Douglas-Rachford splitting for nonconvex feasibility problems

We adapt the Douglas-Rachford (DR) splitting method to solve nonconvex feasibility problems by studying this method for a class of nonconvex optimization problem. While the convergence properties of the method for convex problems have been well studied, far less is known in the nonconvex setting. In this paper, for the direct adaptation of the method … Read more

Global convergence of splitting methods for nonconvex composite optimization

We consider the problem of minimizing the sum of a smooth function $h$ with a bounded Hessian, and a nonsmooth function. We assume that the latter function is a composition of a proper closed function $P$ and a surjective linear map $\M$, with the proximal mappings of $\tau P$, $\tau > 0$, simple to compute. … Read more

Trust-Region Problems with Linear Inequality Constraints: Exact SDP Relaxation, Global Optimality and Robust Optimization

The trust-region problem, which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact semi-definite linear programming relaxation (SDP-relaxation) and strong duality. Unfortunately, such properties do not, in general, hold for an extended trust-region problem having … Read more

Robust Least Square Semidefinite Programming with Applications to Correlation Stress Testing

In this paper, we consider a least square semidefinite programming problem under ellipsoidal data uncertainty. We show that the robustification of this uncertain problem can be reformulated as a semidefinite linear programming problem with an additional second-order cone constraint. We then provide an explicit quantitative sensitivity analysis on how the solution under the robustification depends … Read more

New Fractional Error Bounds for Nonconvex Polynomial Systems with Applications to Holderian Stability in Optimization and Spectral Theory of Tensors

In this paper we derive new fractional error bounds for nonconvex polynomial systems with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. The results obtained do not require any regularity assumptions and resolve, in particular, some open questions posed in the literature. The developed techniques are … Read more

Holder Metric Subregularity with Applications to Proximal Point Method

This paper is mainly devoted to the study and applications of H\”older metric subregularity (or metric $q$-subregularity of order $q\in(0,1]$) for general set-valued mappings between infinite-dimensional spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive neighborhood and pointbased sufficient conditions as well as necessary conditions for $q$-metric subregularity with evaluating the exact … Read more