Error bounds for infinite systems of convex inequalities without Slater's condition (Q1584002)

The article under review is a is a valuable contribution to a field of nonlinear optimization which plays a significant role for the development of numerical methods: error bounds. Focussing on convex programs, the author admits an infinite number of inequality constraints. The set of (continuous and convex) inequality constraints is assumed to be compact. For such a semi-infinite optimization problem, the paper is concerned with upper bounds for the distance of a given point from the feasible set in terms of a constraint function (at this point) which is maximally violated (times a constant). For measuring that distance, some projection is implied; \(\varepsilon\)-relaxing contraints leads to uniform (asymptotical) considerations. While this is a Lipschitzian estimation, a Hölderian one is studied, too. Here, the residual (left-hand side) is raised to some power \(\beta\geq 1\). The author's research is well-embedded into the reseach on error bounds made by other anthors, which is improved in two directions: On the one hand, he turns from finitely constrained to semi-infinite programming. On the other hand, he does no longer assume the Slater condition. The paper's presentation with its eight sections is thorough, clear and well-organized. A central result of the paper consists in sufficient -- or even characterizing -- conditions for the validity of an error bound. In the Lipschitz case \((\beta=1)\), there is the weak Slater condition being sufficient and, under a continuity assumption on one-sided directional derivatives and among five equivalent conditions, uniform validity of both a Mangasarian-Fromovitz type constraint qualification and boundedness of dual solutions are characterizing. Another result compares the latter two conditions in the general Hölder case, partially assuming Fréchet differentiability. The careful investigation utilizes some measure theory. Nine examples, given throughout or concluding the article, are interesting from the viewpoint of motivation, but also to explain the conditions and their interrelations.