A Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity Constraints

To reflect uncertain data in practical problems, stochastic versions of the mathematical program with complementarity constraints (MPCC) have drawn much attention in the recent literature. Our concern is the detailed analysis of convergence properties of a regularization sample average approximation (SAA) method for solving a stochastic mathematical program with complementarity constraints (SMPCC). The analysis of this regularization method is carried out in three steps: First, the almost sure convergence of optimal solutions of the regularized SAA problem to that of the true problem is established by the notion of epiconvergence in variational analysis. Second, under MPCC-MFCQ, which is weaker than MPCC-LICQ, we show that any accumulation point of Karash-Kuhn-Tucker points of the regularized SAA problem is almost surely a kind of stationary point of SMPCC as the sample size tends to infinity. Finally, some numerical results are reported to show the efficiency of the method proposed

Entropic approximation for mathematical programs with robust equilibrium constraints

In this paper, we consider a class of mathematical programs with robust equilibrium constraints represented by a system of semi-infinite complementarity constraints (SIC C). We propose a numerical scheme for tackling SICC. Specific ally, by relaxing the complementarity constraints and then randomizing the index set of SICC, we employ the well-known entropic risk measure to approximate the semi-infinite onstraints with a finite number of stochastic inequality constraints. Under some moderate conditions, we quantify the approximation in term s of the feasible set and the optimal value. The approximation scheme is then applied to a class of two stage stochastic mathematical programs with complementarity constraints in combination with the polynomial decision rules. Finally, we extend the discussion to a mathematical program with distributionally robust equilibrium constraints which is essentially a one stage stochastic program with semi-infinite stochastic constraints indexed by some probability measures from an ambiguity set defined through the KL-divergence

DECENTRALIZED ALGORITHMS FOR NASH EQUILIBRIUM PROBLEMS – APPLICATIONS TO MULTI-AGENT NETWORK INTERDICTION GAMES AND BEYOND

Nash equilibrium problems (NEPs) have gained popularity in recent years in the engineering community due to their ready applicability to a wide variety of practical problems ranging from communication network design to power market analysis. There are strong links between the tools used to analyze NEPs and the classical techniques of nonlinear and combinatorial optimization. However, there remain significant challenges in both the theoretical and algorithmic analysis of NEPs. This dissertation studies certain special classes of NEPs, with the overall purpose of analyzing theoretical properties such as existence and uniqueness, while at the same time proposing decentralized algorithms that provably converge to solutions. The subclasses are motivated by relevant application examples

Optimal Energy Storage Strategies in Microgrids

Microgrids are small-scale distribution networks that provide a template for large-scale deployment of renewable energy sources, such as wind and solar power, in close proximity to demand. However, the inherent variability and intermittency of these sources can have a significant impact on power generation and scheduling decisions. Distributed energy resources, such as energy storage systems, can be used to decouple the times of energy consumption and generation, thereby enabling microgrid operators to improve scheduling decisions and exploit arbitrage opportunities in energy markets. The integration of renewable energy sources into the nation's power grid, by way of microgrids, holds great promise for sustainable energy production and delivery; however, operators and consumers both lack effective strategies for optimally using stored energy that is generated by renewable energy sources. This dissertation presents a comprehensive stochastic optimization framework to prescribe optimal strategies for effectively managing stored energy in microgrids, subject to the inherent uncertainty of renewable resources, local demand and electricity prices. First, a Markov decision process model is created to characterize and illustrate structural properties of an optimal storage strategy and to assess the economic value of sharing stored energy between heterogeneous, demand-side entities. Second, a multistage stochastic programming (MSP) model is formulated and solved to determine the optimal storage, procurement, selling and energy flow decisions in a microgrid, subject to storage inefficiencies, distribution line losses and line capacity constraints. Additionally, the well-known stochastic dual dynamic programming (SDDP) algorithm is customized and improved to drastically reduce the computation time and significantly improve solution quality when approximately solving this MSP model. Finally, and more generally, a novel nonconvex regularization scheme is developed to improve the computational performance of the SDDP algorithm for solving high-dimensional MSP models. Specifically, it is shown that these nonconvex regularization problems can be reformulated as mixed-integer programming problems with provable convergence guarantees. The benefits of this regularization scheme are illustrated by way of a computational study that reveals significant improvements in the convergence rate and solution quality over the standard SDDP algorithm and other regularization schemes