Existence of Solutions for Generalized Vector Equilibrium Problems

AbstractThe concepts of aF(C,D)-pseudomonotone mapping and of a (C,D)-pseudomonotone pair of mappings are introduced. By employing Fan's lemma, we establish several existence results for generalized vector equilibrium problems. The new results extend and modify various existence theorems for similar problems

A general class of relative optimization problems

We consider relative or subjective optimization problems where the goal function and feasible set are dependent on the current state of the system under consideration. In general, they are formulated as quasi-equilibrium problems, hence finding their solutions may be rather difficult. We describe a rather general class of relative optimization problems in metric spaces, which in addition depend on the starting state. We also utilize quasi-equilibrium type formulations of these problems and show that they admit rather simple descent solution methods. This approach gives suitable trajectories tending to a relatively optimal state. We describe several examples of applications of these problems. Preliminary results of computational experiments confirmed efficiency of the proposed method

Exact penalties for decomposable convex optimization problems

We consider a general decomposable convex optimization problem. By using right-hand side allocation technique, it can be transformed into a collection of small dimensional optimization problems. The master problem is a convex non-smooth optimization problem. We propose to apply the exact non-smooth penalty method, which gives a solution of the initial problem under some fixed penalty parameter and provides the consistency of lower level problems. The master problem can be solved with a suitable non-smooth optimization method. The simplest of them is the custom subgradient projection method using the divergent series step-size rule without line-search, whose convergence may be, however, rather low. We suggest to enhance its step-size selection by using a two-speed rule. Preliminary results of computational experiments confirm efficiency of this technique

Decomposable penalty method for generalized game problems with joint constraints

© 2020, © 2020 Informa UK Limited, trading as Taylor & Francis Group. We consider an extension of a non-cooperative game problem where players have joint binding constraints. In this case, justification of a generalized equilibrium point needs a reasonable mechanism for attaining this state. We combine a penalty method and shares allocation of right-hand sides, which replaces the initial problem with a sequence of the usual Nash equilibrium problems together with an upper level variational inequality as a master problem. In order to obtain a completely decomposable problem at the lower level, we apply its additional equivalent transformation. Convergence of solutions of these auxiliary penalized problems to a solution of the initial game problem is established under weak coercivity conditions

Decentralized multi-agent optimization based on a penalty method

We propose a decentralized penalty method for general convex constrained multi-agent optimization problems. Each auxiliary penalized problem is solved approximately with a special parallel descent splitting method. The method can be implemented in a computational network where each agent sends information only to the nearest neighbours. Convergence of the method is established under rather weak assumptions. We also describe a specialization of the proposed approach to the feasibility problem

A little about models

We discuss several aspects of creation of adequate mathematical models in other sciences. In particular, many difficulties stem from great complexity of the source systems and the presence of a variety of uncertain factors. We illustrate the effect of uncertainty on the known consumer demand model. We conclude that not every uncertainty can be represented by a random variable, and that these concepts are not equivalent. We discuss also the role of different information concepts in mathematical models. We give additional illustrative examples of models of quite complex systems