Homotopy Methods to Compute Equilibria in Game Theory

This paper presents a complete survey of the use of homotopy methods in game theory.Homotopies allow for a robust computation of game-theoretic equilibria and their refinements. Homotopies are also suitable to compute equilibria that are selected by variousselection theories. We present all relevant techniques underlying homotopy algorithms.We give detailed expositions of the Lemke-Howson algorithm and the Van den Elzen-Talman algorithm to compute Nash equilibria in 2-person games, and the Herings-Vanden Elzen, Herings-Peeters, and McKelvey-Palfrey algorithms to compute Nash equilibriain general n-person games.operations research and management science;

Hierarchical Structures in Equilibrium Problems

Department of Probability and Mathematical StatisticsKatedra pravděpodobnosti a matematické statistikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Robot Motion Planning Under Topological Constraints

My thesis addresses the the problem of manipulation using multiple robots with cables. I study how robots with cables can tow objects in the plane, on the ground and on water, and how they can carry suspended payloads in the air. Specifically, I focus on planning optimal trajectories for robots. Path planning or trajectory generation for robotic systems is an active area of research in robotics. Many algorithms have been developed to generate path or trajectory for different robotic systems. One can classify planning algorithms into two broad categories. The first one is graph-search based motion planning over discretized configuration spaces. These algorithms are complete and quite efficient for finding optimal paths in cluttered 2-D and 3-D environments and are widely used [48]. The other class of algorithms are optimal control based methods. In most cases, the optimal control problem to generate optimal trajectories can be framed as a nonlinear and non convex optimization problem which is hard to solve. Recent work has attempted to overcome these shortcomings [68]. Advances in computational power and more sophisticated optimization algorithms have allowed us to solve more complex problems faster. However, our main interest is incorporating topological constraints. Topological constraints naturally arise when cables are used to wrap around objects. They are also important when robots have to move one way around the obstacles rather than the other way around. Thus I consider the optimal trajectory generation problem under topological constraints, and pursue problems that can be solved in finite-time, guaranteeing global optimal solutions. In my thesis, I first consider the problem of planning optimal trajectories around obstacles using optimal control methodologies. I then present the mathematical framework and algorithms for multi-robot topological exploration of unknown environments in which the main goal is to identify the different topological classes of paths. Finally, I address the manipulation and transportation of multiple objects with cables. Here I consider teams of two or three ground robots towing objects on the ground, two or three aerial robots carrying a suspended payload, and two boats towing a boom with applications to oil skimming and clean up. In all these problems, it is important to consider the topological constraints on the cable configurations as well as those on the paths of robot. I present solutions to the trajectory generation problem for all of these problems

Computing equilibria in general equilibrium models via interior-point method

In this paper we study new computational methods to find equilibria in general equilibrium models. We first survey the algorithms to compute equilibria that can be found in the literature on computational economics and we indicate how these algorithms can be improved from the computational point of view. We also provide alternative algorithms that are able to compute the equilibria in an efficient manner even for large-scale models, based on interior-point methods. We illustrate the proposed methods with some examples taken from the literature on general equilibrium modelsPublicad

A globally convergent neurodynamics optimization model for mathematical programming with equilibrium constraints

summary:This paper introduces a neurodynamics optimization model to compute the solution of mathematical programming with equilibrium constraints (MPEC). A smoothing method based on NPC-function is used to obtain a relaxed optimization problem. The optimal solution of the global optimization problem is estimated using a new neurodynamic system, which, in finite time, is convergent with its equilibrium point. Compared to existing models, the proposed model has a simple structure, with low complexity. The new dynamical system is investigated theoretically, and it is proved that the steady state of the proposed neural network is asymptotic stable and global convergence to the optimal solution of MPEC. Numerical simulations of several examples of MPEC are presented, all of which confirm the agreement between the theoretical and numerical aspects of the problem and show the effectiveness of the proposed model. Moreover, an application to resource allocation problem shows that the new method is a simple, but efficient, and practical algorithm for the solution of real-world MPEC problems

MPCC: Strong stability of M-stationary points

In this paper we study the class of mathematical programs with complementarity constraints MPCC. Under the Linear Independence constraint qualification MPCC-LICQ we state a topological as well as an equivalent algebraic characterization for the strong stability (in the sense of Kojima) of an M-stationary point for MPCC. By allowing perturbations of the describing functions up to second order, the concept of strong stability refers here to the local existence and uniqueness of an M-stationary point for any sufficiently small perturbed problem where this unique solution depends continuously on the perturbation. Finally, some relations to S- and C-stationarity are briefly discussed.publishedVersio

Numerical Techniques for Optimization Problems with PDE Constraints

The development, analysis and implementation of efficient and robust numerical techniques for optimization problems associated with partial differential equations (PDEs) is of utmost importance for the optimal control of processes and the optimal design of structures and systems in modern technology. The successful realization of such techniques invokes a wide variety of challenging mathematical tasks and thus requires the application of adequate methodologies from various mathematical disciplines. During recent years, significant progress has been made in PDE constrained optimization both concerning optimization in function space according to the paradigm ’Optimize first, then discretize’ and with regard to the fast and reliable solution of the large-scale problems that typically arise from discretizations of the optimality conditions. The contributions at this Oberwolfach workshop impressively reflected the progress made in the field. In particular, new insights have been gained in the analysis of optimal control problems for PDEs that have led to vastly improved numerical solution methods. Likewise, breakthroughs have been made in the optimal design of structures and systems, for instance, by the socalled ’all-at-once’ approach featuring simultaneous optimization and solution of the underlying PDEs. Finally, new methodologies have been developed for the design of innovative materials and the identification of parameters in multi-scale physical and physiological processes