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Published in:
2020 | OriginalPaper | Chapter
Global and Local Search Methods for D.C. Constrained Problems
Author : Alexander S. Strekalovsky
Published in: Mathematical Optimization Theory and Operations Research
Publisher: Springer International Publishing
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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 (\(\mathcal P\)).