in this paper, using clarke’s generalized directional derivative and di-invexity we introduce new concepts of nonsmooth k-α-di-invex and generalized type i univex functions over cones for a nonsmooth vector optimization problem with cone constraints. we obtain some sufficient optimality conditions and mond-weir type duality results under the foresaid generalized invexity and type i cone-univexi...

In this paper, using Clarke’s generalized directional derivative and dI-invexity we introduce new concepts of nonsmooth K-α-dI-invex and generalized type I univex functions over cones for a nonsmooth vector optimization problem with cone constraints. We obtain some sufficient optimality conditions and Mond-Weir type duality results under the foresaid generalized invexity and type I cone-univexi...

In this work we consider the stochastic minimization of nonsmooth convex loss functions, a central problem in machine learning. We propose a novel algorithm called Accelerated Nonsmooth Stochastic Gradient Descent (ANSGD), which exploits the structure of common nonsmooth loss functions to achieve optimal convergence rates for a class of problems including SVMs. It is the first stochastic algori...

It has long been known that variational inequality problems can be reformulated as nonsmooth equations. Recently, locally high-order convergent nonsmooth Newton methods for nonsmooth equations have been well established via the concept of semismoothness. In this paper, we focus our discussions on a way of globalizing nonsmooth Newton methods based on a smoothing-nonsmooth reformulation of nonsm...

This paper investigates the potential behavior, both good and bad, of the well-known BFGS algorithm for smooth minimization, when applied to nonsmooth functions. We consider three very particular examples. We first present a simple nonsmooth example, illustrating how BFGS (in this case with an exact line search) typically succeeds despite nonsmoothness. We then study, computationally, the behav...

We consider minimizing a nonsmooth objective subject to nonsmooth constraints. The nonsmooth functions are approximated by a bundle of subgradients. The novel idea of a filter is used to promote global convergence.

In this paper, we introduce a new class of nonsmooth pseudo-invex and nonsmooth quasi-invex functions to non-smooth variational problems. By using these concepts, numbers of necessary and sufficient conditions are established for a nonsmooth variational problem wherein Clarke’s generalized gradient is used. Also, weak, strong and converse duality are established. Keywords—Variational problem; N...

It is shown that the existence of a continuous controlLyapunov function (CLF) is necessary and sufficient for null asymptotic controllability of nonlinear finitedimensional control systems. The CLF condition is expressed in terms of a concept of generalized derivative that has been studied in set-valued analysis and the theory of differential inclusions with various names such as “upper conting...