Mathematical Programming

, Volume 136, Issue 1, pp 183–207

The adaptive convexification algorithm for semi-infinite programming with arbitrary index sets

Full Length Paper Series B

DOI: 10.1007/s10107-012-0556-5

Cite this article as:
Stein, O. & Steuermann, P. Math. Program. (2012) 136: 183. doi:10.1007/s10107-012-0556-5


A numerical solution method for semi-infinite optimization problems with arbitrary, not necessarily box-shaped, index sets is presented. Following the ideas of Floudas and Stein (SIAM J Optim 18:1187–1208, 2007), convex relaxations of the lower level problem are adaptively constructed and then reformulated as mathematical programs with complementarity constraints and solved. Although the index set is arbitrary, this approximation produces feasible iterates for the original problem. The convex relaxations and needed parameters are constructed with ideas of the αBB method of global optimization and interval methods. It is shown that after finitely many steps an \({\epsilon}\)-stationary point of the original semi-infinite problem is reached. A numerical example illustrates the performance of the proposed method.


Semi-infinite programming αBB Global optimization Convex optimization Mathematical programming with complementarity constraints Bilevel optimization 

Mathematics Subject Classification

90C34 90C33 90C26 65K05 

Copyright information

© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  1. 1.Karlsruhe Institute of Technology (KIT)KarlsruheGermany