We correct an error in Martínez-Legaz et al. (On farthest Bregman Voronoi cells. Optimization. 2022;71:937–947,Theorem 4.1).
In this article, we study the class of increasing and convex along rays (ICAR) functions over a cone. Apart from studying its basic properties, we study them from the point of view of Abstract Convexity. Further, we study the relation between the ICAR and Lipschitz functions and the properties under which an ICAR function has a Lipschitz behaviour. We also study the class of decreasing and convex along rays functions (DCAR).
In this paper we introduce a new approach to representing both TU-games and NTU-games as special economic structures. Instead of representing a game as a market – an exchange economy with concave utility functions – as in the extant literature, we represent an arbitrary game as a coalition production economy with a public good. The economy provides an indirect description of the game. Our model uses the idea of a social planner or arbitrager who, through arbitrage on productive activities, seeks to minimize the net cost of providing the public good subject to the constraint that the society satis…es its reservation welfare level. Note that in constrast to the prior literature on representing games as economic structures, we require neither that the game be balanced nor that there are large numbers of players. Our approach to representing a game as an economic structure uses techniques from consumer demand theory and, in particular, the notion of a compensated demand correspondence. The main results of this paper exhibit a relationship between cooperative game theory and consumer demand theory
A weighted average worth per capita formula is presented for any semivalue of a TU game. Further, this formula is used to derive a characterisation of the class of games with the property that a given semivalue belongs to the power core of the game, by means of a linear system of inequalities. It is shown that for the Shapley value, the only efficient semivalue, this system reduces to the system already obtained by Inarra and Usategui. The potential approach is also used even for the more general case of values possessing a potential. A direct proof shows that for a value possessing a potential, the value of a game is in the power core relative to this value, if and only if the potential game is weak average convex. From this result, it follows that for a game and each of its subgames the value possessing a potential is in the corresponding power cores, if and only if the potential game relative to the value is average convex. This is an extension of the result obtained by Marin–Solano and Rafels for the Shapley value, proved by using the dividend form of the game.
In 1956 Marguerite Frank and Paul Wolfe proved that a quadratic function which is bounded below on a polyhedron $P$ attains its infimum on $P$. In this work we search for larger classes of sets $F$ with this Frank-and-Wolfe property. We establish the existence of non-polyhedral Frank-and-Wolfe sets, obtain internal characterizations by way of asymptotic properties, and investigate stability of the Frank-and-Wolfe class under various operations.
Minimax fractional programming problems are analyzed from the view- point of lower subdifferentiability, obtaining Kuhn-Tucker type optimality conditions. Multiobjective optimization problems with fractional objectives are also studied.
In this paper we introduce and study a subdifferential that is related to the quasi- convex functions, much as the Fenchel--Moreau subdifferential is related to the convex ones. It is defined for any lower semicontinuous function, through an appropriate combination of an abstract subdifferential and the normal cone to sublevel sets. We show that this "quasiconvex" subdifferential is always a cyclically quasimonotone operator that coincides with the Fenchel--Moreau subdifferential whenever the function is convex, and that under mild assumptions, the density of its domain in the domain of the function is equivalent to the quasiconvexity of the function. We also show that the "quasiconvex" subdifferential of a lower semicontinuous function contains the derivatives of its differentiable quasiaffine supports. As a consequence, it contains the subdifferential introduced by Martinez-Legaz and Sach in a recent paper [J. Convex Anal., 6 (1999), pp. 1--12]. Several other properties and calculus rules are also established.
