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

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.

Ekeland-type variational principle with applications to nonconvex minimization and equilibrium problems

The aim of the present paper is to establish a variational principle in metric spaces without assumption of completeness when the involved function is not lower semicontinuous. As consequences, we derive many fixed point results, nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem in noncomplete metric spaces. Examples are also given to illustrate and to show that obtained results are proper generalizations.