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The authors would like to correct the errors caused by a wrong equation (A=B+z), at the end of the proof of Lemma 2.1 in the original article, which affects this lemma, two subsequent examples, as well as other statements spread along the paper.

The radius of robust feasibility provides a numerical value for the largest possible uncertainty set that guarantees feasibility of a robust counterpart of a mathematical program with uncertain constraints. The objective of this review of the state-of-the-art in this field is to present this useful tool of robust optimization to its potential users and to avoid undesirable overlapping of research works on the topic as those we have recently detected. In this paper we overview the existing literature on the radius of robust feasibility in continuous and mixed-integer linearly constrained programs, linearly constrained semi-infinite programs, convexly constrained programs, and conic linearly constrained programs. We also analyze the connection between the radius of robust feasibility and the distance to ill-posedness for different types of uncertain mathematical programs. This research was partially supported by the Australian Research Council, Discovery Project grant and the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22.

Since its appearance, even convexity has become a remarkable notion in convex analysis. In the fifties, W. Fenchel introduced the evenly convex sets as those sets solving linear systems containing strict inequalities. Later on, in the eighties, evenly quasiconvex functions were introduced as those whose sublevel sets are evenly convex. The significance of even convexity relies on the different areas where it enjoys applications, ranging from convex optimization to microeconomics. In this paper, we review some of the main properties of evenly convex sets and evenly quasiconvex functions, provide further characterizations of evenly convex sets, and present some new results for evenly quasiconvex functions. This research has been partially supported by MINECO of Spain and ERDF of EU, Grants PGC2018-097960-B-C22 and ECO2016-77200-P.

Given an infinite family of extended real-valued functions fi, i∈I, and a family H of nonempty finite subsets of I, the H-partial robust sum of fi, i∈I, is the supremum, for J∈H, of the finite sums ∑j∈Jfj. These infinite sums arise in a natural way in location problems as well as in functional approximation problems, and include as particular cases the well-known sup function and the so-called robust sum function, corresponding to the set H of all nonempty finite subsets of I, whose unconstrained minimization was analyzed in previous papers of three of the authors (https://doi.org/10.1007/s11228-019-00515-2 and https://doi.org/10.1007/s00245-019-09596-9). In this paper, we provide ordinary and stable zero duality gap and strong duality theorems for the minimization of a given H-partial robust sum under constraints, as well as closedness and convex criteria for the formulas on the subdifferential of the sup-function. This research was supported by the National Foundation for Science & Technology Development (NAFOSTED), Vietnam, Project 101.01-2018.310, and by Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI), and European Regional Development Fund (ERDF), Project PGC2018-097960-B-C22.

Evenly convex sets - Linear systems containing strict inequalities.- Evenly convex polyhedra - Finite linear systems containing strict inequalities.- Evenly quasiconvex functions.- Evenly convex functions.- Extensions to infinite-dimensional spaces.

This chapter is devoted to the study of those functions whose lower level sets are evenly convex, the so-called evenly quasiconvex functions. In Sect. 3.1 we define this class of functions, which provides greater minorants than the smaller class of the lower semicontinuous quasiconvex functions. Section 3.2 introduces the evenly quasiconvex hull providing the largest evenly quasiconvex minorant of a given function. Section 3.3 analyzes conjugates and subdifferentials for evenly quasiconvex functions, while Sect. 3.4 provides a sketch of quasiconvex duality theory. Finally, Sect. 3.5 describes an application in mathematical economy.