Computation of Approximate Welfare-Maximizing Correlated Equilibria and Pareto-Optima with Applications to Wireless Communication

In a wireless application with multiple communication links, the data rate of each link is subject to degradation due to transmitting interference from other links. A competitive wireless game then arises as each link acts as a player maximizing its own data rate. The game outcome can be evaluated using the solution concept of game equilibria. However, when significant interference among the links arises, uniqueness of equilibrium is not guaranteed. To select among multiple equilibria, the sum of network rate or social welfare is used as the selection criterion. This thesis aims to offer the theoretical foundation and the computational tool for determining approximate correlated equilibria with global maximum expected social welfare in polynomial games. Using sum of utilities as the global objective, we give two theoretical and two wireless-specific contributions. 1. We give a problem formulation for computing near-exact ε -correlated equilibria with highest possible expected social welfare. We then give a sequential Semidefinite Programming (SDP) algorithm that computes the solution. The solution consists of bounds information on the social welfare. 2. We give a novel reformulation to arrive at a leaner problem for computing near-exact ε -correlated equilibria using Kantorovich polynomials with sparsity. 3. Forgoing near-exactness, we consider approximate correlated equilibria. To account for the loss in precision, we introduce the notion of regret. We give theoretical bounds on the regrets at any iteration of the sequential SDP algorithm. Moreover, we give a heuristic procedure for extracting a discrete probability distribution. Subject to players’ acceptance of the regrets, the computed distributions can be used to implement central arbitrators to facilitate real-life implementation of the correlated equilibrium concept. 4. We demonstrate how to compute Pareto-optimal solutions by dropping the correlated equilibria constraints. For demonstration purpose, we focus only on Pareto-optima with equal weights among the players

A cutting-plane method for Mixed-Logical Semidefinite Programs with an application to multi-vehicle robust path planning

The usual approach to dealing with Mixed Logical Semidefinite Programs (MLSDPs) is through the “Big-M” or the convex hull reformulation. The Big-M approach is appealing for its ease of modeling, but it leads to weak convex relaxations when used in a Branch & Bound framework. The convex hull reformulation, on the other hand, introduces a significant number of auxiliary variables and constraints and is only applicable if the feasible region consists of several disjunctive bounded polyhedra. This paper aims to circumvent these shortcomings by leveraging on Combinatorial Benders Cuts due to Codato & Fischetti and by constructing linear cuts based on a Farkas Lemma for Semidefinite Programming (SDP) within a Cutting-Plane framework. We employ the resulting Cutting-Plane algorithm in a Robust Model Predictive Control (RMPC) test application for multi-vehicle robust path planning with obstacle and inter-vehicle collision avoidance, taking into consideration exogenous (eg external wind gusts) and endogenous (eg internal noise in the system gain) uncertainty. We formulate this problem as an MLSDP model using minimax approaches by Löfberg and by El Ghaoui et al. and Big-M formulations due to Richards & How