A Proximal Algorithm with Convergence Guarantee for a Nonconvex Minimization Problem Based on Reproducing Kernel Hilbert Space

Liang-Jian Deng,Hong-Xia Dou

doi:10.3390/sym13122393

Abstract

The underlying function in reproducing kernel Hilbert space (RKHS) may be degraded by outliers or deviations, resulting in a symmetry ill-posed problem. This paper proposes a nonconvex minimization model with ℓ0-quasi norm based on RKHS to depict this degraded problem. The underlying function in RKHS can be represented by the linear combination of reproducing kernels and their coefficients. Thus, we turn to estimate the related coefficients in the nonconvex minimization problem. An efficient algorithm is designed to solve the given nonconvex problem by the mathematical program with equilibrium constraints (MPEC) and proximal-based strategy. We theoretically prove that the sequences generated by the designed algorithm converge to the nonconvex problem’s local optimal solutions. Numerical experiment also demonstrates the effectiveness of the proposed method.

Highlights

The reproducing kernel Hilbert space (RKHS, denote as H) has been widely studied in many studies [1,2,3,4,5,6,7]
We proposed a new nonconvex modeling to deal with a challenging symmetry ill-posed problem
An efficient algorithm for solving the given nonconvex problem is designed by mathematical program with equilibrium constraints (MPEC) and the proximal-based regularization

Summary

Introduction

The reproducing kernel Hilbert space (RKHS, denote as H) has been widely studied in many studies [1,2,3,4,5,6,7]. In [8], Papageorfiou et al, considered that the function g can be linearly represented by coefficient α and constant c as g = Kα + c1, and proposed a kernel regularized orthogonal maching pursuit (KROMP) method to solve the nonconvex problem. This paper mainly establishes the nonconvex optimization model for the degraded problem (2) in RKHS, and gives the designed algorithm whose convergence can be guaranteed, shows the effectiveness of the proposed method in some simulation experiments. We mainly focus on the above mentioned difficulties of nonconvex minimization problem (3) to design an efficient algorithm with convergence guarantee theoretically. New nonconvex minimization modeling based on RKHS; (2) Convergence guarantee of the designed algorithm for the nonconvex problem.

The Solution for the Nonconvex Minimization Problem

Convergence Analysis

Numerical Result

Conclusions