We study a bilevel noncooperative game-theoretic model of restructured electricity markets, with locational marginal prices. Each player in this game faces a bilevel optimization problem that we remodel as a mathematical program with equilibrium constraints, MPEC. The corresponding game is an example of an EPEC, equilibrium problem with equilibrium constraints. We establish sufficient conditions for … Read more
In this paper, we propose a homogeneous model for solving monotone mixed complementarity problems over symmetric cones, by extending the results in \cite{YOSHISE04} for standard form of the problems. We show that the extended model inherits the following desirable features: (a) A path exists, is bounded and has a trivial starting point without any regularity … Read more
We propose a new approach to convex nonlinear multiobjective optimization that captures the geometry of the Pareto set by generating a discrete set of Pareto points optimally. We show that the problem of finding an optimal representation of the Pareto surface can be formulated as a mathematical program with complementarity constraints. The complementarity constraints arise … Read more
We consider a regularization proximal method with variable metric to solve the nonlinear complementarity problem (NCP) for P0-functions. We establish global convergence properties when the solution set is non empty and bounded. Furthermore, we prove, without boundedness of the solution set, that the sequence generated by the algorithm is a minimizing sequence for the implicit … Read more
Multi-leader-follower games arise when modeling competition between two or more dominant firms and lead in a natural way to equilibrium problems with equilibrium constraints (EPECs). We examine a variety of nonlinear optimization and nonlinear complementarity formulations of EPECs. We distinguish two broad cases: problems where the leaders can cost-differentiate and problems with price-consistent followers. We … Read more
The elastic-mode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first- and second-order necessary optimality conditions for the original problem are also first- and second-order points of the elastic-mode formulation. Here we study … Read more
We formulate an optimal control problem of magnetization in a ferromagnet as a mathematical program with evolutionary equilibrium constraints. The evolutionary nature of the equilibrium is due to the hysteresis behavior of the respective magnetization process. To solve the problem numerically, we adapted the implicit programming technique. The adjoint equations, needed to compute the subgradients … Read more
We present a method to compute the implied volatility of American options as a mathematical program with equilibrium constraints. The formulation we present is new, as are the convergence results we prove. The algorithm holds the promise of being practical to implement, and we demonstrate some preliminary numerical results to this end. CitationPrinceton University working … Read more
In this paper, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate piecewise structure and directional differentiability of both — the lower level equilibrium solution and objective integrant. We show almost sure convergence of optimal values, optimal solutions … Read more
We study analyticity, differentiability, and semismoothness of Lowner’s operator and spectral functions under the framework of Euclidean Jordan algebras. In particular, we show that many optimization-related classical results in the symmetric matrix space can be generalized within this framework. For example, the metric projection operator over any symmetric cone defined in a Euclidean Jordan algebra … Read more