Improved dc programming approaches for solving the quadratic eigenvalue complementarity problem

Abstract

In this paper, we discuss the solution of a Quadratic Eigenvalue Complementarity Problem (QEiCP) by using Difference of Convex (DC) programming approaches. We first show that QEiCP can be represented as dc programming problem. Then we investigate different dc programming formulations of QEiCP and discuss their dc algorithms based on a well-known method -DCA. A new local dc decomposition is proposed which aims at constructing a better dc decomposition regarding to the specific feature of the target problem in some neighborhoods of the iterates. This new procedure yields faster convergence and better precision of the computed solution. Numerical results illustrate the efficiency of the new dc algorithms in practice.

On Difference-of-SOS and Difference-of-Convex-SOS Decompositions for Polynomials

“…The Holy Grail which motivates the research in this paper is to solve polynomial optimization via DC programming approaches. As an application, we have applied in [29] our DCSOS decomposition Algorithm 4.2 to solve the quadratic eigenvalue complementarity problem (a fourthordered polynomial optimization) using DCA, the numerical results demonstrated a good performance of DCA on both the convergence rate and the quality of the computed solutions which outperformed many existing methods. The next step, we are going to develop DC programming algorithms for solving general polynomial optimization (involving convex constrained polynomial optimization and nonconvex polynomial constrained polynomial optimization).…”

“…Another open question is how to find the best DC decomposition for DCA. We have understood by now in our recent work [29] that a best DC decomposition for DCA must be an undominated DC decomposition whose DC components are undominated convex functions. However, how to generate undominated DC decomposition for polynomials is still an open question which requires more investigations in future.…”

“…This decomposition requires computing an upper bound for the spectral radius of the Hessian matrix of f over the convex set D, which can be easily computed for some special structured D such as a standard simplex or a box. They applied this decomposition to higher moment portfolio optimization [36] and eigenvalue complementarity problems [29,30,31]. This approach provides an unified DC decomposition method for any real valued C 2 function with bounded spectral radius of its Hessian matrix.…”

“…Meanwhile, Y.S. Niu et al proposed in [29] a DC decomposition in form of difference-of-convexsos polynomials for quadratic eigenvalue complementarity problem. This new DC decomposition yields better numerical results in the quality of the computed solution and the computing time than the previous techniques in [30,31].…”

High-order Moment Portfolio Optimization via An Accelerated Difference-of-Convex Programming Approach and Sums-of-Squares

“…The reason is that when ρ is too big, the DC components ρ 2 x 2 and ρ 2 x 2 −f (x) are more convex. We have proved in (Niu, 2018;Niu et al, 2019) that a better DC decomposition must be an undominated DC decomposition whose DC components should be less convex as possible.…”