Weakly homogeneous variational inequalities and solvability of nonlinear equations over cones

Given a closed convex cone C in a finite dimensional real Hilbert space H, a weakly homogeneous map f:C–>H is a sum of two continuous maps h and g, where h is positively homogeneous of (positive) degree gamma on C and g(x)/||x||^gamma–>0 as ||x||–>infinity in C. Given such a map f, a nonempty closed convex … Read more

Uniform nonsingularity and complementarity problems over symmetric cones

We study the uniform nonsingularity property recently proposed by the authors and present its applications to nonlinear complementarity problems over a symmetric cone. In particular, by addressing theoretical issues such as the existence of Newton directions, the boundedness of iterates and the nonsingularity of B-subdifferentials, we show that the non-interior continuation method proposed by Xin … Read more

Homogeneous algorithms for monotone complementarity problems over symmetric cones

In \cite{aYOSHISE06}, the author proposed a homogeneous model for standard monotone nonlinear complementarity problems over symmetric cones and show that the following properties hold: (a) There is a path that is bounded and has a trivial starting point without any regularity assumption concerning the existence of feasible or strictly feasible solutions. (b) Any accumulation point … Read more

Using EPECs to model bilevel games in restructured electricity markets

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

A Homogeneous Model for Mixed Complementarity Problems over Symmetric Cones

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

Interior Point Trajectories and a Homogeneous Model for Nonlinear Complementarity Problems over Symmetric Cones

We study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones. While the analysis in Faybusovich (1997) depends on the optimization theory of convex log-barrier functions, our approach is based on the paper of Monteiro and Pang (1998), where a vast set of conclusions concerning continuous trajectories is shown for monotone … Read more

Extension of Quasi-Newton Methods to Mathematical Programs with Complementarity Constraints

Quasi-Newton methods in conjunction with the piecewise sequential quadratic programming are investigated for solving mathematical programming with equilibrium constraints, in particular for problems with complementarity constraints. Local convergence as well as superlinear convergence of these quasi-Newton methods can be established under suitable assumptions. In particular, several well-known quasi-Newton methods such as BFGS and DFP are … Read more