We examine various methods for data clustering and data classification that are based on the minimization of the so-called cluster function and its modifications. These functions are nonsmooth and nonconvex. We use Discrete Gradient methods for their local minimization. We consider also a combination of this method with the cutting angle method for global minimization. We present and discuss re...

Abstract: In this paper, we generalize the (V, ρ)-invexity defined for nonsmooth multiobjective fractional programming by Mishra, Rueda and Giorgi (2003) to variational programming problems by defining new classes of vector-valued functions called (V, ρ)B-type I and generalized (V, ρ)-B-type I. Then we use these new classes to derive various sufficient optimality conditions and mixed type duali...

We propose a descent method via gap functions for solving nonsmooth variational inequalities with a locally Lipschitz operator. Assuming monotone operator (not necessarily strongly monotone) and bounded domain, we show that the method with an Armijo-type line search is globally convergent. Finally, we report some numerical experiments.

The main result of this note is an external stabilizability theorem for discontinuous systems affine in the control (with solutions intended in the Filippov’s sense). In order to get it we first prove a sufficient condition for external stability which makes use of nonsmooth Lyapunov-like functions.

The family of optimization problems with value function objectives includes the minmax programming problem and the bilevel optimization problem. In this paper, we derive necessary optimality conditions for this class of problems. The main focus is on the case where the functions involved are nonsmooth and the constraints are the very general operator constraints.

We consider a nonsmooth semi-infinite interval-valued vector programming problem, where the objectives and constraint functions need not to be locally Lipschitz. Using Abadie’s qualification convexificators, we provide Karush–Kuhn–Tucker necessary optimality conditions by converting initial problem into bi-criteria optimization problem. Furthermore, establish sufficient under asymptotic convexi...

Nonsmooth semidefinite programming problems is concerned in this paper where the objective functions is quasidifferentiable. Necessary optimality conditions for quasidifferentiable optimization is proposed. The advantage of Kuhn-Tucker necessary optimality condition in terms of quasidifferential over that in terms of Clarke generalized gradient, is discussed.

The paper reports on a recent construction ofM -functions and Krĕın resolvent formulas for general closed extensions of an adjoint pair, and their implementation to boundary value problems for second-order strongly elliptic operators on smooth domains. The results are then extended to domains with C Hölder smoothness, by use of a recently developed calculus of pseudodifferential boundary operat...