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SEMI-INFINITE PROGRAMMING MARCO L OPEZ, GEORG STILL
 

Summary: SEMI-INFINITE PROGRAMMING
MARCO L ŽOPEZ, GEORG STILL
ABSTRACT. A semi-infinite programming problem is an optimization problem in which
finitely many variables appear in infinitely many constraints. This model naturally arises
in an abundant number of applications in different fields of mathematics, economics and
engineering. The paper, which intends to make a compromise between an introduction
and a survey, treats the theoretical basis, numerical methods, applications and historical
background of the field.
1. INTRODUCTION
1.1. Problem formulation. A semi-infinite program (SIP) is an optimization problem in
finitely many variables x = .x1;:::; xn/ Rn on a feasible set described by infinitely many
constraints:
(1) P : min
x
f .x/ s.t. g.x;t/ 0 t T;
where T is an infinite index set. For the sake of shortness, we omit additional equality
constraints hi.x/ = 0; i = 1;2;:::;m:
By F we denote the feasible set of P, whereas v := inf{ f .x/ | x F } is the optimal value,
and S := {x F | f .x/ = v} is the optimal set or set of minimizers of the problem. We say
that P is feasible or consistent if F = ; and set v = + when F = : With the only ex-

  

Source: Al Hanbali, Ahmad - Department of Applied Mathematics, Universiteit Twente
Still, Georg - Department of Applied Mathematics, Universiteit Twente

 

Collections: Engineering; Mathematics