Mathematical programs with complementarity constraints: convergence properties of a smoothing method

In this paper, optimization problems $P$ with complementarity constraints are considered. Characterizations for local minimizers $\bar{x}$ of $P$ of Orders 1 and 2 are presented. We analyze a parametric smoothing approach for solving these programs in which $P$ is replaced by a perturbed problem $P_{\tau}$ depending on a (small) parameter $\tau$. We are interested in the convergence behavior of the feasible set $\cal{F}_{\tau}$ and the convergence of the solutions $\bar{x}_{\tau}$ of $P_{\tau}$ for $\tau\to 0.$ In particular, it is shown that, under generic assumptions, the solutions $\bar{x}_{\tau}$ are unique and converge to a solution $\bar{x}$ of $P$ with a rate $\cal{O}(\sqrt{\tau})$. Moreover, the convergence for the Hausdorff distance $d(\cal{F}_{\tau}$, $\cal{F})$ between the feasible sets of $P_{\tau}$ and $P$ is of order $\cal{O}(\sqrt{\tau})$

Solving Mathematical Programs with Equilibrium Constraints as Nonlinear Programming: A New Framework

We present a new framework for the solution of mathematical programs with equilibrium constraints (MPECs). In this algorithmic framework, an MPECs is viewed as a concentration of an unconstrained optimization which minimizes the complementarity measure and a nonlinear programming with general constraints. A strategy generalizing ideas of Byrd-Omojokun's trust region method is used to compute steps. By penalizing the tangential constraints into the objective function, we circumvent the problem of not satisfying MFCQ. A trust-funnel-like strategy is used to balance the improvements on feasibility and optimality. We show that, under MPEC-MFCQ, if the algorithm does not terminate in finite steps, then at least one accumulation point of the iterates sequence is an S-stationary point

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

Sequential Convex Programming Methods for Solving Nonlinear Optimization Problems with DC constraints

This paper investigates the relation between sequential convex programming (SCP) as, e.g., defined in [24] and DC (difference of two convex functions) programming. We first present an SCP algorithm for solving nonlinear optimization problems with DC constraints and prove its convergence. Then we combine the proposed algorithm with a relaxation technique to handle inconsistent linearizations. Numerical tests are performed to investigate the behaviour of the class of algorithms.Comment: 18 pages, 1 figur