INVITED REVIEW Semi-infinite programming, duality, discretization and optimality conditionsy

Here is a (possibly infinite) index set, R 1⁄4 R [ fþ1g [ f 1g denotes the extended real line, f : R ! R and g : R !R. The above optimization problem is performed in the finite-dimensional space R and, if the index set is infinite, is a subject to an infinite number of constraints, therefore it is referred to as a SIP problem. There are numerous applications which lead to SIP problems. We can refer the interested reader to survey papers [11,12,21,27] where many such examples are described. There are also several books where SIP is discussed from theoretical and computational points of view (e.g. [4,9,10,22,23,31]). Compared with recent surveys [11,21], we use a somewhat different approach, although, of course, there is a certain overlap with these papers. For some of the presented results, for the sake of completeness, we outline proofs while more involved assertions will be referred to the literature. It is convenient to view the objective function f(x) as an extended real-valued function which is allowed to take þ1 or 1 values. In fact, we always assume in the subsequent analysis that f(x) is proper, i.e. its domain dom f 1⁄4 x 2 R : f ðxÞ5þ1