期刊
OPTIMIZATION
卷 60, 期 10-11, 页码 1291-1308出版社
TAYLOR & FRANCIS LTD
DOI: 10.1080/02331934.2010.527970
关键词
SIP; NLP; Slater point; nonconvex; global optimization
资金
- Deutsche Forschungsgemeinschaft (German Research Association) [GSC 111]
An algorithm is proposed for the global solution of semi-infinite programs (SIPs) without convexity assumptions. It terminates finitely with a guaranteed feasible point, and a certificate of epsilon(f)-optimality. The only assumptions are continuity of the functions and existence of an epsilon(f)-optimal SIP-Slater point. The lower and upper bounds are obtained by the solution of two nonconvex nonlinear programs (NLP) each, thus shifting the nonconvexity to the global NLP solver. The main contribution of this article is a new procedure for the generation of feasible points. These are obtained by a restriction of the constraints right-hand side by epsilon(g)>0 and a successively finer discretization of the parameter set. A suitable decreasing sequence of epsilon(g) -> 0 leads to convergence of the generated upper bound. The converging lower bound is obtained by successively tighter discretization, following the principle of Blankenship and Falk [Infinitely constrained optimization problems, J. Optim. Theory Appl. 19 (1976), pp. 261-281]. A proof of convergence is given along with a prototype implementation in general algebraic modelling system. The algorithm is extended to problems with multiple constraints as well as mixed-integer problems. Literature and new problems are solved numerically. The algorithm is extremely simple to implement, but nevertheless very efficient compared to existing methods for nonconvex lower level programs.
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