Generalized semi-infinite programming: A tutorial

  • Authors:

  • F. Guerra Vázquez;J. -J. Rückmann;O. Stein;G. Still

  • Affiliations:

  • Universidad de las Américas, Department of Actuarial Science and Mathematics, San Andrés Cholula 72820, Puebla, México;The University of Birmingham, School of Mathematics, Edgbaston, Birmingham B152TT, UK;RWTH Aachen University, 52056 Aachen, Germany;University of Twente, Enschede, The Netherlands

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2008

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Abstract

This tutorial presents an introduction to generalized semi-infinite programming (GSIP) which in recent years became a vivid field of active research in mathematical programming. A GSIP problem is characterized by an infinite number of inequality constraints, and the corresponding index set depends additionally on the decision variables. There exist a wide range of applications which give rise to GSIP models; some of them are discussed in the present paper. Furthermore, geometric and topological properties of the feasible set and, in particular, the difference to the standard semi-infinite case are analyzed. By using first-order approximations of the feasible set corresponding constraint qualifications are developed. Then, necessary and sufficient first- and second-order optimality conditions are presented where directional differentiability properties of the optimal value function of the so-called lower level problem are used. Finally, an overview of numerical methods is given.