# LGO

## Contents

## Introduction

### Overview

Welcome to the TOMLAB /LGO User's Guide. TOMLAB /LGO includes the LGO solver from Pintér Consulting Services and an interface to MATLAB, by MathWorks.

The *Lipschitz-Continuous Global Optimizer *(LGO) solver suite 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 smaller LP models. LGO is documented elsewhere in detail: see for example, Pintér (1996, 2001, 2003). \[1, 2, 3\] TOMLAB /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:
- Constrained local search, based on a generalized reduced gradient approach (GRG).

The overall solution approach followed by TOMLAB /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.

TOMLAB /LGO does not rely on any sub-solvers, and it does not require any in-depth 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. TOMLAB /LGO needs only computable function values (without a need for higher order analytical information). TOMLAB /LGO can even solve models having constraints involving continuous, but non-differentiable functions. Thus, within TOMLAB, LGO is well suited to solve nonlinear - global and convex - optimization models.

TOMLAB /LGO can also be used in conjunction with other TOMLAB solvers. Local solvers (when available) can be used to verify the solution found by LGO, and to provide additional local information.

### Contents of this manual

- #Introduction provides a basic overview of the TOMLAB /LGO solver package.
- #Using the Matlab Interface provides an overview of the Matlab interface to LGO.
- #Setting LGO Options describes how to set LGO solver options from Matlab.
- #LGO Test Examples provides information regarding TOMLAB /LGO test examples.
- #LGO Solver Reference gives detailed information about the interface routine
*lgoTL*.

### More information

Please visit the following links for more information and see the illustrative references at the end of this manual.

### Prerequisites

In this concise manual we assume that the user is familiar with global optimization and nonlinear programming, setting up problems in TOMLAB (in particular global constrained nonlinear (**glc**) problems) and with the Matlab language in general.

## Using the Matlab Interface

The LGO solver package is accessed via the *tomRun *driver routine, which calls the *lgoTL *interface routine. The solver itself is located in the MEX file *lgo*.

Function | Description |
---|---|

lgoTL |
The interface routine called by the TOMLAB driver routine tomRun.
This routine then calls the MEX file |

## Setting LGO Options

All control parameters can be set directly from Matlab.

### Setting options using the Prob.LGO structure

The parameters can be set as subfields in the *Prob.LGO *structure. The following example shows how to set a limit on the maximum number of merit function evaluations before termination of global search phase.

Prob = glcAssign(...) % Setup problem, see help glcAssign for more information Prob.LGO.options.g_maxfct = 10000; % Setting the maximum number of global search % phase model function evaluations.

In the cases where a solver specific parameter has a corresponding TOMLAB general parameter, the latter is used only if the user has not given the solver specific parameter.

A complete description of the available LGO parameters can be found in #lgoTL.

## LGO Test Examples

There are several LGO test examples included in the TOMLAB /LGO distribution. The examples are located in the *testprob *folder in TOMLAB. *lgo1 prob *contains one dimensional test problems while *lgo2 prob *includes two- and higher-dimensional.

To test the solution of these problem sets by LGO, the following type of code can be used:

Prob = probInit('lgo1_prob', 1); Result = tomRun('lgo', Prob, 1);

It is also possible to run all the problems located in *glb prob *and *glc prob *if no integer variables are defined.

## LGO Solver Reference

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

### lgoTL

#### Purpose

Solves global and local (convex and mildly non-convex) constrained nonlinear programming problems. LGO solves problems of the form

where and .

#### Calling Syntax

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

#### Description of Inputs

*Prob *Problem description structure. The following fields are used:

Input | Description |
---|---|

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.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 #Description of Prob.LGO.options |

#### Description of Outputs

*Result *Structure with result from optimization. The following fields are set:

Output | Description |
---|---|

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 as needed. |
-1 |

l_maxfct |
Maximal number of function evaluations in local search phase. | |

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 |

g_target |
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 g_target. If a good estimate is known, then its usage may results in a considerably faster search. | -1.0E+10 |

l_target |
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 0 default.
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 |