sqrt{u}
Feasibility Phase iteration limit50*or 5(n + mL)
Optimality Phase iteration limit50*or 5(n + mL)
Hessian rowsn*
Infinite bound size1.0e+20*Plus infinity
Infinite step size1.0e+20*
Iteration limit50*or 5(n + mL)
List*
Maximum degrees of freedomn*
Min sumNo*
Optimality tolerance1.1e-8*\sqrt{u}
Print file9*
Print level10*
Problem typeQP*or QP2
Rank tolerance1.1e-14*100u
Summary file6*

Other options may be set as follows:

Defaults
Nolist
Warm start

The Summary File

The Summary file records an iteration log and error messages. The file name is set in Prob.SOL.SummFile.

Constraint numbering and status

For items Jdel and Jadd in the iteration log, indices 1 through n refer to the bounds on the variables, and indices n + 1 through n + nclin refer to the general constraints.

When the status of a constraint changes, the index of the constraint is printed, along with the designation L (lower bound), U (upper bound), E (equality), F (temporarily fixed variable) or A (artificial constraint).

The iteration log

The following items are printed after each iteration.

Itnis the iteration count (including those from the feasibility phase).
Jdelis the index of the constraint deleted from the working set. If Jdel is zero, no constraint was deleted.
Jaddis the index of the constraint added to the working set. If Jadd is zero, no constraint was added.
Stepis the step taken along the computed search direction. If a constraint is added during the current iteration (i.e., Jadd is positive), Step will be the step to the nearest constraint. During the optimality phase, the step can be greater than one only if the reduced Hessian is not positive definite.
Ninfis the number of violated constraints (infeasibilities). This number will be zero during the optimality phase.

Sinf/Objective is the value of the current objective function. If x is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If x is feasible, Objective is the value of the objective function. The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point.

During the feasibility phase, the number of constraint infeasibilities will not increase until either a feasible point is found, or the optimality of the multipliers implies that no feasible point exists. Note that the sum of the infeasibilities may increase or decrease during this part of the feasibility phase. However, once optimal phase-one multipliers are obtained, the number of infeasibilities can increase, but the sum of infeasibilities must either remain constant or be reduced until the minimum sum of infeasibilities is found.

In the optimality phase, the value of the objective is non-increasing.

Norm gZis ||Z_R^Tg||, the Euclidean norm of the reduced gradient with respect to ZR . During the optimality phase, this norm will be approximately zero after a unit step.
Zris the number of columns of ZR (see #Introduction). Zr is the dimension of the subspace in which the objective is currently being minimized. The value of Zr is the number of variables minus the number of constraints in the working set.
Artis the number of artificial constraints in the working set, i.e., the number of columns of ZA (see #Further Details of the Method). At the start of the optimality phase, Art provides an estimate of the number of nonpositive eigenvalues in the reduced Hessian.

Summary file from the example problem

Following is a Summary file example.

  
                     QPOPT  ---  Version 1.0-10     Sep  1995
                     ========================================

 Itn Jdel  Jadd     Step Ninf  Sinf/Objective Norm gZ   Zr  Art
   0    0     0  0.0E+00    0  0.00000000E+00 0.0E+00    0    6
 Itn     0 -- Feasible point found.
   0    0     0  0.0E+00    0  1.51638000E+03 9.8E+01    1    5
   1    0     8U 2.8E-01    0  1.72380000E+02 0.0E+00    0    5
   2    1L   10L 3.1E-03    0  1.68083225E+02 0.0E+00    0    5
   3    5A   11L 1.2E-02    0  1.57176475E+02 0.0E+00    0    4


 Itn Jdel  Jadd     Step Ninf  Sinf/Objective Norm gZ   Zr  Art
   4    4A   12L 3.2E-02    0  1.38528925E+02 0.0E+00    0    3
   5    3A   13L 6.9E-02    0  1.11295925E+02 0.0E+00    0    2
   6    2A   14L 1.3E-01    0  7.41228000E+01 0.0E+00    0    1
   7    1A    1U 8.4E-01    0 -5.85162625E+01 0.0E+00    0    0
   8   13L    0  1.0E+00    0 -8.72144740E+01 1.3E-15    1    0


 Itn Jdel  Jadd     Step Ninf  Sinf/Objective Norm gZ   Zr  Art
   9    1U    6U 2.5E+00    0 -3.12744888E+02 1.4E+02    1    0
  10    0     1L 1.4E-01    0 -5.62265012E+02 0.0E+00    0    0
  11   14L    7U 1.3E-01    0 -6.21487825E+02 0.0E+00    0    0

 Exit from QP problem after   11 iterations.  Inform =  0


                     QPOPT  ---  Version 1.0-10     Sep  1995
                     ========================================

 Itn Jdel  Jadd     Step Ninf  Sinf/Objective Norm gZ   Zr  Art
   0    0     0  0.0E+00    3  2.35500000E+01 1.7E+00    0    3
   1    2U   10L 4.0E+00    2  1.96000000E+01 1.4E+00    0    3
   2    4U   12L 7.8E+00    1  1.17500000E+01 1.0E+00    0    3
   3    6U   14L 1.2E+01    0  0.00000000E+00 0.0E+00    0    3
 Itn     3 -- Feasible point found.
   3    0     0  0.0E+00    0  8.66526437E+02 1.5E+02    1    2


 Itn Jdel  Jadd     Step Ninf  Sinf/Objective Norm gZ   Zr  Art
   4    0     9L 1.0E-01    0  4.98244375E+01 0.0E+00    0    2
   5    2A   11L 4.5E-01    0 -5.62265013E+02 0.0E+00    0    1
   6    1A    6U 5.7E-13    0 -5.62265013E+02 0.0E+00    0    0
   7   14L    7U 1.3E-01    0 -6.21487825E+02 0.0E+00    0    0

 Exit from QP problem after    7 iterations.  Inform =  0

The Print File

The Print file records specified options, error messages, a detailed iteration log, and the final solution. The print file is specified in Prob.SOL.PrintFile.

Constraint numbering and status

Items Jdel and Jadd in the iteration log are the same as in the Summary file. Please see #Constraint numbering and status.

The iteration log

When PrintLevel >= 5, a line of output is produced at every iteration. The quantities printed are those in effect on completion of the iteration. Several items are the same as in the Summary file. Please see #The iteration log.

ItnSame as Summary file.
JdelSame as Summary file.
JaddSame as Summary file.
StepSame as Summary file.
NinfSame as Summary file.
Sinf/ObjectiveSame as Summary file.
Bndis the number of simple bound constraints in the current working set.
Linis the number of general linear constraints in the current working set.
ArtSame as Summary file.
ZrSame as Summary file. Zr = n - (Bnd + Lin + Art).

The number of columns of Z (see #Introduction) can be calculated as Nz = n-(Bnd+Lin) = Zr+Art.

If Nz is zero, x lies at a vertex of the feasible region.

Norm gZSame as Summary file.
NOptis the number of nonoptimal Lagrange multipliers at the current point. NOpt is not printed if the current x is infeasible or no multipliers have been calculated. At a minimizer, NOpt will be zero.
Min LMis the value of the Lagrange multiplier associated with the deleted constraint. If the Min LM is negative, a lower bound constraint has been deleted, if Min LM is positive, an upper bound constraint has been deleted. If no multipliers are calculated during a given iteration, Min LM will be zero.
Cond Tis a lower bound on the condition number of the working-set matrix W .
Cond Rzis a lower bound on the condition number of the triangular factor R (the Cholesky factor of the current reduced Hessian HR , whose dimension is Zr). If the problem type is LP, Cond Rz is not printed.
Rzzis the last diagonal element ω of the matrix D associated with the RTDR factorization of the reduced Hessian HR (see #Introduction). Rzz is only printed if HR is not positive definite (in which case ω = 1). If the printed value of Rzz is small in absolute value, then HR is approximately singular. A negative value of Rzz implies that the objective function has negative curvature on the current working set.

Printing the solution

When PrintLevel = 1 or PrintLevel >= 10, the final output from qpopt includes a listing of the status of every variable and constraint. Numerical values that are zero are printed as ".". In the "Variables" section, the following output is given for each variable xj (j = 1 to n).

Variablegives j, the number of the variable.
Stategives the state of the variable. The possible states are as follows, where δis the Feasibility tolerance.
FRThe variable lies between its upper and lower bound.
EQThe variable is a fixed variable, with xj equal to its upper and lower bound.
LLThe variable is active at its lower bound (to within δ).
ULThe variable is active at its upper bound (to within δ).
TFThe variable is temporarily fixed at its current value.
--The lower bound is violated by more than δ.
++The upper bound is violated by more than δ.

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

AAlternative optimum possible. The variable is active at one of its bounds, but its Lagrange multiplier is essentially zero. This means that if the variable were allowed to start moving away from its bound, there would be no change to the objective function. The values of the other free variables might change, giving a genuine alternative solution. However, if there are any degenerate variables (labeled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case, the values of the Lagrange multipliers might also change.
DDegenerate. The variable is free, but it is equal to (or very close to) one of its bounds.
IInfeasible. The variable is currently violating one of its bounds by more than δ.
Valueis the final value of the variable xj .
Lower boundis the lower bound specified for xj . "None" indicates that bl(j) <= -bigbnd.
Upper boundis the upper bound specified for xj . "None" indicates that bu(j) >= bigbnd.
Lagr multiplieris the Lagrange multiplier for the associated bound. This will be zero if State is FR. If x is optimal, the multiplier should be non-negative if State is LL, and non-positive if State is UL.
Slackis the difference between the variable "Value" and the nearer of its (finite) bounds bl(j) and bu(j). A blank entry indicates that the associated variable is not bounded (i.e., bl(j) <= -bigbnd and bu(j) >= bigbnd).

In the "Constraints" section, similar output is given for each constraint aiTx, i = 1 to nclin. The word "variable" must be replaced by "constraint", and xj should be changed to aiTx, and (j) should be changed to (nclin + i). "Movement off a constraint" means allowing the entry in the slack column to become positive.

Interpreting the printout

The input data for qpopt should always be checked (even if it terminates with inform = 0!). Two common sources of error are uninitialized variables and incorrectly dimensioned array arguments. The user should check that all components of A, bl, bu and x are defined on entry to qpopt, and that qpHess computes all relevant components of Hx.

In the following, we list the different ways in which qpopt terminates abnormally and discuss what further action may be necessary.

UnderflowA single underflow will always occur if machine constants are computed automatically (as in the dis- tributed version of QPOPT). Other floating-point underflows may occur occasionally, but can usually be ignored.
OverflowIf the printed output before the overflow error contains a warning about serious ill-conditioning in the working set when adding the jth constraint, it may be possible to avoid the difficulty by increasing the Feasibility tolerance. If the message recurs, the offending linearly dependent constraint (with index "j") must be removed from the problem. If a warning message did not precede the fatal overflow, contact the authors.
inform = 3The problem appears to have no feasible point. Check that there are no conflicting constraints, such as x1 = 1, x2 = 2 and x1 + x2 = 0. If the data for the constraints are accurate to the absolute precision s, make sure that the Feasibility tolerance is greater than s. For example, if all elements of A are of order unity and are accurate to only three decimal places, the Feasibility tolerance should be at least 10-3.
inform = 4One of the iteration limits may be too small. (See Feasibility Phase and Optimality Phase.) Increase the appropriate limit and rerun qpopt.
inform = 5The Maximum Degrees of Freedom is too small. Rerun qpopt with a larger value (possibly using the warm start facility to specify the initial working set).
inform = 6An input parameter is invalid. The printed output will indicate which parameter(s) must be redefined.

Rerun with corrected values.

inform = 7The specified problem type was not FP, LP, QP1, QP2, QP3, or QP4. Rerun qpopt with Problem type set to one of these values.
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