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Global and Local Search Methods for D.C. Constrained Problems

  • Alexander S. StrekalovskyEmail author
Conference paper
  • 213 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12095)

Abstract

This paper addresses the general optimization problem (\(\mathcal P\)) with equality and inequality constraints and the cost function given by d.c. functions. We reduce the problem to a penalized problem (\(\mathcal P_{\sigma }\)) without constraints with the help of the Exact Penalization Theory. Further, we show that the reduced problem is also a d.c. minimization problem. This property allows us to prove the Global Optimality Conditions (GOCs), which reduce the study of the penalized problem to an investigation of a family of linearized (convex) problems tractable with the help of standard convex optimization methods and software.

In addition, we propose a new Local Search Scheme (LSS1) which produces a sequence of vectors converging to a so-called critical point. On the other hand, the vector satisfying the GOCs turns out to be also a critical point.

On the basis of the GOCs for finding a global solution to (\(\mathcal P_{\sigma }\)), we develop a Global Search Scheme, including the LSS1 with an update of the penalty parameter, and a special stopping criteria allowing detection of a feasible vector in the original problem (\(\mathcal P\)), and, consequently, a global solution to the original Problem (\(\mathcal P\)).

Keywords

Difference of convex functions Equality and inequality constraints Exact penalty Linearized problem Local search Critical vector Global search 

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Matrosov Institute for System Dynamics and Control Theory of SB of RASIrkutskRussia

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