In this paper a new algorithm to locally minimize nonsmooth, nonconvex functions is developed. We introduce the notion of secants and quasisecants for nonsmooth functions. The quasisecants are applied to find descent directions of locally Lipschitz functions. We design a minimization algorithm which uses quasisecants to find descent directions. We prove that this algorithm converges to Clarke s...

We consider the minimization of a nonsmooth convex function over a compact convex set subject to a nonsmooth convex constraint. We work in the setting of derivative-free optimization (DFO), assuming that the objective and constraint functions are available through a black-box that provides function values for lower-C2 representation of the functions. Our approach is based on a DFO adaptation of...

Steepest descent is central in variational mathematics. We present a new transparent existence proof for curves of near-maximal slope — an influential notion of steepest descent in a nonsmooth setting. We moreover show that for semi-algebraic functions — prototypical nonpathological functions in nonsmooth optimization — such curves are precisely the solutions of subgradient dynamical systems.

Nonconvex nonsmooth potential functions have superior restoration performance for the images with neat boundaries. However, several difficulties emerge from the numerical computation. Thus the graduated nonconvex (GNC) method is suggested to deal with these problems. To improve the performance of the GNC method further, a class of nonconvex nonsmooth approximate potential functions have been co...

The singular values of a rectangular matrix are nonsmooth functions of its entries. In this work we study the nonsmooth analysis of functions of singular values. In particular we give simple formulae for the regular subdifferential, the limiting subdifferential, and the horizon subdifferential, of such functions. Along the way to the main result we give several applications and in particular de...

this paper is concerned with the study of the existence of positive solutions for a navier boundaryvalue problem involving the p-biharmonic operator; the right hand side of problem is a nonsmoothfunctional with variable parameters. the existence of at least three positive solutions is establishedby using nonsmooth version of a three critical points theorem for discontinuous functions. our resul...

We study nonsmooth multiobjective programming problems involving locally Lipschitz functions and support functions. Two types of Karush-Kuhn-Tucker optimality conditions with support functions are introduced. Sufficient optimality conditions are presented by using generalized convexity and certain regularity conditions. We formulate Wolfe-type dual and Mond-Weirtype dual problems for our nonsmo...