\(\) In this work, we address a class of nonconvex nonsmooth optimization problems where the objective function is the sum of two smooth functions (one of which is proximable) and two nonsmooth functions (one weakly convex and proximable and the other continuous and weakly concave). We introduce a new splitting algorithm that extends the Davis-Yin … Read more
\(\) This paper considers the problem of minimizing the sum of a smooth function and the Schatten-\(p\) norm of the matrix. Our contribution involves proposing accelerated iteratively reweighted nuclear norm methods designed for solving the nonconvex low-rank minimization problem. Two major novelties characterize our approach. Firstly, the proposed method possesses a rank identification property, enabling … Read more
We first consider the convex composite optimization models without globally Lipschitz condition imposed on the gradient of the differentiable term. The classical method which is proximal gradient will be studied with our new strategy of stepsize selection. The idea for constructing such a stepsize is motivated by the one in \cite{hoai} that used for gradient … Read more
Goldstein’s 1977 idealized iteration for minimizing a Lipschitz objective fixes a distance – the step size – and relies on a certain approximate subgradient. That “Goldstein subgradient” is the shortest convex combination of objective gradients at points within that distance of the current iterate. A recent implementable Goldstein-style algorithm allows a remarkable complexity analysis (Zhang … Read more
In nonsmooth, nonconvex stochastic optimization, understanding the uniform convergence of subdifferential mappings is crucial for analyzing stationary points of sample average approximations of risk as they approach the population risk. Yet, characterizing this convergence remains a fundamental challenge. This work introduces a novel perspective by connecting the uniform convergence of subdifferential mappings to that of subgradient … Read more
In this paper we propose an approach for solving systems of nonlinear equations without computing function derivatives. Motivated by the application area of tomographic absorption spectroscopy, which is a highly-nonlinear problem with variables coupling, we consider a situation where straightforward translation to a fixed point problem is not possible because the operators that represent the … Read more
We analyze the Douglas-Rachford splitting method for weakly convex optimization problems, by the token of the Douglas-Rachford envelope, a merit function akin to the Moreau envelope. First, we use epi-convergence techniques to show that this artifact approximates the original objective function via epigraphs. Secondly, we present how global convergence and local linear convergence rates for … Read more
\(\) This paper investigates constrained nonsmooth multiobjective fractional programming problem (NMFP) in real Banach spaces. It derives a quotient calculus rule for computing the first- and second-order Clarke derivatives of fractional functions involving locally Lipschitz functions. A novel second-order Abadie-type regularity condition is presented, defined with the help of the Clarke directional derivative and the … Read more
\(\) This paper is devoted to study the convergence rates of a second-order dynamical system and its corresponding discretizations associated with a nonsmooth bilinearly coupled convex-concave saddle point problem. We derive the convergence rate of the primal-dual gap for the second-order dynamical system with asymptotically vanishing damping term. Based on the implicit discretization, we propose … Read more
Recent works have developed new projection-free first-order methods based on utilizing linesearches and normal vector computations to maintain feasibility. These oracles can be cheaper than orthogonal projection or linear optimization subroutines but have the drawback of requiring a known strictly feasible point to do these linesearches with respect to. In this work, we develop new … Read more