SOL MINOS
This page is part of the SOL Manual. See SOL. |
Direct Solver Call
A direct solver call is not recommended unless the user is 100 % sure that no other solvers will be used for the problem. Please refer to subsection Using TOMLAB for information on how to use MINOS with TOMLAB.
Purpose
minos solves nonlinear optimization problems defined as
where , , , and .
or quadratic optimization problems defined as
where , , , and .
The full input matrix A has three parts A = [d/dx g(x); A; c'];
Calling Syntax
The file 'funfdf.m' must be defined and contain: function [mode, f, g] = funfdf(x, Prob, mode, nstate) to compute the objective function f and the gradient g at the point x.
The file 'funcdc.m' must be defined and contain: function [mode ,c ,dcS] = funcdc(x, Prob, mode, nstate) to compute the nonlinear constraint value c and the constraint Jacobian dcS for the nonlinear constraints at the point x.
NOTE: The matrix dcS MUST be a SPARSE MATLAB matrix. Do dcS = sparse(dcS); after dcS has been computed.
[hs, xs, pi, rc, Inform, nS, nInf, sInf, Obj, iwCount, gObj, fCon, gCon] = minos(H, A, bl, bu, nnCon, nnObj, nnJac, Prob, iObj, optPar, Warm, hs, xs, pi, nS, SpecsFile, PrintFile, SummFile, PriLev, ObjAdd, moremem, ProbName );
Description of Inputs
The following fields are used:
Input | Description |
---|---|
H | Matrix n x n in a quadratic programming (QP) problem. DENSE or SPARSE. Leave empty if LP, or NLP problem. |
A | Constraint matrix, m x n SPARSE (nonlinear, linear and objective) m > 0 always!!! Define dummy constraint for unconstrained problems. |
bl | Lower bounds on (x,g(x),Ax,c'). |
bu | Upper bounds on (x,g(x),Ax,c').
NOTE! The bl and bu values for the last nonlinear constraint c must have reverse signs and be put in each other places: If cL <= c(x) <= cU , then bl = -cU and bu = -cL . This is because the bounds acts as the constraints on the slack variables for the nonlinear constraints. |
nnCon | Number of nonlinear constraints. |
nnObj | Number of nonlinear objective variables. |
nnJac | Number of nonlinear Jacobian variables. |
Prob | Must be a structure. No check is made in the MEX interface. If TOMLAB calls minos, then Prob is the standard TOMLAB problem structure, otherwise the user should set:
Prob.P = ProblemNumber, where ProblemNumber is some integer. If the problem is a LP or QP problem (H defined), the user does not have to specify anything else in the structure. For a general nonlinear objective or nonlinear constraints names of two user written routines must be given: funfdf, actual name stored in Prob.FUNCS.fg, with syntax [mode, f, g] = funfdf(x, Prob, mode, nstate). funcdc, actual name stored in Prob.FUNCS.cdc, with syntax [mode, c, dcS] = funcdc(x, Prob, mode, nstate). MINOS is calling the TOMLAB routines nlp fg.m and nlp cdcS.m in the call- back, and they call funfdf and funcdc, respectively. If these fields in Prob are empty (Prob.FUNCS.fg, Prob.FUNCS.cdc), the TOMLAB callback routines calls the usual function routines. Then the Prob struct should be normally defined, and the fields Prob.FUNCS.f, Prob.FUNCS.g, Prob.FUNCS.c, Prob.FUNCS.dc be set in the normal way (e.g. by the routine mFiles.m, or one of the Assign-routines like conAssign.m). If the mode parameter is 0, funfdf should return f, otherwise both f and the gradient vector g. If the mode parameter is 0, funcdc should return c, otherwise both c and dcS. Note that each row in dcS corresponds to a constraint, and that dcS must be a SPARSE matrix. The user could also write his own versions of the routines nlp fg.m and nlp cdcS.m and put them before in the path. |
iObj | Says which row of A is a free row containing a linear objective vector c. If there is no such vector, iObj = 0. Otherwise, this row must come after any nonlinear rows, so that nnCon <= iObj <= m. |
optPar | Vector with optimization parameters overriding defaults and the optionally specified SPECS file. If using only default options, set optPar as an empty matrix. |
Warm | Flag, if true: warm start. Default cold start (if empty). If 'Warm Start' xs, nS and hs must be supplied with correct values. |
hs | Basis status of variables + constraints (n+m x 1 vector). State of vari- ables: 0=nonbasic (on bl), 1=nonbasic (on bu) 2=superbasic (between bounds), 3=basic (between bounds). |
xs | Initial vector, optionally including m slacks at the end. If warm start, full xs must be supplied. |
pi | Lagrangian multipliers for the nnCon nonlinear constraints. If empty, set as 0. |
nS | # of superbasics. Only used if calling again with a Warm Start. |
SpecsFile | Name of the SPECS input parameter file, see TOMLAB. |
PrintFile | Name of the Print file. Name includes the path, maximal number of characters = 500. |
SummFile | Name of the Summary file. Name includes the path, maximal number of char- acters = 500. |
PriLev | Printing level in the minos m-file and minos MEX-interface.
= 0 Silent = 1 Summary information = 2 More detailed information |
ObjAdd | Constant added to the objective for printing purposes, typically 0. |
moremem | Add extra memory for the sparse LU, might speed up the optimization. 1E6 is 10MB of memory. If empty, set as 0. |
ProbName | Name of the problem. ¡=100 characters are used in the MEX interface. In the MINOS solver the first 8 characters are used in the printed solution and in some routines that output BASIS files. Blank is OK. |
Description of Outputs
The following fields are used:
Output | Description |
---|---|
hs | Basis status of variables + constraints (n+m x 1 vector). State of vari- ables: 0=nonbasic (on bl), 1=nonbasic (on bu) 2=superbasic (between bounds),
3=basic (between bounds). Basic and superbasic variables may be outside their bounds by as much as the Feasibility tolerance. Note that if scaling is specified, the Feasibility tolerance applies to the variables of the scaled problem. In this case, the variables of the original problem may be as much as 0.1 outside their bounds, but this is unlikely unless the problem is very badly scaled. Check the "Primal infeasibility" printed after the EXIT message. Very occasionally some nonbasic variables may be outside their bounds by as much as the Feasibility tolerance, and there may be some nonbasics for which xn(j) lies strictly between its bounds. If ninf > 0, some basic and superbasic variables may be outside their bounds by an arbitrary amount (bounded by sinf if scaling was not used). |
xs | Solution vector (n+m by 1) with n decision variable values together with the m slack variables. |
pi | Lagrangian multipliers (dual solution vector) (m x 1 vector) |
rc | Vector of reduced costs, g - ( A I )Tp, where g is the gradient of the objective function if xn is feasible, or the gradient of the Phase-1 objective otherwise. If ninf = 0, the last m entries are -p. Reduced costs vector is of n+m length. |
Inform | Result of MINOS run.
0 Optimal solution found. 1 The problem is infeasible. 2 The problem is unbounded (or badly scaled). 3 Too many iterations. 4 Apparent stall. The solution has not changed for a large number of iterations (e.g. 1000). 5 The Superbasics limit is too small. 6 User requested termination (by returning bad value). 7 Gradient seems to be giving incorrect derivatives. 8 Jacobian seems to be giving incorrect derivatives. 9 The current point cannot be improved. 10 Numerical error in trying to satisfy the linear constraints (or the linearized nonlinear constraints). The basis is very ill-conditioned. 11 Cannot find a superbasic to replace a basic variable. 12 Basis factorization requested twice in a row. Should probably be treated as inform = 9. 13 Near-optimal solution found. Should probably be treated as inform = 9. 20 Not enough storage for the basis factorization. 21 Error in basis package. 22 The basis is singular after several attempts to factorize it (and add slacks where necessary). 30 An OLD BASIS file had dimensions that did not match the current problem. 32 System error. Wrong number of basic variables. 40 Fatal errors in the MPS file. 41 Not enough storage to read the MPS file. 42 Not enough storage to solve the problem. |
nS | # of superbasics. |
nInf | Number of infeasibilities. |
sInf | Sum of infeasibilities. |
Obj | Objective function value at optimum. |
iwCount | Number of iterations (major and minor), function and constraint calls. |
gObj | Gradient of the nonlinear objective. |
fCon | Nonlinear constraint vector. |
gCon | Gradient vector (non-zeros) of the nonlinear constraint vector. |
Using TOMLAB
Purpose
minosTL solves nonlinear optimization problems defined as
where , , , and .
Calling Syntax
Using the driver routine tomRun :
Prob = ''o''Assign( ... ); Result = tomRun('minos', Prob, 1);
Description of Inputs
Prob, The following fields are used: | |
---|---|
x_L, x_U | Bounds on variables. |
b_L, b_U | Bounds on linear constraints. |
c_L, c_U | Bounds on nonlinear constraints. |
A | Linear constraint matrix. |
QP.c | Linear coefficients in objective function. |
PriLevOpt | Print level. |
WarmStart | If true, use warm start, otherwise cold start. |
SOL.xs | Solution and slacks from previous run. |
SOL.hs | State for solution and slacks from previous run. |
SOL.nS | Number of superbasics from previous run. |
SOL.moremem | Add more memory if MINOS stops with not enough storage message. 1E6 is 10MB of memory. Default n+m1+m2 (number of variables + linear + nonlin- ear constraints). |
SOL.SpecsFile | Name of user defined SPECS file, read BEFORE optPar() is used. |
SOL.PrintFile | Name of SOL Print file. Amount and type of printing determined by SPECS parameters or optPar parameters. |
SOL.SummFile | Name of SOL Summary File. |
SOL.optPar | Elements > -999 takes precedence over corresponding TOMLAB params. See Table in SOL Using the SOL Solvers in TOMLAB. |
Description of Outputs
Result, The following fields are used: | |
---|---|
Result | The structure with results (see ResultDef.m). |
f_k | Function value at optimum. |
x_k | Solution vector. |
x_0 | Initial solution vector. |
g_k | Gradient of the function. |
c_k | Nonlinear constraint residuals. |
cJac | Nonlinear constraint gradients. |
xState | State of variables. Free == 0; On lower == 1; On upper == 2; Fixed == 3; |
bState | State of linear constraints. Free == 0; Lower == 1; Upper == 2; Equality == 3; |
cState | State of nonlinear constraints. Free == 0; Lower == 1; Upper == 2; Equality == 3; |
v_k | Lagrangian multipliers (for bounds + dual solution vector). |
ExitFlag | Exit status from minos.m (similar to TOMLAB). |
Inform | Result of MINOS run.
0 Optimal solution found. 1 The problem is infeasible. 2 The problem is unbounded (or badly scaled). 3 Too many iterations. 4 Apparent stall. The solution has not changed for a large number of iterations (e.g. 1000). 5 The Superbasics limit is too small. 6 User requested termination (by returning bad value). 7 Gradient seems to be giving incorrect derivatives. 8 Jacobian seems to be giving incorrect derivatives. 9 The current point cannot be improved. 10 Numerical error in trying to satisfy the linear constraints (or the linearized nonlinear constraints). The basis is very ill-conditioned. 11 Cannot find a superbasic to replace a basic variable. 12 Basis factorization requested twice in a row. Should probably be treated as inform = 9. 13 Near-optimal solution found. Should probably be treated as inform = 9. 20 Not enough storage for the basis factorization. 21 Error in basis package. 22 The basis is singular after several attempts to factorize it (and add slacks where necessary). 30 An OLD BASIS file had dimensions that did not match the current problem. 32 System error. Wrong number of basic variables. 40 Fatal errors in the MPS file. 41 Not enough storage to read the MPS file. 42 Not enough storage to solve the problem. |
rc | Vector of reduced costs, g-(AI)Tp, where g is the gradient of the objective function if xn is feasible, or the gradient of the Phase-1 objective otherwise. If ninf = 0, the last m entries are -p. Reduced costs vector is of n+m length. |
Iter | Number of iterations. |
FuncEv | Number of function evaluations. GradEv Number of gradient evaluations. |
ConstrEv | Number of constraint evaluations. |
QP.B | Basis vector in TOMLAB QP standard. |
MinorIter | Number of minor iterations. |
Solver | Name of the solver (minos). |
SolverAlgorithm | Description of the solver. |
SOL.xs | Solution and slack variables. |
SOL.hs | Basis status of variables + constraints (n+m x 1 vector). State of vari- ables: 0=nonbasic (on bl), 1=nonbasic (on bu) 2=superbasic (between bounds), 3=basic (between bounds).
Basic and superbasic variables may be outside their bounds by as much as the Feasibility tolerance. Note that if scaling is specified, the Feasibility tolerance applies to the variables of the scaled problem. In this case, the variables of the original problem may be as much as 0.1 outside their bounds, but this is unlikely unless the problem is very badly scaled. Check the "Primal infeasibility" printed after the EXIT message. Very occasionally some nonbasic variables may be outside their bounds by as much as the Feasibility tolerance, and there may be some nonbasics for which xn(j) lies strictly between its bounds. If ninf > 0, some basic and superbasic variables may be outside their bounds by an arbitrary amount (bounded by sinf if scaling was not used). |
SOL.nS | # of superbasics. |
SOL.nInf | # of infeasibilities. |
SOL.sInf | Sum of infeasibilities. |
optPar
Description
Use missing value (-999 or less), when no change of parameter setting is wanted. The default value will then be used by MINOS, unless the value is altered in the SPECS file.
Definition: nnL = max(nnObj,nnJac))
Description of Inputs
Table: The following fields are used:
# | SPECS keyword text | Lower | Default | Upper | Comment |
---|---|---|---|---|---|
Printing | |||||
1. | PRINT LEVEL | 0 | 0 | 11111 | JFLXB: Jac, fCon, lambda, x, B=LU stats |
Frequencies I | |||||
5. | PRINT FREQUENCY | 0 | 100 | ||
6. | SUMMARY FREQUENCY | 0 | 100 | ||
7. | SOLUTION YES/NO | 0 | 1 | 1 | 1 = YES; 0 = NO |
8. | SUPPRESS PARAMETERS | 0 | 0 | 1 | 1 = True |
Convergence Tolerances | |||||
9. | ROW TOLERANCE | > 0 | 1E-6 | ||
10. | OPTIMALITY TOLERANCE | > 0 | max(1E - 6, (10eps_{R} )^{0.5} ) = 1.73E-6 | ||
11. | FEASIBILITY TOLERANCE | > 0 | 1E-6 | ||
Derivative checking | |||||
13. | VERIFY LEVEL | -1 | -1 | 3 | -1,0,1,2,3 |
14. | START OBJECTIVE CHECK AT COL | 0 | 1 | nnObj | |
15. | STOP OBJECTIVE CHECK AT COL | 0 | nnObj | nnObj | |
16. | START CONSTRAINT CHECK AT COL | 0 | 1 | nnJac | |
17. | STOP CONSTRAINT CHECK AT COL | 0 | nnJac | nnJac | |
Scaling | |||||
18. | SCALE OPTION
See Section 6.2 for more information. |
0 | 1 or 2 | 2 | 2 if LP,1 if NLP |
19. | SCALE TOLERANCE | > 0 | 0.9 | < 1 | |
20. | SCALE PRINT | 0 | 0 | 1 | 1 = True |
21. | CRASH TOLERANCE | 0 | 0.1 | < 1 | |
Other Tolerances | |||||
22. | LINESEARCH TOLERANCE | > 0 | 0.1 | < 1 | |
LU I | |||||
23. | LU FACTORIZATION TOLERANCE | 1 | 100 or 5 | 100 if LP | |
24. | LU UPDATE TOLERANCE | 1 | 10 or 5 | 10 if LP | |
25. | LU SWAP TOLERANCE | > 0 | 1.22E-4 | eps^^{}1/4 | |
26. | LU SINGULARITY TOLERANCE | > 0 | 3.25E-11 | eps^^{}0.67 | |
LP or LC subproblems | |||||
27. | PIVOT TOLERANCE | > 0 | 3.25E-11 | eps^^{}0.67 | |
28. | CRASH OPTION | 0 | 3 | 3 | 0,1,2,3 |
29. | WEIGHT ON LINEAR OBJECTIVE | 0.0 | 0.0 | during Phase 1 | |
30. | ITERATIONS LIMIT
m3=1 if length(Prob.QP.c) > 0, otherwise m3=0. TOMLAB default: max(10000,3(m+m3) + 10nnL). |
0 | 3(m+m3) + 10nnL | ||
SLC method | |||||
32. | MAXIMIZE | 0 | 0 | 1 | 1=maximize |
33. | LAGRANGIAN | 0 | 1 | 1 | 1=YES, 0=NO |
34. | PENALTY PARAMETER | 0.0 | 1.0 | ||
35. | MAJOR ITERATIONS LIMIT | > 0 | 50 | ||
36. | MINOR ITERATIONS LIMIT | > 0 | 40 | ||
37. | MAJOR DAMPING PARAMETER | > 0 | 2.0 | ||
38. | MINOR DAMPING PARAMETER | > 0 | 2.0 | ||
39. | DERIVATIVE LEVEL
Is always set by minosTL dependent on Prob.ConsDiff, Prob.NumDiff. |
0 | 3 | 3 | 0,1,2,3 |
40. | RADIUS OF CONVERGENCE | 0.0 | 0.01 | ||
41. | FUNCTION PRECISION | > 0 | 3.0E-13 | eps^{}0.8 = eps_{R} | |
42. | DIFFERENCE INTERVAL | > 0 | 5.48E-7 | eps^{}0.4 | |
43. | CENTRAL DIFFERENCE INTERVAL | > 0 | 6.69E-5 | eps^{}0.8/3 | |
44. | COMPLETION | 0 | 1 LC, 0 1
NC |
0=PARTIAL
1=FULL | |
45. | UNBOUNDED STEP SIZE | > 0 | 1E10 | ||
46. | UNBOUNDED OBJECTIVE | > 0 | 1E20 | ||
Hessian approximation | |||||
47. | HESSIAN DIMENSION | 1 | 50 | 1+nnL | |
48. | SUPERBASICS LIMIT | 1 | 50 | 1+nnL
TOMLAB default (to avoid termination with Superbasics Limit too small): If n <= 5000: max(50, n + 1) If n > 5000: max(500, n + 200 - size(A, 1) - length(cL )) Avoid setting REDUCED HESSIAN (number of columns in reduced Hessian). It will then be set to the same value as the SUPERBASICS LIMIT by MINOS. Frequencies II | |
51. | CHECK FREQUENCY | > 0 | 60 | ||
52. | EXPAND FREQUENCY | > 0 | 10000 | ||
53. | FACTORIZATION FREQUENCY | > 0 | 50 | ||
LU II | |||||
63. | LU PARTIAL PIVOTING
or LU COMPLETE PIVOTING or LU ROOK PIVOTING |
0 | 0 | 3 | 0=partial
1=complete 2=rook |
Additional parameters | |||||
67. | AIJ TOLERANCE
Elements |a(i, j)| < AI J TOLERANCE are set as 0 |
0 | 1E-10 | ||
70. | SUBSPACE
Convergence tolerance in current subspace before consider moving off another constraint. |
> 0 | 0.5 | 1 | Subspace tolerance |