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5  TOMLAB /LGO Solver Reference

A detailed description of the TOMLAB /LGO solver interface is given below. Also see the M-file help for lgoTL.m.

5.1  lgoTL

Purpose
Solves global and local (convex and mildly non-convex) constrained nonlinear programming problems.

LGO solves problems of the form where x,x_L,x_U Rⁿ,  A R^{m₁ × n},  b_L,b_U R^m₁ and c(x), c_L, c_U R^m₂.


Calling Syntax
Prob = glcAssign( ... );
Result = tomRun('lgo',Prob,...);


Description of Inputs

Prob	Problem description structure. The following fields are used:
	A	Linear constraints coefficient matrix.
	x_L, x_U	Bounds on variables.
	b_L, b_U	Bounds on linear constraints.
	c_L, c_U	Bounds on nonlinear constraints. For equality constraints (or fixed variables), set e.g. b_L(k) == b_U(k).
	PriLevOpt	Print level in MEX interface.
	optParam	Structure with optimization parameters.
	MaxFunc	Maximum number of function evaluations. (Prob.LGO.options.g_maxfct)
	LGO.PrintFile	Name of LGO Print file. Amount and type of printing determined by PriLevOpt.
	LGO.SummFile	Name of LGO summary file.
	LGO.options	Structure with special fields for LGO optimization parameters. See Table 5.1.

Description of Outputs

Result	Structure with result from optimization. The following fields are set:
	f_k	Function value at optimum.
	x_k	Solution vector.
	x_0	Initial solution vector.
	c_k	Nonlinear constraint residuals.
	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;
	Ax	Values of linear constraints.
	ExitFlag	Exit status from LGO (TOMLAB standard).
	ExitText	Exit text from LGO.
	Inform	LGO information parameter.
		1 = Normal completion.
		2 = Iteration interrupt.
		3 = Time limit exceeded.
		4 = Terminated by solver.
		7 = Size limitation.
		Other = Optimal solution found.
	FuncEv	Number of function evaluations.
	ConstrEv	Number of constraint evaluations. In the context of LGO, ConstrEv = FuncEv.
	QP.B	Basis vector in TOMLAB QP standard.
	Solver	Name of the solver (LGO).
	SolverAlgorithm	Description of the solver.
	LGO	Subfield with LGO specific results.
	sstat	Solver status.
	mstat	Model status.
	runtime	Time spent, measured by LGO solver.

The following table shows all the options that the user can set for the solver. The TOMLAB /LGO options are divided into two main categories:

• General options
• Limits and tolerances

Description of Prob.LGO.options

User options for the TOMLAB /LGO solver. The following fields are used:
Option	Description	Default
General options
opmode	Specifies the algorithm to be used.	3
0	Local search from the given nominal solution without a preceding local search (LS)
1	Global branch-and-bound search and local search (BB+LS)
2	Global adaptive random search and local search (GARS+LS)
3	Global multistart random search and local search (MS+LS)
	Note that an option of 0 will work for many slightly nonconvex, as well as convex models. See note below this table.
tlimit	Time limit in seconds.	600
Limits and Tolerances
g_maxfct	Maximum number of merit function evaluations before termination of global search phase (BB, GARS, or MS). The value of -1 uses 500(nvars+ncons), where nvars is the number of variables and ncons the number of constraints. The difficulty of global optimization models varies greatly: for difficult models, g_maxfct can be increased to as needed.	-1
max_nosuc	Maximum number of merit function evaluations in global search phase (BB, GARS, or MS) where no improvement is made. Global search terminates upon reaching this limit. The value of -1 uses 100(nvars+ncons), where nvars is the number of variables and ncons the number of constraints. For difficult models, this value can also be increased as deemed necessary.	-1
penmult	Constraint penalty multiplier. Global search phase merit function is defined as objective + the violated constraints weighted by penmult.	100
acc_tr	Global search termination criterion parameter (acceptability threshold). The global search phase (BB, GARS, or MS) ends, if an overall merit function value found in the global search phase is less than acc_tr. If a good estimate is known, then its usage may results in a considerably faster search.	-1.0E+10
fct_trg	Target merit function value in local search phase.	-1.0E+10
fi_tol	Local search merit function improvement tolerance.	1.0E-06
con_tol	Maximal admissible constraint violation in local search.	1.0E-06
kt_tol	Kuhn-Tucker local optimality condition tolerance.	1.0E-06
irngs	Random number seed. An arbitrary integer value can be used instead of the default.	0

Note that the local search operational mode (Opmode 0) is the fastest, and that it will work for convex, as well as for many mildly non-convex models. If the model has a highly non-convex (multiextremal) structure, then at least one of the global search modes should be used. In difficult or complex models, it may be a good idea to apply all three global search modes, to verify the global solution, or perhaps to find alternative good solutions. Usually, Opmode 3 is the safest (and slowest), since it applies several local searches; Opmodes 1 and 2 launch only a single local search from the best point found in the global search phase.
Note that if model-specific information is known (more sensible target objective/merit function value, tolerances, tighter variable bounds), then such information should always be used, since it may help to solve the model far more efficiently than by using 'blind' defaults.

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