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.
To evaluate system capacity, past works on MultipleInput-Multiple-Output (MIMO) systems with mutually interfering links have focused on maximizing the sum of mutual information as the objective criterion. Since the ultimate goal of a MIMO system is to support network applications used by consumers, we consider the non-concave sigmoid utility function, which is the recommended choice according to Shenker for modeling consumer satisfaction in applications with inelastic network traffic. We formulate the sum of utilities maximization as a global optimization problem with polynomial constraints and a rational objective function. Using a technique known as moment relaxation, we derive a sequence of Semidefinite Programming (SDP) problems whose optimal objective values converge to the global maximum sum of utilities. In our simulation examples, we employ our optimization model to determine the average global maxima sum of utilities by optimizing the covariance matrices of the transmitters. We then compare the results with those attainable by the alternative non-uniform optimal power control model that optimizes only the eigenvalues of the covariance matrices. By examining performance differences between the two models, we obtain insights about how interference and excessive data-rate requirements imposed by the application can impede link-consumers' ability to maximize their sum of utilities.Index Terms-MIMO, non-concave sigmoid utility, global optimization of rational function, moment relaxation, semidefinite programming.
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