Decision making under uncertainty: A semi-infinite programming approach (Q760341)

Consider the semi-infinite linear vector optimization problem \[ (SIVP)\quad \max \sum^{N}_{k=1}c_ k(t)x_ k\quad for\quad all\quad t\in T\quad s.t.\quad \sum^{N}_{k=1}a_ k(s)x_ k\leq b(s)\quad for\quad all\quad s\in S, \] where T and S are two nonempty, arbitrary subsets of \({\mathbb{R}}^ m\), \(C_ 0(T)\) and \(C_ 0(S)\) are the vector spaces of all continuous functions on T, resp. S, \(c_ 1(t),...,c_ N(t)\in C_ 0(T)\) and \(a_ 1(s),...,a_ N(s)\), \(b(s)\in C_ 0(S)\). This problem specializes to a linear vector optimization problem if \(| T|\) and \(| S|\) are finite and to a semi-infinite programming problem in the case of \(| T| =1\). The author derives theoretical results for (SIVP) based on known basic results in vector optimization and semi-infinite programming theory. The (SIVP) is interpreted as a decision model under uncertainty where the objectives as well as the constraints themselves are affected by uncertain parameters and the author treats the case of an infinite number of occuring states of nature.