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3  TOMVIEW /SOL Solver Reference

The SOL solvers are a set of Fortran solvers that were developed by the Stanford Systems Optimization Laboratory (SOL). Table 1 lists the solvers included in TOMVIEW /SOL. The solvers are called using a solver VI developed as part of TOMVIEW. All functionality of the SOL solvers are available and changeable in the TOMVIEW framework in LabVIEW.

Detailed descriptions of the TOMVIEW /SOL solvers are given in the following sections.

The solvers reference guides for the TOMVIEW /SOL solvers are available for download from the TOMVIEW home page http://tomopt.com/tomview/. There is also detailed instruction for using the solvers in Section 4.

TOMVIEW /SOL solves nonlinear optimization problems (con) defined as
 
min
x
f(x)
   
s/t
xL ≤ x ≤ xU,
bL ≤ A x ≤ bU
cL ≤ c(x) ≤ cU
    (1)
where x, xL, xU ∈ Rn, f(x) ∈ R, A ∈ Rm1 × n, bL,bU ∈ Rm1 and cL,c(x),cU ∈ Rm2.

quadratic programming (qp) problems defined as
 
min
x
f(x) =
1
2
xT F x + cT x
   
s/t
xL ≤ x ≤ xU,
bL ≤ A x ≤ bU
    (2)
where c, x, xL, xU ∈ Rn, F ∈ Rn × n, A ∈ Rm1 × n, and bL,bU ∈ Rm1.

linear programming (lp) problems defined as
 
min
x
f(x) = cT x
   
s/t
xL ≤ x ≤ xU,
bL ≤ A x ≤ bU
    (3)
where c, x, xL, xU ∈ Rn, A ∈ Rm1 × n, and bL,bU ∈ Rm1.

linear least squares (lls) problems defined as
 
min
x
f(x) =
1
2
|| C x − d ||
   
s/t
xL ≤ x ≤ xU,
bL ≤ A x ≤ bU
    (4)
where x, xL, xU ∈ Rn, d ∈ RM, C ∈ RM × n, A ∈ Rm1 × n, bL,bU ∈ Rm1.

and constrained nonlinear least squares problems defined as
 
min
x
f(x) =
1
2
r(x)T r(x)
   
s/t
xL ≤ x ≤ xU,
bL ≤ A x ≤ bU
cL ≤ c(x) ≤ cU
    (5)
where x, xL, xU ∈ Rn, r(x) ∈ RM, A ∈ Rm1 × n, bL,bU ∈ Rm1 and cL,c(x),cU ∈ Rm2.



Table 1: The SOL optimization solvers in TOMVIEW /SOL.
Function Description Reference
MINOS 5.5 Sparse linear and nonlinear programming with linear and nonlinear constraints. [27]
QPOPT 1.0-10 Non-convex quadratic programming with dense constraint matrix and sparse or dense quadratic matrix. [16]
LSSOL 1.05-4 Dense linear and quadratic programs (convex), and constrained linear least squares problems. [15]
NLSSOL 5.0-2 Constrained nonlinear least squares. NLSSOL is based on NPSOL. No reference except for general NPSOL [20]
NPSOL 5.02 Dense linear and nonlinear programming with linear and nonlinear constraints. [20]
SNOPT 7.1-1 Large, sparse linear and nonlinear programming with linear and nonlinear constraints. [19, 17]
SQOPT 7.1-1 Sparse convex quadratic programming. [18]


3.1  MINOS

3.1.1  Reference

Purpose
minos solves nonlinear optimization problems defined as

 
min
x
f(x)
   
s/t
xL ≤ x ≤ xU,
bL ≤ A x ≤ bU
cL ≤ c(x) ≤ cU
    (6)


where x, xL, xU ∈ Rn, f(x) ∈ R, A ∈ Rm1 × n, bL,bU ∈ Rm1 and cL,c(x),cU ∈ Rm2.


Calling Syntax
Using the driver routine tomRun:

assign_*.vi
tomRun.vi



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.
 
 
 

Description of Outputs
Result, The following fields are used:
 
 
Result The structure with results.
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.
 
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 − ( A I )Tπ, 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 −π. 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 TOMVIEW QP standard.
 
MinorIter Number of minor iterations.
 
Solver Name of the solver (minos).
SolverAlgorithm Description of the solver.
 

3.1.2  Options

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 Blocks and Parameters

The following fields are used:
SPECS keyword text Lower Default Upper Comment
 
 
Printing
 
PRINT FILE File name for printing
SUMM FILE File name for summary file
SPECS FILE File to read options from
 
 
Print Levels
 
PRINT LEVEL 0 0 11111 JFLXB: Jac, fCon, lambda, x, B=LU stats
PRINT FREQUENCY 0 100  
SUMMARY FREQUENCY 0 100  
SOLUTION 0 1 1 1 = YES; 0 = NO
 
 
SLC Method 1
 
ROW TOLERANCE >0 1E-6
OPTIMALITY TOLERANCE >0 max(1E−6,(10epsR)0.5) = 1.73E-6
FEASIBILITY TOLERANCE >0 1E-6
LINESEARCH TOLERANCE >0 0.1 <1
MAXIMIZE 0 0 1 1=maximize
LAGRANGIAN 0 1 1 1=YES, 0=NO
PENALTY PARAMETER 0.0 1.0
MAJOR ITERATIONS LIMIT >0 50
MINOR ITERATIONS LIMIT >0 40
DERIVATIVE LEVEL 0 3 3 0,1,2,3
Is always set by minos dependent on Prob.ConsDiff, Prob.NumDiff.
 
RADIUS OF CONVERGENCE 0.0 0.01
FUNCTION PRECISION >0 3.0E-13   eps0.8=epsR
 
 
SLC Method 2
 
DIFFERENCE INTERVAL >0 5.48E-7   eps0.4
CENTRAL DIFFERENCE INTERVAL >0 6.69E-5   eps0.8/3
COMPLETION 0 1 LC,0 NC 1 0=PARTIAL 1=FULL
UNBOUNDED STEP SIZE >0 1E10
UNBOUNDED OBJECTIVE >0 1E20
SUPERBASICS LIMIT 1 50 1+nnL
TOMVIEW 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.
 
HESSIAN DIMENSION 1 50 1+nnL
 
 
LP Subproblem
 
PIVOT TOLERANCE >0 3.25E-11   eps0.67
CRASH OPTION 0 3 3 0,1,2,3
WEIGHT ON LINEAR OBJECTIVE 0.0 0.0   during Phase 1
ITERATIONS LIMIT 0 3(m+m3) + 10nnL
m3=1 if length(Prob.QP.c) > 0, otherwise m3=0.
TOMVIEW default: max(10000,3(m+m3) + 10nnL).
 
PARTIAL PRICE 1 10 or 1   10 for LP
 
 
LU Options
 
LU FACTORIZATION TOLERANCE 1 100 or 5   100 if LP
LU UPDATE TOLERANCE 1 10 or 5   10 if LP
LU SWAP TOLERANCE >0 1.22E-4   eps1/4
LU SINGULARITY TOLERANCE >0 3.25E-11   eps0.67
LU PARTIAL PIVOTING 0 0 3 0=partial
or LU COMPLETE PIVOTING       1=complete
or LU ROOK PIVOTING       2=rook
 
 
Other options
 
CHECK FREQUENCY >0 60
EXPAND FREQUENCY >0 10000
FACTORIZATION FREQUENCY >0 50
AIJ TOLERANCE 0 1E-10
Elements |a(i,j)| < AIJ TOLERANCE are set as 0
 
SUBSPACE >0 0.5 1 Subspace tolerance
Convergence tolerance in current subspace before consider moving off
another constraint.
 
VERIFY LEVEL -1 -1 3 -1,0,1,2,3
 
 

3.2  QPOPT

3.2.1  Reference

Purpose
qpopt solves quadratic optimization problems defined as

 
min
x
f(x) =
1
2
xT F x + cT x
   
s/t
xL ≤ x ≤ xU,
bL ≤ A x ≤ bU
    (7)
where c, x, xL, xU ∈ Rn, F ∈ Rn × n, A ∈ Rm1 × n, and bL,bU ∈ Rm1.


Calling Syntax
Using the driver routine tomRun:

assign_qp.vi
tomRun.vi



Description of Inputs
Prob, The following fields are used:
 
x_L, x_U Bounds on variables.
 
b_L, b_U Bounds on linear constraints.
 
A Linear constraint matrix.
 
QP.c Linear coefficients in objective function.
 
QP.F Quadratic matrix of size nnObj x nnObj. nnObj < n is OK.
 
 
 

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 Exact gradient computed at optimum.
 
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;
 
v_k Lagrangian multipliers (for bounds + dual solution vector).
 
ExitFlag Exit status from qpopt.
 
Inform Result of QPOPT run. 0 = Optimal solution found.
 
  0: x is a unique local minimizer. This means that x is feasible (it satisfies the constraints to the accuracy requested by the Feasibility tolerance), the reduced gradient is negligible, the Lagrange multipliers are optimal, and the reduced Hessian is positive definite. If H is positive definite or positive semidefinite, x is a global minimizer. (All other feasible points give a higher objective value.) Otherwise, the solution is a local minimizer, which may or may not be global. (All other points in the immediate neighborhood give a higher objective.)
 
  1: A dead-point was reached. This might occur if H is not sufficiently positive definite. If H is positive semidefinite, the solution is a weak minimizer. (The objective value is a global optimum, but there may be infinitely many neighboring points with the same objective value.) If H is indefinite, a feasible direction of decrease may or may not exist (so the point may not be a local or weak minimizer).
 
  At a dead-point, the necessary conditions for optimality are satisfied (x is feasible, the reduced gradient is negligible, the Lagrange multipliers are optimal, and the reduced Hessian is positive semidefinite.) However, the reduced Hessian is nearly singular, and/or there are some very small multipliers. If H is indefinite, x is not necessarily a local solution of the problem. Verification of optimality requires further information, and is in general an NP-hard problem [39].
 
  2: The solution appears to be unbounded. The objective is not bounded below in the feasible region, if the elements of x are allowed to be arbitrarily large. This occurs if a step larger than Infinite Step would have to be taken in order to continue the algorithm, or the next step would result in a component of x having magnitude larger than Infinite Bound. It should not occur if H is sufficiently positive definite.
 
  3: The constraints could not be satisfied. The problem has no feasible solution.
 
  4: One of the iteration limits was reached before normal termination occurred. See Feasibility Phase Iterations and Optimality Phase Iterations.
 
  5: The Maximum degrees of freedom is too small. The reduced Hessian must expand if further progress is to be made.
 
  6: An input parameter was invalid.
 
  7: The Problem type was not recognized.
 
rc Reduced costs. If ninf=0, last m == -v_k.
 
Iter Number of iterations.
FuncEv Number of function evaluations. Set to Iter.
GradEv Number of gradient evaluations. Set to Iter.
ConstrEv Number of constraint evaluations. Set to 0.
 
QP.B Basis vector in TOMVIEW QP standard.
 
MinorIter Number of minor iterations. Not Set.
 
Solver Name of the solver (QPOPT).
SolverAlgorithm Description of the solver.
 

3.2.2  Options

Description
Use missing value (-999 or less), when no change of parameter setting is wanted. The default value will then be used by QPOPT, unless the value is altered in the SPECS file.


Description of Blocks and Parameters

The following fields are used:
SPECS keyword text Lower Default Upper Comment
 
 
Printing
 
PRINT FILE File name for printing
SUMM FILE File name for summary file
SPECS FILE File to read options from
 
 
Print Levels
 
PRINT LEVEL 0 10   0,1,5,10,20,30
 
 
Convergence Tolerances
 
OPTIMALITY TOLERANCE >0 1.1E-8   sqrt(eps)
FEASIBILITY TOLERANCE >0 1.1E-8   sqrt(eps)
 
 
Other options
 
CRASH TOLERANCE 0 0.01 <1
RANK TOLERANCE >0 1.1E-14   100*eps
ITERATION LIMIT >0 max(50,5(n+m))
MIN SUM YES (or NO) 0 0 1 1=min infeas.
IF 1 (MIN SUM YES), minimize the infeasibilities before return.
 
FEASIBILITY PHASE ITERATIONS >0 max(50,5(n+m))
INFINITE STEP SIZE >0 1E20
HESSIAN ROWS 0 n n 0 if FP or LP
Implicitly given by the dimensions of H in the call from LabVIEW
 
MAX DEGREES OF FREEDOM 0 n n
ONLY USED IF HESSIAN ROWS == N
CHECK FREQUENCY >0 50
EXPAND FREQUENCY >0 5
 
 

3.3  LSSOL

3.3.1  Reference

Purpose
lssol solves least squares optimization problems defined as

 
min
x
f(x) =
1
2
|| C x − d ||
   
s/t
xL ≤ x ≤ xU,
bL ≤ A x ≤ bU
    (8)
where x, xL, xU ∈ Rn, d ∈ RM, C ∈ RM × n, A ∈ Rm1 × n, bL,bU ∈ Rm1.


Calling Syntax
Using the driver routine tomRun:

assign_lls.vi
tomRun.vi



Description of Inputs
Prob, The following fields are used:
 
x_L, x_U Bounds on variables.
 
b_L, b_U Bounds on linear constraints.
 
A Linear constraint matrix.
 
QP.c Linear coefficients in objective function.
 
QP.F Quadratic matrix of size nnObj x nnObj.
 
 
 

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 Exact gradient computed at optimum.
 
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;
 
v_k Lagrangian multipliers (for bounds + dual solution vector).
 
ExitFlag Exit status from lssol.
Inform LSSOL information parameter.
  0 = Optimal solution with unique minimizer found.
  1 = Weak local solution (nonunique) was reached.
  2 = The solution appears to be unbounded.
  3 = The constraints could not be satisfied. The problem has no feasible solution.
  4 = Too many iterations, in either phase.
  5 = 50 changes of working set without change in x, cycling?
  6 = An input parameter was invalid.
  Other = UNKNOWN LSSOL Inform value.
 
 
rc Reduced costs. If ninf=0, last m == -v_k.
 
Iter Number of iterations.
FuncEv Number of function evaluations. Set to Iter.
GradEv Number of gradient evaluations. Set to Iter.
ConstrEv Number of constraint evaluations. Set to 0.
 
QP.B Basis vector in TOMVIEW QP standard.
 
MinorIter Number of minor iterations. NOT SET.
 
Solver Name of the solver (LSSOL).
SolverAlgorithm Description of the solver.
 

3.3.2  Options

Description
Use missing value (-999 or less), when no change of parameter setting is wanted. The default value will then be used by LSSOL, unless the value is altered in the SPECS file.


Description of Blocks and Parameters

The following fields are used:
SPECS keyword text Lower Default Upper Comment
 
 
Printing
 
PRINT FILE File name for printing
SUMM FILE File name for summary file
SPECS FILE File to read options from
 
 
Print Levels
 
PRINT LEVEL 0 10   0,1,5,10,20,30
 
 
Convergence Tolerances
 
OPTIMALITY TOLERANCE >0 1.1E-8   sqrt(eps)
FEASIBILITY TOLERANCE >0 1.1E-8   sqrt(eps)
 
 
Other options
 
CRASH TOLERANCE >0 0.01 <1
RANK TOLERANCE >0 1.1E-14   100*eps
ITERATIONS LIMIT >0 max(50,5(n+m))
FEASIBILITY PHASE ITERATIONS >0 max(50,5(n+m))
INFINITE STEP SIZE >0 1E20
INFINITE BOUND SIZE >0 1E20
 
 

3.4  NLSSOL

3.4.1  Reference

Purpose
nlssol solves constrained nonlinear least squares problems defined as

 
min
x
f(x) =
1
2
r(x)T r(x)
   
s/t
xL ≤ x ≤ xU,
bL ≤ A x ≤ bU
cL ≤ c(x) ≤ cU
    (9)
where x, xL, xU ∈ Rn, r(x) ∈ RM, A ∈ Rm1 × n, bL,bU ∈ Rm1 and cL,c(x),cU ∈ Rm2.


Calling Syntax
Using the driver routine tomRun:

assign_cls.vi
tomRun.vi



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.
 
 
 

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.
 
r_k Residual vector.
J_k Jacobian matrix.
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 nlssol.
Inform NLSSOL information parameter.
  0 = Optimal solution found.
  1 = Optimal solution found but not to requested accuracy.
  2 = No feasible point for the linear constraints.
  3 = No feasible point for the nonlinear constraints.
  4 = Too many major iterations.
  5 = Problem is unbounded, or badly scaled.
  6 = The current point cannot be improved on.
  7 = Large errors found in the derivatives.
  9 = An input parameter is invalid.
  Other = User requested termination
 
 
rc Reduced costs. If ninf=0, last m == -v_k.
 
Iter Number of iterations.
FuncEv Number of function evaluations.
GradEv Number of gradient evaluations.
ConstrEv Number of constraint evaluations.
 
QP.B Basis vector in TOMVIEW QP standard.
 
MinorIter Number of minor iterations.
 
Solver Name of the solver (nlssol).
SolverAlgorithm Description of the solver.
 

3.4.2  Options

Description
Use missing value (-999 or less), when no change of parameter setting is wanted. The default value will then be used by NLSSOL, if not the value is altered in the SPECS file (input SpecsFile)


Description of Blocks and Parameters

The following fields are used:
SPECS keyword text Lower Default Upper Comment
 
 
Printing
 
PRINT FILE File name for printing
SUMM FILE File name for summary file
SPECS FILE File to read options from
 
 
Print Levels
 
PRINT LEVEL 0 10   0,1,5,10,20,30
MINOR PRINT LEVEL 0 0   0,1,5,10,20,30
 
 
Convergence Tolerances
 
NONLINEAR FEASIBILITY TOLERANCE >0 1.1E-8   sqrt(eps)
OPTIMALITY TOLERANCE >0 3.0E-13   eps0.8
LINEAR FEASIBILITY TOLERANCE >0 1.1E-8   sqrt(eps)
 
 
Other options 1
 
CRASH TOLERANCE >0 0.01 <1
LINESEARCH TOLERANCE >0 0.9 <1
ITERATIONS LIMIT >0 max(50,3(n+m_L)+10*m_N)
MINOR ITERATIONS LIMIT >0 max(50,3(n+m_L+m_N))
STEP LIMIT >0 2
DERIVATIVE LEVEL 0 3 3 0,1,2,3
FUNCTION PRECISION >0 3.0E-13   eps0.8=epsR
DIFFERENCE INTERVAL >0 5.48E-8   eps0.4
CENTRAL DIFFERENCE INTERVAL >0 6.70E-5   eps0.8/3
INFINITE STEP SIZE >0 max(BIGBND,1E10)
INFINITE BOUND SIZE >0 1E10   = BIGBND
INITIAL HESSIAN (JTJ) 0 1   0 = UNIT
 
 
Other options 2
 
RESET FREQUENCY 0 2    
HESSIAN YES or NO 0 0   1 = YES
VERIFY LEVEL -1 -1 3 -1,0,1,2,3
 
 

3.5  NPSOL

3.5.1  Reference

Purpose
npsol solves dense nonlinear optimization problems defined as

 
min
x
f(x)
   
s/t
xL ≤ x ≤ xU,
bL ≤ A x ≤ bU
cL ≤ c(x) ≤ cU
    (10)


where x, xL, xU ∈ Rn, f(x) ∈ R, A ∈ Rm1 × n, bL,bU ∈ Rm1 and cL,c(x),cU ∈ Rm2.


Calling Syntax
Using the driver routine tomRun:

assign_*.vi
tomRun.vi



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.
 
 
 

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 npsol.
Inform NPSOL information parameter.
  0 = Optimal solution found.
  1 = Optimal solution found but not to requested accuracy.
  2 = No feasible point for the linear constraints.
  3 = No feasible point for the nonlinear constraints.
  4 = Too many major iterations.
  6 = The current point cannot be improved on.
  7 = Large errors found in the derivatives.
  9 = An input parameter is invalid.
  Other = User requested termination
 
rc Reduced costs. If ninf=0, last m == -v_k.
 
Iter Number of iterations.
FuncEv Number of function evaluations.
GradEv Number of gradient evaluations.
ConstrEv Number of constraint evaluations.
 
QP.B Basis vector in TOMVIEW QP standard.
 
MinorIter Number of minor iterations.
 
Solver Name of the solver (npsol).
SolverAlgorithm Description of the solver.
 

3.5.2  Options

Description
Use missing value (-999 or less), when no change of parameter setting is wanted. The default value will then be used by NPSOL, if not the value is altered in the SPECS file (input SpecsFile).


Description of Inputs

The following fields are used:
SPECS keyword text Lower Default Upper Comment
 
 
Printing
 
PRINT FILE File name for printing
SUMM FILE File name for summary file
SPECS FILE File to read options from
 
 
Print Levels
 
PRINT LEVEL 0 10   0,1,5,10,20,30
MINOR PRINT LEVEL 0 0   0,1,5,10,20,30
 
 
Convergence Tolerances
 
NONLINEAR FEASIBILITY TOLERANCE >0 1.1E-8   sqrt(eps)
OPTIMALITY TOLERANCE >0 3.0E-13   eps0.8
LINEAR FEASIBILITY TOLERANCE >0 1.1E-8   sqrt(eps)
 
 
Other options 1
 
CRASH TOLERANCE >0 0.01 <1
Note: Decision variables will be set to the closest bound to the starting point
based on this tolerance before running the optimization.
LINESEARCH TOLERANCE >0 0.9 <1
 
ITERATIONS LIMIT >0 max(50,3(n+m_L)+10*m_N)
MINOR ITERATIONS LIMIT >0 max(50,3(n+m_L+m_N))
STEP LIMIT >0 2
DERIVATIVE LEVEL 0 3 3 0,1,2,3
Is set by npsol dependent on Prob.ConsDiff, Prob.NumDiff
FUNCTION PRECISION >0 3.0E-13   eps0.8=epsR
DIFFERENCE INTERVAL >0 5.48E-8   eps0.4
CENTRAL DIFFERENCE INTERVAL >0 6.70E-5   eps0.8/3
INFINITE STEP SIZE >0 max(BIGBND,1E10)
INFINITE BOUND SIZE >0 1E10   = BIGBND
HESSIAN YES or NO 0 0   1 = YES
 
 
Other options 1
 
VERIFY LEVEL -1 -1 3 -1,0,1,2,3
 

3.6  SNOPT

3.6.1  Reference

Purpose
snopt solves nonlinear optimization problems defined as

 
min
x
f(x)
   
s/t
xL ≤ x ≤ xU,
bL ≤ A x ≤ bU
cL ≤ c(x) ≤ cU
    (11)


where x, xL, xU ∈ Rn, f(x) ∈ R, A ∈ Rm1 × n, bL,bU ∈ Rm1 and cL,c(x),cU ∈ Rm2.


Calling Syntax
Using the driver routine tomRun:

assign_*.vi
tomRun.vi



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.
 
 
 

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 snopt.
Inform Result of SNOPT run.
 
  Finished successfully
  1 optimality conditions satisfied
  2 feasible point found
  3 requested accuracy could not be achieved
 
  The problem appears to be infeasible
  11 infeasible linear constraints
  12 infeasible linear equalities
  13 nonlinear infeasibilities minimized
  14 infeasibilities minimized
 
  The problem appears to be unbounded
  21 unbounded objective
  22 constraint violation limit reached
 
  Resource limit error
  31 iteration limit reached
  32 major iteration limit reached
  33 the superbasics limit is too small
 
  Terminated after numerical difficulties
  41 current point cannot be improved
  42 singular basis
  43 cannot satisfy the general constraints
  44 ill-conditioned null-space basis
 
  Error in the user-supplied functions
  51 incorrect objective derivatives
  52 incorrect constraint derivatives
 
  Undefined user-supplied functions
  61 undefined function at the first feasible point
  62 undefined function at the initial point
  63 unable to proceed into undefined region
 
  User requested termination
  72 terminated during constraint evaluation
  73 terminated during objective evaluation
  74 terminated from monitor routine
 
  Insufficient storage allocated
  81 work arrays must have at least 500 elements
  82 not enough character storage
  83 not enough integer storage
  84 not enough real storage
 
  Input arguments out of range
  91 invalid input argument
  92 basis file dimensions do not match this problem
 
  System error
  141 wrong number of basic variables
  142 error in basis package
 
rc Vector of reduced costs, g − (  A −I  )Tπ, where g is the gradient of the objective if xs is feasible (or the gradient of the Phase-1 objective otherwise). The last m entries are π. The vector is n+m. If nInf=0, last m == pi.
 
Iter Number of iterations.
FuncEv Number of function evaluations.
GradEv Number of gradient evaluations.
ConstrEv Number of constraint evaluations.
 
QP.B Basis vector in TOMVIEW QP standard.
 
MinorIter Number of minor iterations.
 
Solver Name of the solver (snopt).
SolverAlgorithm Description of the solver.
 

3.6.2  Options

Description
Use missing value (-999 or less), when no change of parameter setting is wanted. The default value will then be used by SNOPT, if not the value is altered in the SPECS file (input SpecsFile).
Definition: nnL = max(nnObj,nnJac)).

Description of Inputs
The following fields are used:
SPECS keyword text Lower Default Upper Comment
 
 
Printing
 
PRINT FILE File name for printing
SUMM FILE File name for summary file
SPECS FILE File to read options from
 
 
Print Levels
 
MAJOR PRINT LEVEL 0 1 11111
MINOR PRINT LEVEL 0 1 10 0, 1 or 10
PRINT FREQUENCY 0 100
SUMMARY FREQUENCY 0 100
SOLUTION YES/NO 0 1 1 1 = YES; 0 = NO
SUPPRESS OPTIONS LISTING 0 0 1 1 = True
 
 
SQP Method 1
 
MAJOR FEASIBILITY TOLERANCE >0 1E-6
MAJOR OPTIMALITY TOLERANCE >0 max(2E−6,(10epsR)0.5) = 1.73E-6
eps_R == optPar(41), Default relative function precision eps_R gives (10*epsR)0.5=1.73E−6.
LINESEARCH TOLERANCE >0 0.9 <1
MAXIMIZE 0 0 1 1=maximize
FEASIBLE POINT 0 0 1 1=feasible pnt
VIOLATION LIMIT >0 1e6
MAJOR ITERATIONS LIMIT >0 max(1000,3*max(n,m))
Maximal number of major iterations
 
MINOR ITERATIONS LIMIT >0 500
Maximal number of minor iterations, i.e. in the solution of QP or simplex
 
STEP LIMIT >0 2
DERIVATIVE LEVEL 0 3 3 0,1,2,3
DERIVATIVE LINESEARCH 0 1 1 0=NONDERIVATIVE
0 is quadratic - gives quadratic, without gradient values
1 is cubic - gives cubic, always using gradient values
Default: 0 if numerical derivatives, otherwise 1
 
FUNCTION PRECISION >0 3.0E-13   eps0.8=epsR
 
 
SQP Method 2
 
DIFFERENCE INTERVAL >0 5.48E-7   eps0.4
CENTRAL DIFFERENCE INTERVAL >0 6.70E-5   eps0.8/3
PROXIMAL POINT METHOD 1 1 2 1,2
Minimize the 1-norm (or 2-norm) of ||(x−x0)|| to find an initial point
that is feasible subject to simple bounds and linear constraints.
UNBOUNDED STEP SIZE >0 1E20
UNBOUNDED OBJECTIVE >0 1E15
SUPERBASICS LIMIT >0 max(500,n+1)
TOMVIEW extension (to avoid termination with Superbasics Limit too small):
Set =n+1 if n−size(A,1)−length(cL) > 450 and n <= 5000
If n > 5000: max(500,n−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 SNOPT.
 
PENALTY PARAMETER >=0 0.0
Initial penalty parameter.
 
HESSIAN DIMENSION >0 min(2000,nnL+1)   =1 if LP problem (n upper limit)
also called REDUCED HESSIAN. Number of columns in Reduced Hessian.
 
 
QP Sub problem 1
 
MINOR FEASIBILITY TOLERANCE >0 1E-6
Feasibility tolerance on linear constraints.
MINOR OPTIMALITY TOLERANCE >0 1E-6
SCALE OPTION 0 0 or 2 2 2 if LP,0 if NLP
SCALE TOLERANCE >0 0.9 <1
SCALE PRINT 0 0 1 1 = True
CRASH TOLERANCE 0 0.1 <1
PIVOT TOLERANCE >0 3.25E-11   eps(0.67)
CRASH OPTION 0 3 3 0,1,2,3
ELASTIC WEIGHT 0 10000.0
ITERATIONS LIMIT 0 10000   or 20m, if more
Maximal sum of minor iterations
PARTIAL PRICE 0 10 or 1   10 for LP
NEW SUPERBASICS >0 99
Also MINOR SUPERBASICS. Maximal number of new superbasics per major iteration.
 
 
QP Sub problem 2
 
QPSOLVER CHOLESKY 0 0 2 0=Cholesky
or QPSOLVER CG       1=CG
or QPSOLVER QN       2=Quasi-Newton CG
CG TOLERANCE >0 1e−2
CG ITERATIONS >0 100   Max number of CG iters
QG PRECONDITIONING 0 0 1 QN preconditioned CG
Default 1 if QPSOLVER QN.
SUBSPACE 0 0.1 1 Subspace tolerance
Quasi-Newton QP rg tolerance.
 
 
LU options
 
LU FACTORIZATION TOLERANCE 1 100/3.99   100 if LP
LU UPDATE TOLERANCE 1 10/3.99   10 if LP
LU SWAP TOLERANCE >0 1.22E-4   eps(1/4)
LU SINGULARITY TOLERANCE >0 3.25E-11   eps0.67
LU PARTIAL PIVOTING 0 0 3 0=partial
or LU COMPLETE PIVOTING       1=complete
or LU ROOK PIVOTING       2=rook
or LU DIAGONAL PIVOTING       3=diagonal
 
 
Other options
 
HESSIAN FREQUENCY >0 99999999
HESSIAN FULL MEMORY 0 1 1 =1 if nnL <= 75
or HESSIAN LIMITED MEMORY =0 if nnL > 75
HESSIAN UPDATES >0 20
Maximum number of QN (Quasi-Newton) updates.
If HESSIAN FULL MEMORY, default is 99999999, otherwise 20.
HESSIAN FLUSH >0 99999999
CHECK FREQUENCY >0 60
EXPAND FREQUENCY >0 10000
FACTORIZATION FREQUENCY >0 50
VERIFY LEVEL -1 -1 3 -1,0,1,2,3
 
 

3.7  SQOPT

3.7.1  Reference

Purpose
sqopt solves nonlinear optimization problems defined as

 
min
x
f(x) =
1
2
xT F x + cT x
   
s/t
xL ≤ x ≤ xU,
bL ≤ A x ≤ bU
    (12)
where c, x, xL, xU ∈ Rn, F ∈ Rn × n, A ∈ Rm1 × n, and bL,bU ∈ Rm1.


Calling Syntax
Using the driver routine tomRun:

assign_qp.vi
tomRun.vi



Description of Inputs
Prob, The following fields are used:
 
x_L, x_U Bounds on variables.
 
b_L, b_U Bounds on linear constraints.
 
A Linear constraint matrix.
 
QP.c Linear coefficients in objective function.
 
QP.F Quadratic matrix of size nnObj x nnObj. nnObj < n is OK.
 
 

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.
 
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;
 
v_k Lagrangian multipliers (for bounds + dual solution vector).
 
ExitFlag Exit status from sqopt.
 
Inform Result of SQOPT run.
 
  0: Optimal solution found. (The reduced gradients are optimal, and xs satisfies the constraints to the accuracy requested.
 
  1: The problem is infeasible.
 
  2: The problem is unbounded (or badly scaled).
 
  3: Too many iterations.
 
  4: The QP Hessian H appears to be indefinite (the QP is non-convex).
 
  5: The Superbasics limit is too small.
 
  6: A weak solution has been found—i.e., the solution is not unique.
 
  10: Numerical error in trying to satisfy the constraints Ax − s=0. The basis is very ill-conditioned.
 
  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.
 
  42: Not enough 8-character workspace to solve the problem.
 
  43: Not enough integer workspace to solve the problem.
 
  44: Not enough real workspace to solve the problem.
 
rc A vector of reduced costs, g − (  A −I  )Tπ, where g is the gradient of the objective if xs is feasible (or the gradient of the Phase-1 objective otherwise). The last m entries are π.
 
Iter Number of iterations.
FuncEv Number of function evaluations. Set to Iter.
GradEv Number of gradient evaluations. Set to Iter.
ConstrEv Number of constraint evaluations. Set to 0.
 
QP.B Basis vector in TOMVIEW QP standard.
 
Solver Name of the solver (sqopt).
SolverAlgorithm Description of the solver.
 

3.7.2  Options

Description
Use missing value (-999 or less), when no change of parameter setting is wanted. The default value will then be used by SQOPT, unless the value is altered in the SPECS file (input SpecsFile).



Description of Inputs

The following fields are used:
SPECS keyword text Lower Default Upper Comment
 
 
Printing
 
PRINT FILE File name for printing
SUMM FILE File name for summary file
SPECS FILE File to read options from
 
 
Print Levels
 
PRINT LEVEL 0 0 10 0, 1 or 10
PRINT FREQUENCY 0 100
SUMMARY FREQUENCY 0 100
SOLUTION YES/NO 0 1 1 1 = YES; 0 = NO
SUPPRESS OPTIONS LISTING 0 0 1 1 = True
 
 
QP Parameters 1
 
FEASIBILITY TOLERANCE >0 1E-6
OPTIMALITY TOLERANCE >0 1E-6
SCALE OPTION 0 2 2
SCALE TOLERANCE >0 0.9 <1
SCALE PRINT 0 0 1 1 = True
CRASH TOLERANCE 0 0.1 <1
PIVOT TOLERANCE >0 3.25E-11   eps0.67
CRASH OPTION 0 0 3 0,1,2,3
ELASTIC WEIGHT 0 1
ITERATIONS LIMIT 0 10000
PARTIAL PRICE 1 10
MAXIMIZE 0 0 1 1=maximize
 
 
QP Parameters 2
 
UNBOUNDED STEP SIZE >0 1E20
SUPERBASICS LIMIT >0 min(500,1+nnObj)
ELASTIC MODE 0 1   0,1,2
ELASTIC OBJECTIVE 0 2   0,1,2
 
 
LU options
 
LU FACTORIZATION TOLERANCE 1 100
LU UPDATE TOLERANCE 1 10
LU SWAP TOLERANCE >0 1.22E-4   eps1/4
LU SINGULARITY TOLERANCE >0 3.25E-11   eps0.67
LU COMPLETE PIVOTING or LU PARTIAL PIVOTING 0 0 1 1=complete, 0=partial
 
 
Other options
 
CHECK FREQUENCY >0 60
EXPAND FREQUENCY >0 10000
FACTORIZATION FREQUENCY >0 50
 
 

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