Nonlinear transformations for the simplification of unconstrained nonlinear optimization problems

Elvira Antal, T. Csendes, János Virágh

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Formalization decisions in mathematical programming could significantly influence the complexity of the problem, and so the efficiency of the applied solver methods. This widely accepted statement induced investigations for the reformulation of optimization problems in the hope of getting easier to solve problem forms, e.g. in integer programming. These transformations usually go hand in hand with relaxation of some constraints and with the increase in the number of the variables. However, the quick evolution and the widespread use of computer algebra systems in the last few years motivated us to use symbolic computation techniques also in the field of global optimization. We are interested in potential simplifications generated by symbolic transformations in global optimization, and especially in automatic mechanisms producing equivalent expressions that possibly decrease the dimension of the problem. As it was pointed out by Csendes and Rapcsák (J Glob Optim 3(2):213-221, 1993), it is possible in some cases to simplify the unconstrained nonlinear objective function by nonlinear coordinate transformations. That means mostly symbolic replacement of redundant subexpressions expecting less computation, while the simplified task remains equivalent to the original in the sense that a conversion between the solutions of the two forms is possible. We present a proper implementation of the referred theoretical algorithm in a modern symbolic programming environment, and testing on some examples both from the original publications and from the set of standard global optimization test problems to illustrate the capabilities of the method.

Original languageEnglish
Pages (from-to)665-684
Number of pages20
JournalCentral European Journal of Operations Research
Volume21
Issue number4
DOIs
Publication statusPublished - Dec 2013

Fingerprint

Optimization problem
Nonlinear optimization
Nonlinear transformation
Global optimization
Computer systems
Formalization
Replacement
Objective function
Integer programming
Mathematical programming
Programming
Testing

Keywords

  • Maple
  • Reformulation
  • Symbolic computation
  • Unconstrained nonlinear optimization

ASJC Scopus subject areas

  • Management Science and Operations Research

Cite this

Nonlinear transformations for the simplification of unconstrained nonlinear optimization problems. / Antal, Elvira; Csendes, T.; Virágh, János.

In: Central European Journal of Operations Research, Vol. 21, No. 4, 12.2013, p. 665-684.

Research output: Contribution to journalArticle

@article{85ab37835f4d460cbca584ebd215151d,
title = "Nonlinear transformations for the simplification of unconstrained nonlinear optimization problems",
abstract = "Formalization decisions in mathematical programming could significantly influence the complexity of the problem, and so the efficiency of the applied solver methods. This widely accepted statement induced investigations for the reformulation of optimization problems in the hope of getting easier to solve problem forms, e.g. in integer programming. These transformations usually go hand in hand with relaxation of some constraints and with the increase in the number of the variables. However, the quick evolution and the widespread use of computer algebra systems in the last few years motivated us to use symbolic computation techniques also in the field of global optimization. We are interested in potential simplifications generated by symbolic transformations in global optimization, and especially in automatic mechanisms producing equivalent expressions that possibly decrease the dimension of the problem. As it was pointed out by Csendes and Rapcs{\'a}k (J Glob Optim 3(2):213-221, 1993), it is possible in some cases to simplify the unconstrained nonlinear objective function by nonlinear coordinate transformations. That means mostly symbolic replacement of redundant subexpressions expecting less computation, while the simplified task remains equivalent to the original in the sense that a conversion between the solutions of the two forms is possible. We present a proper implementation of the referred theoretical algorithm in a modern symbolic programming environment, and testing on some examples both from the original publications and from the set of standard global optimization test problems to illustrate the capabilities of the method.",
keywords = "Maple, Reformulation, Symbolic computation, Unconstrained nonlinear optimization",
author = "Elvira Antal and T. Csendes and J{\'a}nos Vir{\'a}gh",
year = "2013",
month = "12",
doi = "10.1007/s10100-013-0310-y",
language = "English",
volume = "21",
pages = "665--684",
journal = "Central European Journal of Operations Research",
issn = "1435-246X",
publisher = "Physica-Verlag",
number = "4",

}

TY - JOUR

T1 - Nonlinear transformations for the simplification of unconstrained nonlinear optimization problems

AU - Antal, Elvira

AU - Csendes, T.

AU - Virágh, János

PY - 2013/12

Y1 - 2013/12

N2 - Formalization decisions in mathematical programming could significantly influence the complexity of the problem, and so the efficiency of the applied solver methods. This widely accepted statement induced investigations for the reformulation of optimization problems in the hope of getting easier to solve problem forms, e.g. in integer programming. These transformations usually go hand in hand with relaxation of some constraints and with the increase in the number of the variables. However, the quick evolution and the widespread use of computer algebra systems in the last few years motivated us to use symbolic computation techniques also in the field of global optimization. We are interested in potential simplifications generated by symbolic transformations in global optimization, and especially in automatic mechanisms producing equivalent expressions that possibly decrease the dimension of the problem. As it was pointed out by Csendes and Rapcsák (J Glob Optim 3(2):213-221, 1993), it is possible in some cases to simplify the unconstrained nonlinear objective function by nonlinear coordinate transformations. That means mostly symbolic replacement of redundant subexpressions expecting less computation, while the simplified task remains equivalent to the original in the sense that a conversion between the solutions of the two forms is possible. We present a proper implementation of the referred theoretical algorithm in a modern symbolic programming environment, and testing on some examples both from the original publications and from the set of standard global optimization test problems to illustrate the capabilities of the method.

AB - Formalization decisions in mathematical programming could significantly influence the complexity of the problem, and so the efficiency of the applied solver methods. This widely accepted statement induced investigations for the reformulation of optimization problems in the hope of getting easier to solve problem forms, e.g. in integer programming. These transformations usually go hand in hand with relaxation of some constraints and with the increase in the number of the variables. However, the quick evolution and the widespread use of computer algebra systems in the last few years motivated us to use symbolic computation techniques also in the field of global optimization. We are interested in potential simplifications generated by symbolic transformations in global optimization, and especially in automatic mechanisms producing equivalent expressions that possibly decrease the dimension of the problem. As it was pointed out by Csendes and Rapcsák (J Glob Optim 3(2):213-221, 1993), it is possible in some cases to simplify the unconstrained nonlinear objective function by nonlinear coordinate transformations. That means mostly symbolic replacement of redundant subexpressions expecting less computation, while the simplified task remains equivalent to the original in the sense that a conversion between the solutions of the two forms is possible. We present a proper implementation of the referred theoretical algorithm in a modern symbolic programming environment, and testing on some examples both from the original publications and from the set of standard global optimization test problems to illustrate the capabilities of the method.

KW - Maple

KW - Reformulation

KW - Symbolic computation

KW - Unconstrained nonlinear optimization

UR - http://www.scopus.com/inward/record.url?scp=84884531198&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884531198&partnerID=8YFLogxK

U2 - 10.1007/s10100-013-0310-y

DO - 10.1007/s10100-013-0310-y

M3 - Article

AN - SCOPUS:84884531198

VL - 21

SP - 665

EP - 684

JO - Central European Journal of Operations Research

JF - Central European Journal of Operations Research

SN - 1435-246X

IS - 4

ER -