LGO

János D. Pintér, Pintér Consulting Services, Inc. Halifax, NS, Canada, B3M 1J2 jdpin.nosp@m.ter@.nosp@m.hfx.e.nosp@m.astl.nosp@m.ink.c.nosp@m.a, http://www.pinterconsulting.com

# Introduction

## The LGO Solver Suite

The Lipschitz-Continuous Global Optimizer 1) (LGO) serves for the analysis and global solution of general nonlinear programming (NLP) models. The LGO solver system has been developed and gradually extended for more than a decade and it now incorporates a suite of robust and efficient global and local nonlinear solvers. It can also handle small LP models.

GAMS/LGO can be used in several search modes, providing a robust, effective, and flexible solver suite approach to a broad range of nonlinear models. The solver suite approach increases the reliability of the overall solution process. GAMS/LGO integrates the following global scope algorithms:

• Branch-and-bound (adaptive partition and sampling) based global search (BB)
• Adaptive global random search (GARS)
• Adaptive multistart global random search (MS)

LGO also includes the following local solver strategies:

• Heuristic global scope scatter search method (HSS)
• Bound-constrained local search, based on the use of an exact penalty function (EPM)
• Constrained local search, based on sequential model linearization (SLP)
• Constrained local search, based on a generalized reduced gradient approach (GRG).

The overall solution approach followed by GAMS/LGO is based on the seamless combination of the global and local search strategies. This allows for a broad range of operations. In particular, a solver suite approach supports the flexible usage of the component solvers: one can execute fully automatic (global and/or local search based) optimization, and can design customized interactive runs.

GAMS/LGO does not rely on any sub-solvers, and it does not require any structural information about the model. It is particularly suited to solve even 'black box' (closed, confidential), or other complex models, in which the available analytical information may be limited. GAMS/LGO needs only computable function values (without a need for higher order analytical information). GAMS/LGO can even solve models having constraints involving continuous, but non-differentiable functions. Thus, within GAMS, LGO is well suited to solve DNLP models.

GAMS/LGO can also be used in conjunction with other GAMS solvers. For instance, the local solver CONOPT can be used after LGO is finished to verify the solution and/or to provide additional information such as marginal values. To call CONOPT, the user can specify the LGO solver option 'callConopt'. See the LGO Options section for details.

The LGO solver suite has been successfully applied to complex, large-scale models both in educational/research and commercial contexts for over a decade. Possible application areas include advanced engineering design, econometrics and finance, medical research and biotechnology, chemical and process industries, and scientific modeling. Tractable model sizes depend only on the available hardware, although LGO has a 3000 variable, 2000 constraint size limit.

## Running GAMS/LGO

GAMS/LGO is capable of solving the following model types: LP, RMIP, NLP, and DNLP. If LGO is not specified as the default solver for these models, it can be invoked by issuing the following command before the solve statement:

 option (modeltype) = lgo;


where (modeltype) stands for LP, RMIP, NLP, or DNLP.

# LGO Options

GAMS/LGO works like other GAMS solvers, and many options can be set directly within the GAMS model. The most relevant GAMS options are reslim, iterlim, and optfile. For details on these and other options, see the section on GAMS Options.

In addition, LGO-specific options can be specified by using a solver option file. For details on creating and using solver options files, see the section basic option file usage. The options supported by the GAMS/LGO solver are detailed below.

## General LGOoptions

Option Description Default
acc_tr Global search termination - acceptability threshold
Global search termination criterion parameter (acceptability threshold). The global search phase (BB, GARS, or MS) ends, if an overall merit function value is found in the global search phase that is not greater than acc_tr.
Range: [-∞, ∞]
-1e10
con_tol Maximal constraint violation tolerance in local search 1e-6
fct_trg Partial stopping criterion in second local search phase
Target objective function value.
Range: [-∞, ∞]
-1e10
fi_tol Local search (merit function improvement) tolerance 1e-6
g_maxfct Maximum number of function evaluations in global search
Maximum number of merit (model) function evaluations before termination of global search phase (BB, GARS, or MS). In the default setting, n is the number of variables and m is the number of constraints. The difficulty of global optimization models varies greatly: for difficult models, g_maxfct can be increased as deemed necessary.
Range: [-1, ∞]
500(n+m)
irngs Random number seed 0
kt_tol Kuhn-Tucker local optimality condition violation tolerance 1e-6
max_nosuc Maximum number of function evaluations without improvement
Maximum number of merit function evaluations in global search phase (BB, GARS, or MS) where no improvement is made. Algorithm phase terminates upon reaching this limit. A value of -1 resets this to the default of 100(n+m).
Range: [-1, ∞]
100(n+m)
opmode Operational mode
0: LS. Local search from the given nominal solution without a preceding local search (LS)
1: BB + LS. Global branch-and-bound search and local search (BB+LS)
2: RS+LS. Global adaptive random search and local search (GARS+LS)
3: MS+LS. Global multistart random search and local search (MS+LS)
3
penmult Constraint penalty multiplier.
Global merit function is defined as objective + the constraint violations weighted by penmult.
100
tlimit Time limit in seconds.
This is equivalent to the GAMS option reslim. If specified, this overrides the GAMS reslim option.
1000

## Gams system interface only

Option Description Default
bad_obj Default value for objective function, if evaluation errors occur 1e8
CallConopt call CONOPT to get duals for final point
Number of seconds given for cleanup phase using CONOPT. CONOPT terminates after at most CallConopt seconds. The cleanup phase determines duals for final solution point.
5
debug Debug option.
Prints out complete LGO status report to listing file.
0
Include Start reading from a new file
log_err Log first log_err errors
Error reported (if applicable) every log_err iterations.
10
log_iter Iteration log iteration interval.
Log output occurs every log_iter iterations.
10
log_time Iteration log time interval in seconds.
Log output occurs every log_time seconds.
0.5
var_lo Smallest (default) lower bound, unless set by user.
Range: [-∞, ∞]
-1e6
var_up Largest (default) upper bound, unless set by user.
Range: [-∞, ∞]
1e6

Note that the local search operational mode (opmode 0) is the fastest, and that it will work for convex, as well as for some 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. 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 would be the case using the defaults.

# The GAMS/LGO Log File

The GAMS/LGO log file gives much useful information about the current solver progress and its individual phases. To illustrate, we use the nonconvex model mhw4d.gms from the GAMS model library:

   $Title Nonlinear Test Problem (MHW4D,SEQ=84)$Ontext
Another popular testproblem for NLP codes.

Wright, M H, Numerical Methods for Nonlinearly Constrained Optimization.
PhD thesis, Stanford University, 1976.
\$Offtext

Variables m, x1, x2, x3, x4, x5;
Equations funct, eq1, eq2, eq3;

funct.. m =e= sqr(x1-1)      + sqr(x1-x2)     + power(x2-x3,3)
+ power(x3-x4,4) + power(x4-x5,4) ;
eq1.. x1 + sqr(x2) + power(x3,3) =e= 3*sqrt(2) + 2 ;
eq2.. x2 - sqr(x3) + x4          =e= 2*sqrt(2) - 2 ;
eq3.. x1*x5 =e= 2 ;

Model wright / all / ;

x1.l = -1; x2.l = 2; x3.l = 1; x4.l = -2; x5.l = -2;
Solve wright using nlp minimizing m;


Note that the solution given by LGO (shown on the next page) corresponds to the global minimum. For comparison, note that local scope nonlinear solvers will not find the global solution, unless started from a suitable neighbourhood (i.e., the model- and solver-specific region of attraction) of that solution.

In this example we use an option file to print out log information every 500 iterations, regardless of the elapsed time. Note that we set the log_time option to 0 to ignore the log_time interval.

   LGO 1.0       May  15, 2003 LNX.LG.NA 21.0 001.000.000.LXI Lib001-030502

LGO Lipschitz Global Optimization
(C) Pinter Consulting Services, Inc.
129 Glenforest Drive, Halifax, NS, Canada B3M 1J2
Website: www.dal.ca/~jdpinter

--- Using option file C:/GAMSPROJECTS/LGODOC/LGO.OPT
> log_iter 500
> log_time 0

3 defined, 0 fixed, 0 free
6 +/- INF bound(s) have been reset
1 LGO equations and 3 LGO variables


The first part prints out information about the model size after presolve. In this particular problem, the original model had 4 rows, 6 columns, and 14 non-zeroes, of which 3 were defined constraints, meaning that they could be eliminated via GAMS/LGO presolve techniques. Note that none of these were fixed or free constraints. Furthermore, LGO presolve reduced the model size further to 1 row (LGO equations) and 3 columns (LGO variables).

The main log gives information for every n iterations about current progress. The main fields are given in the table below:

FieldDescription
IterCurrent iteration.
ObjectiveCurrent objective function value.
SumInfSum of constraint infeasibilities.
MaxInfMaximum constraint infeasibility.
SecondsCurrent elapsed time in seconds.
ErrorsNumber of errors and type. Type can either be
D/E: Evaluation error
B: Bound violation.
      Iter      Objective     SumInf   MaxInf     Seconds  Errors
500   4.515428E-01   5.76E-02  5.8E-02       0.007
1000   6.700705E-01   5.03E-05  5.0E-05       0.014
1500   2.765930E+00   6.25E-04  6.2E-04       0.020
2000   2.710653E+00   1.55E-02  1.6E-02       0.026
2500   4.016702E+00   1.44E-02  1.4E-02       0.032
3000   4.865399E+00   2.88E-04  2.9E-04       0.038
3500   4.858826E+00   3.31E-03  3.3E-03       0.044
4000   1.106472E+01   1.53E-02  1.5E-02       0.050
4500   1.595505E+01   1.56E-06  1.6E-06       0.055
5000   1.618715E+01   2.17E-05  2.2E-05       0.062
5500   1.618987E+01   3.45E-04  3.5E-04       0.067
6000   1.985940E+01   4.03E-04  4.0E-04       0.074
6500   1.624319E+01   5.64E-03  5.6E-03       0.079
7000   1.727653E+01   8.98E-05  9.0E-05       0.086
7500   1.727033E+01   3.03E-03  3.0E-03       0.091
7840   2.933167E-02   0.00E+00  0.0E+00       0.097


LGO then reports the termination status, in this case globally optimal, together with the solver resource time. The resource time is also disaggregated by the total time spent performing function evaluations and the number of milliseconds (ms) spent for each function evaluation.

--- LGO Exit: Terminated by solver - Global solution
0.047 LGO Secs (0.015 Eval Secs, 0.001 ms/eval)


A local solver such as CONOPT can be called to compute marginal values. To invoke a postsolve using CONOPT, the user specifies the callConopt option with a positive value, indicating the number of seconds CONOPT is given to solve. See the LGO Options section for details.

# Illustrative References

R. Horst and P. M. Pardalos, Editors (1995) Handbook of Global Optimization. Vol. 1. Kluwer Academic Publishers, Dordrecht.

P. M. Pardalos and H. E. Romeijn, Editors (2002) Handbook of Global Optimization. Vol. 2. Kluwer Academic Publishers, Dordrecht.

J. D. Pintér (1996)Global Optimization in Action , Kluwer Academic Publishers, Dordrecht.

J. D. Pintér (2001)Computational Global Optimization in Nonlinear Systems: An Interactive Tutorial, Lionheart Publishing, Atlanta, GA.

J. D. Pintér (2002) Global optimization: software, tests and applications. Chapter 15 (pp. 515-569) in:Pardalos and Romeijn, Editors, Handbook of Global Optimization. Vol. 2. Kluwer Academic Publishers, Dordrecht.