Jacobian Consistency of a Smoothing Function for the Weighted Second-Order Cone Complementarity Problem

In this paper, a weighted second-order cone (SOC) complementarity function and its smoothing function are presented. Then, we derive the computable formula for the Jacobian of the smoothing function and show its Jacobian consistency. Also, we estimate the distance between the subgradient of the weighted SOC complementarity function and the gradient of its smoothing function. These results will be critical to achieve the rapid convergence of smoothing methods for weighted SOC complementarity problems.

A nonmonotone inexact smoothing Newton-type method for P0-NCP based on a parametric complementarity function

In this paper, nonlinear complementarity problem with P

Simulation and Analysis of Dynamic Evolution of Electricity Market Based on Bidding Decisions with Heterogeneous Expectations

Based on different bidding decisions with heterogeneous expectations of market participants, a dynamic model of electricity market considering power network constraints is proposed. This model is represented by a discrete difference equations embedded with the optimization problem of market clearing. By using the nonlinear complementarity function, the complex dynamic behaviors of electricity market are simulated and analyzed. The Nash equilibrium and its stability, the periodic and even chaotic dynamic behaviors beyond the stability region of Nash equilibrium are investigated.

VECTOR-VALUED IMPLICIT LAGRANGIAN FOR SYMMETRIC CONE COMPLEMENTARITY PROBLEMS

The implicit Lagrangian was first proposed by Mangasarian and Solodov as a smooth merit function for the nonnegative orthant complementarity problem. It has attracted much attention in the past ten years because of its utility in reformulating complementarity problems as unconstrained minimization problems. In this paper, exploiting the Jordan-algebraic structure, we extend it to the vector-valued implicit Lagrangian for symmetric cone complementary problem (SCCP), and show that it is a continuously differentiable complementarity function for SCCP and whose Jacobian is strongly semismooth. As an application, we develop the real-valued implicit Lagrangian and the corresponding smooth merit function for SCCP, and give a necessary and sufficient condition for the stationary point of the merit function to be a solution of SCCP. Finally, we show that this merit function can provide a global error bound for SCCP with the uniform Cartesian P-property.