Veronica Piccialli

On generalized Nash games and variational inequalities

We show that for a large class of problems a generalized Nash equilibrium can be calculated by solving a variational inequality. We analyze what solutions are found by this reduction procedure and hint at possible applications.

Generalized Nash equilibrium problems and Newton methods

The generalized Nash equilibrium problem, where the feasible sets of the players may depend on the other players’ strategies, is emerging as an important modeling tool. However, its use is limited by its great analytical complexity. We consider several Newton methods, analyze their features and compare their range of applicability. We illustrate in detail the results obtained by applying them to a model for internet switching.

New Classes of Globally Convexized Filled Functions for Global Optimization

We propose new classes of globally convexized filled functions. Unlike the globally convexized filled functions previously proposed in literature, the ones proposed in this paper are continuously differentiable and, under suitable assumptions, their unconstrained minimization allows to escape from any local minima of the original objective function. Moreover we show that the properties of the proposed functions can be extended to the case of box constrained minimization problems. We also report the results of a preliminary numerical experience.

Decomposition algorithms for generalized potential games

We analyze some new decomposition schemes for the solution of generalized Nash equilibrium problems. We prove convergence for a particular class of generalized potential games that includes some interesting engineering problems. We show that some versions of our algorithms can deal also with problems lacking any convexity and consider separately the case of two players for which stronger results can be obtained.

A DIRECT-based approach exploiting local minimizations for the solution of large-scale global optimization problems

In this paper we propose a new algorithm for solving difficult large-scale global optimization problems. We draw our inspiration from the well-known DIRECT algorithm which, by exploiting the objective function behavior, produces a set of points that tries to cover the most interesting regions of the feasible set. Unfortunately, it is well-known that this strategy suffers when the dimension of the problem increases. As a first step we define a multi-start algorithm using DIRECT as a deterministic generator of starting points. Then, the new algorithm consists in repeatedly applying the previous multi-start algorithm on suitable modifications of the variable space that exploit the information gained during the optimization process. The efficiency of the new algorithm is pointed out by a consistent numerical experimentation involving both standard test problems and the optimization of Morse potential of molecular clusters.

A partition-based global optimization algorithm

This paper is devoted to the study of partition-based deterministic algorithms for global optimization of Lipschitz-continuous functions without requiring knowledge of the Lipschitz constant. First we introduce a general scheme of a partition-based algorithm. Then, we focus on the selection strategy in such a way to exploit the information on the objective function. We propose two strategies. The first one is based on the knowledge of the global optimum value of the objective function. In this case the selection strategy is able to guarantee convergence of every infinite sequence of trial points to global minimum points. The second one does not require any a priori knowledge on the objective function and tries to exploit information on the objective function gathered during progress of the algorithm. In this case, from a theoretical point of view, we can guarantee the so-called every-where dense convergence of the algorithm.

An Algorithm Model for Mixed Variable Programming

In this paper we consider a particular class of nonlinear optimization problems involving both continuous and discrete variables. The distinguishing feature of this class of nonlinear mixed variable optimization problems is that the structure and the number of variables of the problem depend on the values of some discrete variables. In particular, we define a general algorithm model for the solution of this class of problems, that draws inspiration from the approach recently proposed by Audet and Dennis [SIAM J. Optim., 11 (2001), pp. 573--594], and is based on the strategy of combining in a suitable way a local search with respect to the continuous variables and a local search with respect to the discrete variables. We prove global convergence of the algorithm model without specifying the local continuous search, but only identifying some reasonable requirements. Moreover, we define a particular derivative-free algorithm for solving mixed variable programming problems where the continuous variables are linearly constrained and derivative information is not available. Finally, we report numerical results obtained by the proposed algorithm in solving a real optimal design problem. These results show the effectiveness of the approach.

A game-theoretic approach to computation offloading in mobile cloud computing

We consider a three-tier architecture for mobile and pervasive computing scenarios, consisting of a local tier of mobile nodes, a middle tier (cloudlets) of nearby computing nodes, typically located at the mobile nodes access points but characterized by a limited amount of resources, and a remote tier of distant cloud servers, which have practically infinite resources. This architecture has been proposed to get the benefits of computation offloading from mobile nodes to external servers while limiting the use of distant servers whose higher latency could negatively impact the user experience. For this architecture, we consider a usage scenario where no central authority exists and multiple non-cooperative mobile users share the limited computing resources of a close-by cloudlet and can selfishly decide to send their computations to any of the three tiers. We define a model to capture the users interaction and to investigate the effects of computation offloading on the users’ perceived performance. We formulate the problem as a generalized Nash equilibrium problem and show existence of an equilibrium. We present a distributed algorithm for the computation of an equilibrium which is tailored to the problem structure and is based on an in-depth analysis of the underlying equilibrium problem. Through numerical examples, we illustrate its behavior and the characteristics of the achieved equilibria.

Euclidean distance matrices, semidefinite programming and sensor network localization

The fundamental problem of distance geometry involves the characterization and study of sets of points based only on given values of some or all of the distances between pairs of points. This problem has a wide range of applications in various areas of mathe- matics, physics, chemistry, and engineering. Euclidean distance matrices play an important role in this context by providing elegant and powerful convex relaxations. They play an important role in problems such as graph realization and graph rigidity. Moreover, by relaxing the embedding dimension restriction, these matrices can be used to approximate the hard problems e‰ciently using semidefinite programming. Throughout this survey we emphasize the interplay between these concepts and problems. In addition, we illustrate this interplay in the context of the sensor network localization problem.

A magnetic resonance device designed via global optimization techniques

Abstract.In this paper we are concerned with the design of a small low-cost, low-field multipolar magnet for Magnetic Resonance Imaging with a high field uniformity. By introducing appropriate variables, the considered design problem is converted into a global optimization one. This latter problem is solved by means of a new derivative free global optimization method which is a distributed multi-start type algorithm controlled by means of a simulated annealing criterion. In particular, the proposed method employs, as local search engine, a derivative free procedure. Under reasonable assumptions, we prove that this local algorithm is attracted by global minimum points. Additionally, we show that the simulated annealing strategy is able to produce a suitable starting point in a finite number of steps with probability one.