On the convergence of a smoothed penalty algorithm for semi-infinite programming. (English) Zbl 1291.90271
A class of smoothed penalty functions for the standard semi-infinite programming (SIP) problem is studied where the semi-infinite constraint is replaced equivalently by a single finite integral constraint with a nonsmooth function and an exact \(l_{p}\) penalty function with \(0<p\leq 1\) for the latter problem is approximated by a smoothed penalty function. This smoothed penalty function is formed by a (possibly also nonconvex) continuously differentiable function having certain well-defined properties. It motivates an algorithm where, in each iteration, an inexact global minimizer of the penalty function has to be determined. Next it is proven that accumulation points of the sequence of iterates generated by the algorithm are global solutions of the SIP problem. Furthermore a necessary and sufficient condition is given so that the total sequence of objective function values produced by the algorithm converges to the optimal value of the SIP program. At last, the convergence of the algorithm is studied for a subclass of convex smoothing functions used. Some preliminary numerical experiments are reported.
Reviewer: Rembert Reemtsen (Cottbus)
Keywords:
semi-infinite programming; penalty algorithm; \(l_{p}\) exact penalty function; smoothed penalty function; smoothing functionReferences:
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