An alternating linearization method with inexact data for bilevel nonsmooth convex optimization
Dan Li Li-Ping Pang Fang-Fang Guo Zun-Quan Xia
An alternating linearization method with inexact data, for the bilevel problem of minimizing a nonsmooth convex function over the optimal solution set of another nonsmooth convex problem, is presented in this paper. The problem is first approximately transformed into an unconstrained optimization with the help of a penalty function and we prove that the penalty function admits exact penalization under some mild conditions. The objective function of this unconstrained problem is the sum of two nonsmooth convex functions and in the algorithm each iteration involves solving two easily solved subproblems. It is shown that the iterative sequence converges to a solution of the original problem by parametric minimization theorem. Numerical experiments validate the theoretical convergence analysis and illustrate the implementation of the alternating linearization algorithm for this bilevel program.
keywords: bilevel programming. exact penalty function convex programming nonsmooth optimization Proximal point
The bundle scheme for solving arbitrary eigenvalue optimizations
Ming Huang Xi-Jun Liang Yuan Lu Li-Ping Pang

Optimization involving eigenvalues arises in a large spectrum of applications in various domains, such as physics, engineering, statistics and finance. In this paper, we consider the arbitrary eigenvalue minimization problems over an affine family of symmetric matrices, which is a special class of eigenvalue function--D.C. function $λ^_{l}$. An explicit proximal bundle approach for solving this class of nonsmooth, nonconvex (D.C.) optimization problem is developed. We prove the global convergence of our method, in the sense that every accumulation point of the sequence of iterates generated by the proposed algorithm is stationary. As an application, we use the proposed bundle method to solve some particular eigenvalue problems. Some encouraging preliminary numerical results indicate that our method can solve the test problems efficiently.

keywords: Eigenvalue optimization nonsmooth optimization D.C. function nonconvex optimization first-order proximal bundle method
An inexact semismooth Newton method for variational inequality with symmetric cone constraints
Shuang Chen Li-Ping Pang Dan Li
In this paper, we consider using the inexact nonsmooth Newton method to efficiently solve the symmetric cone constrained variational inequality (VISCC) problem. It red provides a unified framework for dealing with the variational inequality with nonlinear constraints, variational inequality with the second-order cone constraints, and the variational inequality with semidefinite cone constraints. We get convergence of the above method and apply the results to three special types symmetric cones.
keywords: Jordan algebra. Variational inequality Newton method semismooth symmetric cone
Two approaches for solving mathematical programs with second-order cone complementarity constraints
Xi-De Zhu Li-Ping Pang Gui-Hua Lin
This paper considers a mathematical program with second-order cone complementarity constrains (MPSOCC). We present two approximation methods for solving the MPSOCC. One employs some smoothing functions to approximate the MPSOCC and the other makes use of some techniques to relax the complementarity constrains in the MPSOCC. We investigate the limiting behavior of both methods. In particular, we show that, under mild conditions, any accumulation point of stationary points of the approximation problems must be a Clarke-type stationary point of the MPSOCC.
keywords: convergence. Clarke-type stationarity smoothing function MPSOCC relaxation

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