Convex semi-infinite programming: Implicit optimality criterion based on the concept of immobile indices
The paper of O. I. Kostyukova, T. V. Tchemisova and S. A. Yermalinskayais is located in the area of semi-infinite programming (SIP) which stands in between the classical optimization (with a finite number of (equality and) inequality constraints) and infinite programming (where the optimization takes place in an infinite dimensional space). In fact, in SIP, an infinite number of inequality constraints may occur. Motivations of SIP come, among many other fields, from Chebychev approximation, maneuverability of robots, optimal control, modeling of genetic networks and special cases of infinite kernel learning. Researchers in SIP could develop various concepts of algorithms by the locally approximation (or even representation) of their progam with a much smaller (in fact, finite) number of inequality constraints. The authors contribute to this tradition under the assumption of (in the state variable) convex model functions and by the help of the concept of ``immobility''. The authors state a new implicit optimality criterion for convex semi-infinite programming (SIP) problems. That criterion does not require any constraint qualification but, indeed, it bases on the concept of immobile index and immobility order. For any given convex SIP program with a continuum of constraints, the authors use an information about its immobile indices to construct a nonlinear programming (NLP) problem of some special form. They prove that a feasible point of the original (infinitely constrained) SIP problem is optimal if and only if it is optimal in the corresponding (finitely constrained) NLP problem. With the help of optimality theory of NLP, the previous fact allows the authors to obtain new efficient optimality conditions for convex SIP problems. To construct the NLP problem, the authors use the DIO algorithm. They provide a comparison of their optimality conditions with known results. This article is well written, structured and, by proofs and examples, demonstrated; it advances given knowledge and could serve to support future research and, via more refined and still very efficient algorithms, practical applications as well. The paper consists of Section 1 (introduction), Section 2 on immobility orders and immobile indices, Section 3 on DIO algorithm, Section 4 on new optimality conditions for convex SIP including a comparison between new and known results, and of Section 5 (discussion). The appendix which is given too, gives further details of proofs and it provides, for the problem class under investigation in the paper, the equivalence between some constraint qualifications and, especially, with the absence of immobile indices. In future, further comparisons with other approaches of SIP, but also of convex programming (e.g., of (globally) well-structured convex programs), and further algorithmic refinements may be expected. Moreover, practical applications can certainly be found and become useful in connection with other sciences, with engineering, OR or economics. This will then again underline the importance of this nice study.
scientific article; zbMATH DE number 5770310
scientific article