Addressing Hierarchical Jointly-Convex Generalized Nash Equilibrium Problems with Nonsmooth Payoffs

We consider a Generalized Nash Equilibrium Problem whose joint feasible region is implicitly defined as the solution set of another Nash game. This structure arises e.g. in multi-portfolio selection contexts, whenever agents interact at different hierarchical levels. We consider nonsmooth terms in all players’ objectives, to promote, for example, sparsity in the solution. Under standard … Read more

Solving Multi-Follower Games

We consider bilevel programs where a single leader interacts with multiple followers who are coupled by a Nash equilibrium problem at the lower level. We generalize the value function reformulation to include multiple followers. This allows us to propose a convergent method based on the sequential convex approximation paradigm, and study the (exact or inexact) … Read more

An explicit Tikhonov algorithm for nested variational inequalities

We consider nested variational inequalities consisting in a (upper-level) variational inequality whose feasible set is given by the solution set of another (lower-level) variational inequality. Purely hierarchical convex bilevel optimization problems and certain multi-follower games are particular instances of nested variational inequalities. We present an explicit and ready-to-implement Tikhonov-type solution method for such problems. We … Read more

Effectively managing diagnostic tests to monitor the COVID-19 outbreak in Italy

Urged by the outbreak of the COVID-19 in Italy, this study aims at helping to tackle the spread of the disease by resorting to operations research techniques. In particular, we propose a mathematical program to model the problem of establishing how many diagnostic tests the Italian regions must perform in order to maximize the overall … Read more

Equilibrium selection for multi-portfolio optimization

This paper studies a Nash game arising in portfolio optimization. We introduce a new general multi-portfolio model and state sufficient conditions for the monotonicity of the underlying Nash game. This property allows us to treat the problem numerically and, for the case of nonunique equilibria, to solve hierarchical problems of equilibrium selection. We also give … Read more

The Noncooperative Fixed Charge Transportation Problem

We introduce the noncooperative fixed charge transportation problem (NFCTP), which is a game-theoretic extension of the fixed charge transportation problem. In the NFCTP, competing players solve coupled fixed charge transportation problems simultaneously. Three versions of the NFCTP are discussed and compared, which differ in their treatment of shared social costs. This may be used from … Read more

The Standard Pessimistic Bilevel Problem

Pessimistic bilevel optimization problems, as optimistic ones, possess a structure involving three interrelated optimization problems. Moreover, their finite infima are only attained under strong conditions. We address these difficulties within a framework of moderate assumptions and a perturbation approach which allow us to approximate such finite infima arbitrarily well by minimal values of a sequence … Read more

Numerically tractable optimistic bilevel problems

We consider fully convex lower level standard optimistic bilevel problems. We show that this class of mathematical programs is sufficiently broad to encompass significant real-world applications. We establish that the critical points of a relaxation of the original problem correspond to the equilibria of a suitably defined generalized Nash equilibrium problem. The latter game is … Read more

Parallel Algorithms for Big Data Optimization

We propose a decomposition framework for the parallel optimization of the sum of a differentiable function and a (block) separable nonsmooth, convex one. The latter term is usually employed to enforce structure in the solution, typically sparsity. Our framework is very flexible and includes both fully parallel Jacobi schemes and Gauss-Seidel (i.e., sequential) ones, as … Read more