Calmness of the argmin mapping in linear semi-infinite optimization (Q2250063)

This paper is a valuable contribution to the mathematical theories of linear and continuous optimization, which at the same time can become very helpful for the numerical solution of such programming problems and, hence, for the solution of various modern real-world applications. In fact, this article is located on the important ``contact'' point between standard linear programming (LP) and the complex infinite programming, and on the wider interface of LP where parametric and bilevel optimization may appear, and nonsmooth optimization, calculus of variations and optimal control as well. Herewith, it can easily cover a great generality of optimization needs and challenges. Regarding the state variable, the program is linear, whereas in the manifold of constraints, nonlinearity in the family index is permitted. The article is written with great care for the details and with a valuing of related and earlier results. Explanatory and formal parts and an example are presented in a balanced way. Indeed, the article characterizes the calmness condition of the argmin mapping in the frame of linear semi-infinite programming (LSIP) programs under canonical perturbations; this is a famous (Lipschitz) continuity kind of property. The latter perturbations are continuous perturbations of the right-hand side of the (inequality) constraints and perturbations of the objective function's coefficient vector. The presented characterization is new for semi-infinite problems, not requiring uniqueness of minimizers. For ordinary (i.e., finitely constrained) linear problems, calmness of the argmin mapping always holds, as its graph is piecewise polyhedral (following from a classical result by S. M. Robinson). Furthermore, isolated calmness (corresponding to the case of a unique optimal minimizer of the nominal problem) was characterized already. As a key method of the authors, they appeal to a certain supremum function associated with their nominal problem, not involving problems in a neighborhood, which is related to (sub)level sets. Under Slater constraint qualification, the main result establishes that perturbations of the objective function can be regarded as negligible, when characterizing the calmness of the argmin mapping. That result also states an equivalence between the calmness of the argmin mapping and the calmness of the level set mapping. The paper has five sections, beginning with the introduction, then notations and preliminaries are carefully provided before turning to calmness of the argmin mapping, and via future research coming to the conclusions. The various explanations and example themselves serve for a well-integrated and motivated paper that guides the reader and invites him or her to reflect further on future analysis and utilization, related with this important topic from mathematical optimization, data mining, Operations Research and finance, and towards employing them in various areas of scientific, economial, technological and social life.