Using EPECs to model bilevel games in restructured electricity markets with locational prices

CWPE0619 (EPRG0602) Xinmin Hu and Daniel Ralph (Feb 2006) Using EPECs to model bilevel games in restructured electricity markets with locational prices We study a bilevel noncooperative game-theoretic model of electricity markets with locational marginal prices. Each player faces a bilevel optimization problem that we remodel as a mathematical program with equilibrium constraints, MPEC. This gives an EPEC, equilibrium problem with equilibrium constraints. We establish sufficient conditions for existence of pure strategy Nash equilibria for this class of bilevel games and give some applications. We show by examples the effect of network transmission limits, i.e. congestion, on existence of equilibria. Then we study, for more general EPECs, the weaker pure strategy concepts of local Nash and Nash stationary equilibria. We model the latter via complementarity problems, CPs. Finally, we present numerical examples of methods that attempt to find local Nash or Nash stationary equilibria of randomly generated electricity market games. The CP solver PATH is found to be rather effective in this context

Equitable Efficiency in Multiple Criteria Optimization

Equitable efficiency in multiple criteria optimization was introduced mathematically in the middle of nineteen-nineties. The concept tends to strengthen the notion of Pareto efficiency by imposing additional conditions on the preference structure defining the Pareto preference. It is especially designed to solve multiple criteria problems having commensurate criteria where different criteria values can be compared directly. In this dissertation we study some theoretical and practical aspects of equitably efficient solutions. The literature on equitable efficiency is not very extensive and provides very limited number of ways of generating such solutions. After introducing some relevant notations, we develop some scalarization based methods of generating equitably efficient solutions. The scalarizations developed do not assume any special structure of the problem. We prove an existence result for linear multiple criteria problems. Next, we show how equitably efficient solutions arise in the context of a particular type of linear complementarity problem and matrix games. The set of equitably efficient solutions, in general, is a subset of efficient solutions. The multiple criteria alternative of the linear complementarity problem dealt in our dissertation has identical efficient and equitably efficient solution sets. Finally, we demonstrate the relevance of equitable efficiency by applying it to the problem of regression analysis and asset allocation

On the local stability of semidefinite relaxations

We consider a parametric family of quadratically constrained quadratic programs (QCQP) and their associated semidefinite programming (SDP) relaxations. Given a nominal value of the parameter at which the SDP relaxation is exact, we study conditions (and quantitative bounds) under which the relaxation will continue to be exact as the parameter moves in a neighborhood around the nominal value. Our framework captures a wide array of statistical estimation problems including tensor principal component analysis, rotation synchronization, orthogonal Procrustes, camera triangulation and resectioning, essential matrix estimation, system identification, and approximate GCD. Our results can also be used to analyze the stability of SOS relaxations of general polynomial optimization problems.Comment: 23 pages, 3 figure