Convex and exact games with non-transferable utility

We generalize exactness to games with non-transferable utility (NTU). A game is exact if for each coalition there is a core allocation on the boundary of its payoff set.     

The gap function of a convex program

The gap function expresses the duality gap of a convex program as a function of the primal variables only. Differentiability and convexity properties are derived, and a convergent minimization algorithm is given. An example gives a simple one-variable interpretation of weak and strong duality. Application to user-equilibrium traffic assignment yields an appealing alternative optimization problem.     

Proof of a Conjecture by Deutsch, Li, and Swetits on Duality of Optimization Problems

A conjecture of Deutsch, Li, and Swetits on duality relationships among three optimization problems is shown to hold true. The proof relies on a reformulation of one of the problems in a suitable product space, to which then a version of the classical Fenchel duality theorem applies.     

Subdifferentiability and Inf-Sup Theorems

Subdifferentiability criterions for non necessarily lower semicontinuous convex functions on general locally convex spaces or Fréchet spaces are used to derive inf-sup theorems. The importance of quasicontinuous convex functions is pointed out, and the usual compactness condition relaxed.     

On some families of cooperative fuzzy games

In a fuzzy cooperative game the players may choose to partially participate in a coalition. A fuzzy coalition consists of a group of participating players along with their participation level. The characteristic function of a fuzzy game specifies the worth of each such coalition. This paper introduces well-known properties of classical cooperative games to the theory of fuzzy games, and studies their interrelations. It deals with convex games, exact games, games with a large core, extendable games and games with a stable core.     

凸对策核仁的非空性及合成凸对策核仁的单调性

本文给出了核仁与核及最小核心之间的关系 ,且证明了凸对策核仁的存在性和唯一性 ,证明了凸对策的合成对策仍是凸对策 .最后 ,我们讨论了合成凸对策的核仁不满足单调性 .     

Convex operators: some subdifferentiability results

The subdifferentiability properties of convex operators are intimately related to the properties of the ordered (topological) vector spaces they range in. This relation is investigated for various types of subdifferentiability notions     

On some symmetric dual models in multiobjective programming

Certain omissions have been pointed out in some papers on symmetric duality in multiobjective programming. Corrective measures have also been discussed.     

Convex multi-choice games: Characterizations and monotonic allocation schemes

This paper focuses on new characterizations of convex multi-choice games using the notions of exactness and superadditivity. Furthermore, level-increase monotonic allocation schemes (limas) on the class of convex multi-choice games are introduced and studied. It turns out that each element of the Weber set of such a game is extendable to a limas, and the (total) Shapley value for multi-choice games generates a limas for each convex multi-choice game.     

两个泛函优化定理的证明

在泛函优化理论中,Lagrange乘子定理、对偶定理占有重要地位.建立了带有等式和不等式约束的泛函优化问题,并给出了广义Lagrange乘子定理、广义Lagrange对偶定理的证明.     

Wolfe duality and Mond-Weir duality via perturbations

Considering a general optimization problem, we attach to it by means of perturbation theory two dual problems having in the constraints a subdifferential inclusion relation. When the primal problem and the perturbation function are particularized different new dual problems are obtained. In the special case of a constrained optimization problem, the classical Wolfe and Mond-Weir duals, respectively, follow as particularizations of the general duals by using the Lagrange perturbation. Examples to show the differences between the new duals are given and a gate towards other generalized convexities is opened.     

Aanwendung des erweiterungsprinzips zur lösung konkayer optimierungsaufgaben

For the problem of minimizing a concave function over a convex polyedral-set an algorithm is given, which is based on the extension principle developed by Schoch. This algorithm yields after a finite number of steps an exact optimal solution of the problem. On the other hand one can find out throughout the algorithm an approximate optimal solution with any given precision.     

Regularity via duality in calculus of variations and degenerate elliptic PDEs

A technique based on duality to obtain $H^1$ or other Sobolev regularity results for solutions of convex variational problems is presented. This technique, first developed in order to study the regularity of the pressure in the variational formulation of the Incompressible Euler equation, has been recently re-employed in Mean Field Games. Here, it is shown how to apply it to classical problems in relation with degenerate elliptic PDEs of p-Laplace type. This allows to recover many classical results via a different point of view, and to have inspiration for new ones. The applications include, among others, variational models for traffic congestion and more general minimization problems under divergence constraints, but the most interesting results are obtained in dynamical problems such as Mean Field Games with density constraints or density penalizations.     

A class of problems where dual bounds beat underestimation bounds

We investigate the problem of minimizing a nonconvex function with respect to convex constraints, and we study different techniques to compute a lower bound on the optimal value: The method of using convex envelope functions on one hand, and the method of exploiting nonconvex duality on the other hand. We investigate which technique gives the better bound and develop conditions under which the dual bound is strictly better than the convex envelope bound. As a byproduct, we derive some interesting results on nonconvex duality.     

The conjugate of the difference of convex functions

Given an arbitrary functiong and a convex functionh, we derive the expression of the conjugate ofg —h via a simple proof.     

A convex duality approach for pricing contingent claims under partial information and short selling constraints

We consider the pricing problem facing a seller of a contingent claim. We assume that this seller has some general level of partial information, and that he is not allowed to sell short in certain assets. This pricing problem, which is our primal problem, is a constrained stochastic optimization problem. We derive a dual to this problem by using the conjugate duality theory introduced by Rockafellar. Furthermore, we give conditions for strong duality to hold. This gives a characterization of the price of the claim involving martingale- and super-martingale conditions on the optional projection of the price processes.     

Asymptotic properties of the fenchel dual functional and applications to decomposition problems

We study dual functionals which have two fundamental properties. Firstly, they have a good asymptotical behavior. Secondly, to each dual sequence of subgradients converging to zero, one can associate a primal sequence which converges to an optimal solution of the primal problem. Furthermore, minimal conditions for the convergence of the Gauss-Seidel methods are given and applied to such kinds of functionals.