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3  TOMVIEW /SNOPT Solver Reference

The SNOPT 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 /SNOPT. The solvers are called using a solver VI developed as part of TOMVIEW. All functionality of the SNOPT solvers are available and changeable in the TOMVIEW framework in LabVIEW.

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

The solvers reference guides for the TOMVIEW /SNOPT 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 /SNOPT solves nonlinear optimization problems (con) defined as where x, x_L, x_U Rⁿ, f(x) R, A R^{m₁ × n}, b_L,b_U R^m₁ and c_L,c(x),c_U R^m₂.

quadratic programming (qp) problems defined as where c, x, x_L, x_U Rⁿ, F R^{n × n}, A R^{m₁ × n}, and b_L,b_U R^m₁.

linear programming (lp) problems defined as where c, x, x_L, x_U Rⁿ, A R^{m₁ × n}, and b_L,b_U R^m₁.

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Table 1: The SOL optimization solvers in TOMVIEW /SNOPT.

Function	Description	Reference
MINOS 5.5	Sparse linear and nonlinear programming with linear and nonlinear constraints.	[25]
LP-MINOS	A special version of the MINOS 5.5 MEX-file interface for sparse linear programming.	[25]
QP-MINOS	A special version of the MINOS 5.5 MEX-file interface for sparse quadratic programming.	[25]
LPOPT 1.0-10	Dense linear programming.	[15]
QPOPT 1.0-10	Non-convex quadratic programming with dense constraint matrix and sparse or dense quadratic matrix.	[15]
SNOPT 6.1-1	Large, sparse linear and nonlinear programming with linear and nonlinear constraints.	[18, 16]
SQOPT 6.1-1	Sparse convex quadratic programming.	[17]

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3.1  MINOS

3.1.1  Reference

Purpose
minos solves nonlinear optimization problems defined as



where x, x_L, x_U Rⁿ, f(x) R, A R^{m₁ × n}, b_L,b_U R^m₁ and c_L,c(x),c_U R^m₂.


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

where c, x, x_L, x_U Rⁿ, F R^{n × n}, A R^{m₁ × n}, and b_L,b_U R^m₁.


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 [37].
	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  SNOPT

3.3.1  Reference

Purpose
snopt solves nonlinear optimization problems defined as



where x, x_L, x_U Rⁿ, f(x) R, A R^{m₁ × n}, b_L,b_U R^m₁ and c_L,c(x),c_U R^m₂.


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.

3.3.2  Options

Description
Use missing value (-999 or less), when no change of parameter setting is wanted. The default value will then be used by 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

3.4  SQOPT

3.4.1  Reference

Purpose
sqopt solves nonlinear optimization problems defined as

where c, x, x_L, x_U Rⁿ, F R^{n × n}, A R^{m₁ × n}, and b_L,b_U R^m₁.


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.

3.4.2  Options

Description
Use missing value (-999 or less), when no change of parameter setting is wanted. The default value will then be used by 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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