# TOMLAB

TOMLAB is a general purpose development, modeling and optimal control 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. 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.

## Overall Design

Overall Design presents the general design of TOMLAB.

## Problem Types and Solver Routines

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

## Solving Linear, Quadratic and Integer Programming Problems

Contains examples on the process of defining problems and solving them. All test examples are available as part of the TOMLAB distribution.

## Solving Unconstrained and Constrained Optimization Problems

Contains examples on the process of defining problems and solving them. All test examples are available as part of the TOMLAB distribution.

## Solving Global Optimization Problems

Contains examples on the process of defining problems and solving them. All test examples are available as part of the TOMLAB distribution.

## Solving Least Squares and Parameter Estimation Problems

Contains examples on the process of defining problems and solving them. All test examples are available as part of the TOMLAB distribution.

## Multi Layer Optimization

Shows how to setup and define multi layer optimization problems in TOMLAB.

## Tomhelp - The Help Program

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 are not described, but documentation is available for each solver.

## TOMLAB Utility Functions

Describes the utility functions that can be used, for example tomRun and SolverList.

## Approximation of Derivatives

Introduces the different options for derivatives, automatic differentiation.

## Special Notes and Features

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

## Appendix A: Prob - the Input Problem Structure

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: Result - the Output Result Structure

Contains tables describing all elements defined in the output result structure returned from all solvers and driver routines.

## Appendix D: Global Variables and Recursive Calls

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

## Appendix E: External Interfaces

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

## Appendix F: Motivation and Background to TOMLAB

Gives some motivation for the development of TOMLAB.

## Appendix G: Performance Tests on Linear Programming Solvers

TOMLAB has been discussed in several papers and at several conferences. The main paper on TOMLAB v1.0 is [1] . The use of TOMLAB for nonlinear programming and parameter estimation is presented in [2] , and the use of linear and discrete optimization is discussed in [3] . Global optimization routines are also implemented, one is described in [4].

In all these papers TOMLAB was divided into two toolboxes, the NLPLIB TB and the OPERA TB. TOMLAB v2.0 was discussed in [5] [6] . and [7] . TOMLAB v4.0 and how to solve practical optimization problems with TOMLAB is discussed in [8] .

The use of TOMLAB for costly global optimization with industrial applications is discussed in [9]; costly global optimization with financial applications in [10] [11] [12]. Applications of global optimization for robust control is presented in [13] [14] . The use of TOMLAB for exponential fitting and nonlinear parameter estimation are discussed in e.g. [15] [16] [17] [18] [19] [20] .

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

## References

1. K. Holmström. The TOMLAB Optimization Environment in Matlab. Advanced Modeling and Optimization, 1(1):47-69, 1999.
2. K. Holmström and M. Björkman. The TOMLAB NLPLIB Toolbox for Nonlinear Programming. Advanced Modeling and Optimization, 1(1):70-86, 1999.
3. K. Holmström, M. Björkman, and E. Dotzauer. The TOMLAB OPERA Toolbox for Linear and Discrete Optimization. Advanced Modeling and Optimization, 1(2):1-8, 1999.
4. M. Björkman and K. Holmström. Global Optimization Using the DIRECT Algorithm in Matlab. Advanced Modeling and Optimization, 1(2):17-37, 1999.
5. K. Holmström. The TOMLAB v2.0 Optimization Environment. In E. Dotzauer, M. Björkman, and K. Holmstöm, editors, Sixth Meeting of the Nordic Section of the Mathematical Programming Society. Proceedings, Opuscula 49, ISSN 1400-5468, Västerås, 1999. Mälardalen University, Sweden.
6. K. Holmström. New Optimization Algorithms and Software. Theory of Stochastic Processes, 5(21)(1-2):55-63, 1999.
7. K. Holmström. Solving applied optimization problems using TOMLAB. In G. Osipenko, editor, Proceedings from MATHTOOLS '99, the 2nd International Conference on Tools for Mathematical Modelling, pages 90-98, St.Petersburg, Russia, 1999. St.Petersburg State Technical University.
8. K. Holmström. Practical Optimization with the Tomlab Environment. In T. A. Hauge, B. Lie, R. Ergon, M. D. Diez, G.-O. Kaasa, A. Dale, B. Glemmestad, and A Mjaavatten, editors, Proceedings of the 42nd SIMS Conference, pages 89-108, Porsgrunn, Norway, 2001. Telemark University College, Faculty of Technology.
9. M. Björkman and K. Holmström. Global Optimization of Costly Nonconvex Functions Using Radial Basis Functions. Optimization and Engineering, 1(4):373-397, 2000.
10. T. Hellström and K. Holmström. Parameter Tuning in Trading Algorithms using ASTA. In Y. S. Abu-Mostafa, B. LeBaron, A. W. Lo, and A. S. Weigend, editors, Computational Finance (CF99) - Abstracts of the Sixth International Conference, Leonard N. Stern School of Business, January 1999, Leonard N. Stern School of Business, New York University, 1999. Department of Statistics and Operations Research.
11. T. Hellström and K. Holmström. Parameter Tuning in Trading Algorithms using ASTA. In Y. S. Abu-Mostafa, B. LeBaron, A. W. Lo, and A. S. Weigend, editors, Computational Finance 1999, Cambridge, MA, 1999. MIT Press.
12. T. Hellström and K. Holmström. Global Optimization of Costly Nonconvex Functions, with Financial Applications. Theory of Stochastic Processes, 7(23)(1-2):121-141, 2001.
13. C. M. Fransson, B. Lennartson, T. Wik, and K. Holmström. Multi Criteria Controller Optimization for Uncertain MIMO Systems Using Nonconvex Global Optimization. In Proceedings of the 40th Conference on Decision and Control, Orlando, FL, USA, December 2001.
14. C. M. Fransson, B. Lennartson, T. Wik, K. Holmström, M. Saunders, and P.-O. Gutmann. Global Controller Optimization Using Horowitz Bounds. In Proceedings of the 15th IFAC Conference, Barcelona, Spain, 21th-26th July, 2002.
15. K. Holmström and J. Petersson. A Review of the Parameter Estimation Problem of Fitting Positive Exponential Sums to Empirical Data. Applied Mathematics and Computations, 126(1):31-61, 2002.
16. Jordan M. Berg and K. Holmström. On Parameter Estimation Using Level Sets. SIAM Journal on Control and Optimization, 37(5):1372-1393, 1999.
17. V. N. Fomin, K. Holmström, and T. Fomina. Least squares and Minimax methods for inorganic chemical equilibrium analysis. Research Report 2000-2, ISSN-1404-4978, Department of Mathematics and Physics, Mälardalen University, Sweden, 2000.
18. T. Fomina, K. Holmström, and V. B. Melas. Nonlinear parameter estimation for inorganic chemical equilib- rium analysis. Research Report 2000-3, ISSN-1404-4978, Department of Mathematics and Physics, Mälardalen University, Sweden, 2000.
19. K. Holmström and T. Fomina. Computer Simulation for Inorganic Chemical Equilibrium Analysis. In S.M. Ermakov, Yu. N. Kashtanov, and V.B. Melas, editors, Proceedings of the 4th St.Petersburg Workshop on Simulation, pages 261-266, St.Petersburg, Russia, 2001. NII Chemistry St. Peterburg University Publishers.
20. K. Holmström, T. Fomina, and Michael Saunders. Parameter Estimation for Inorganic Chemical Equilibria by Least Squares and Minimax Models. Optimization and Engineering, 4, 2003. Submitted.