Introduction to optimization

1 Introduction

1.1 What is TOMLAB?

TOMLAB is a general purpose development environment in Matlab for research, teaching and practical solution of optimization problems.

TOMLAB has grown out of the need for advanced, robust and reliable tools to be used in the development of algorithms and software for the solution of many different types of applied optimization problems.

There are many good tools available in the area of numerical analysis, operations research and optimization, but because of the different languages and systems, as well as a lack of standardization, it is a time consuming and complicated task to use these tools. Often one has to rewrite the problem formulation, rewrite the function specifications, or make some new interface routine to make everything work. Therefore the first obvious and basic design principle in TOMLAB is: Define your problem once, run all available solvers. The system takes care of all interface problems, whether between languages or due to different demands on the problem specification.

In the process of optimization one sometimes wants to graphically view the problem and the solution process, especially for ill-conditioned nonlinear problems. Sometimes it is not clear what solver is best for the particular type of problem and tests on different solvers can be of use. In teaching one wants to view the details of the algorithms and look more carefully at the different algorithmic steps. An unexperienced user or a student might want some very easy way to solve the problem, and would like to use a menu system or a graphical user interface (GUI). Using a GUI or a menu system also makes it very easy to change parameters influencing the solution process. When developing new algorithms tests on thousands of problems are necessary to fully access the pros and cons of the new algorithm. One might want to solve a practical problem very many times, with slightly different conditions for the run. Or solve a control problem looping in real-time and solving the optimization problem each time slot.

All these issues and many more are addressed with the TOMLAB optimization environment. TOMLAB gives easy access to a large set of standard test problems, optimization solvers and utilities. Furthermore, it is easy to define new problems in the TOMLAB format, and try to solve them using any solver. To access the user problem in the GUI or menu system, routines converting the problem into the TOMLAB Init File and adding the problems to the GUI data base are available and simple to use. To use TOMLAB in real-time control, the efficient MEX-file interfaces calling fast Fortran solvers are of great importance.

1.2 The Organization of This Guide

Section 2 presents the general design of TOMLAB .

Section 3 contains strict mathematical definitions of the optimization problem types. All solver routines available in TOMLAB are described.

Section 4 describes the two available input formats, the TOMLAB Format (TQ) and the TOMLAB Init File Format (IF).

Sections 5, 6, 7 and 8 contain examples on the process of defining problems and solving them. All test examples are available as part of the TOMLAB distribution.

Section 9 shows how to setup and define multi layer optimization problems in TOMLAB .

Section 10 presents the Graphical User Interface (GUI). The GUI gives the user the possibility to set all kinds of solver parameters that influences the optimization process. It can also be used as a code generator, saving the status of the GUI and generating m-file code to run the current problem. There is also an option to retrieve the saved status of the GUI.

Section 10.5 presents the menu system, tomMenu . The menu system implements some, but not all of the functionality of the GUI. tomRemote but can be useful when running TOMLAB on remote machines over text-only connections.

Section 11 contains detailed descriptions of many of the functions in TOMLAB . The TOM solvers, originally developed by the Applied Optimization and Modeling (TOM) group, are described together with TOMLAB driver routine and utility functions. Other solvers, like the Stanford Optimization Laboratory (SOL) solvers called using MEX-file interfaces are not described, but documentation is available for each solver, e.g. the MINOS User's Guide [60].

Section 12 describes the utility functions that can be used, for example tomRun and SolverList.

Section 13 introduces the different options for derivatives, automatic differentiation.

Section 14 discusses a number of special system features such as partially separable functions and user supplied parameter information for the function computations.

Appendix A contains tables describing all elements defined in the problem structure. Some subfields are either empty, or filled with information if the particular type of optimization problem is defined. To be able to set different parameter options for the optimization solution, and change problem dependent information, the user should consult the tables in this Appendix.

Appendix B contains tables describing all elements defined in the output result structure returned from all solvers and driver routines. An array of such structures are also returned if calling the GUI or menu system with an output variable.

Appendix C is concerned with the global variables used in TOMLAB and routines for handling important global variables enabling recursive calls of any depth.

Appendix D describes in detail how to edit TOMLAB Init Files directly.

Appendix E describes the available set of interfaces to other optimization software, such as CUTE, AMPL, and The Mathworks' Optimization Toolbox.

Appendix F gives some motivation for the development of TOMLAB .

1.3 Further Reading

TOMLAB has been discussed in several papers and at several conferences. The main paper on TOMLAB v1.0 is [45]. The use of TOMLAB for nonlinear programming and parameter estimation is presented in [48], and the use of linear and discrete optimization is discussed in [49]. Global optimization routines are also implemented, one is described in [8].

In all these papers TOMLAB was divided into two toolboxes, the NLPLIB TB and the OPERA TB. This was impractical because of the integration of linear and mixed-integer programming in the GUI and the other menu and driver tools. The first version of the graphical user interface (GUI) is described in [20]. TOMLAB v2.0 was discussed in [46], [43]. and [44]. TOMLAB v4.0 and how to solve practical optimization problems with TOMLAB is discussed in [47].

The use of TOMLAB for costly global optimization with industrial applications is discussed in [9]; costly global optimization with financial applications in [40, 41, 42]. Applications of global optimization for robust control is presented in [28, 29]. The use of TOMLAB for exponential fitting and nonlinear parameter estimation are discussed in e.g. [52, 4, 25, 26, 50, 51].

The manuals for the add-on solver packages are also recommended reading material.

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