For solving a wide class of nonconvex and nonsmooth problems, we propose a proximal linearized iteratively reweighted least squares (PL-IRLS) algorithm. We first approximate the original problem by smoothing methods, and second write the approximated problem into an auxiliary problem by introducing new variables. PL-IRLS is then built on solving the auxiliary problem by utilizing the proximal l...

We introduce basic ideas of a nonsmooth Newton’s method and its application in solving semidefinite optimization (SDO) problems. In particular, the method can be used to solve both linear and nonlinear semidefinite complementarity problems. We also survey recent theoretical results in matrix functions and stability of SDO that are stemed from the research on the matrix form of the nonsmooth New...

We discuss the energy generation expansion planning with environmental constraints, formulated as a nonsmooth convex constrained optimization problem. To solve such problems, methods suitable for constrained nonsmooth optimization need to be employed. We describe a recently developed approach, which applies the usual unconstrained bundle techniques to a dynamically changing “improvement functio...

In this paper we study a nonlinear Dirichlet elliptic differential equation driven by the p-Laplacian and with a nonsmooth potential. The hypotheses on the nonsmooth potential allow resonance with respect to the principal eigenvalue λ1 > 0 of (−∆p,W 1,p 0 (Z)). We prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal.

In this paper, we introduce the concept of a (weak) minimizer of order k for a nonsmooth vector optimization problem over cones. Generalized classes of higher-order cone-nonsmooth (F, ρ)convex functions are introduced and sufficient optimality results are proved involving these classes. Also, a unified dual is associated with the considered primal problem, and weak and strong duality results ar...

In this paper, the existence, multiplicity and noexistence of positive solutions for a class of semipositone discrete boundary value problems with two parameters is studied by applying nonsmooth critical point theory and sub-super solutions method. Keywords—Discrete boundary value problems, nonsmooth critical point theory, positive solutions, semipositone, sub-super solutions method

This paper introduces the normalized and signed gradient dynamical systems associated with a differentiable function. Extending recent results on nonsmooth stability analysis, we characterize their asymptotic convergence properties and identify conditions that guarantee finite-time convergence. We discuss the application of the results to consensus problems in multiagent systems and show how th...

We present a nonsmooth optimization technique for nonconvex maximum eigenvalue functions and for nonsmooth functions which are infinite maxima of eigenvalue functions. We prove global convergence of our method in the sense that for an arbitrary starting point, every accumulation point of the sequence of iterates is critical. The method is tested on several problems in feedback control synthesis.

In this paper, we prove the existence of at least two nontrivial solutions for a nonlinear elliptic problem involving p(x)-Laplacian-like operator and nonsmooth potentials. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.

An important aspect in the solution process of constraint satisfaction problems is to identify exclusion boxes which are boxes that do not contain feasible points. This paper presents a certificate of infeasibility for finding such boxes by solving a linearly constrained nonsmooth optimization problem. Furthermore, the constructed certificate can be used to enlarge an exclusion box by solving a...