Approximate optimality conditions for a class of nonconvex semi-infinite programs involving support functions are given. The objective function and the constraint functions are locally Lipschitz functions on n . By using a Karush-Kuhn-Tucker KKT condition, we deduce a necessary optimality condition for local approximate solutions. Then, generalized KKT conditions for the problems are proposed. ...

We consider the problem of computing a critical point of a nonconvex locally Lipschitz function over a convex compact constraint set given an inexact oracle that provides an approximate function value and an approximate subgradient. We assume that the errors in function and subgradient evaluations are merely bounded, and in particular need not vanish in the limit. After some discussion on how t...

The notion of a secant for locally Lipschitz continuous functions is introduced and a new algorithm to locally minimize nonsmooth, nonconvex functions based on secants is developed. We demonstrate that the secants can be used to design an algorithm to find descent directions of locally Lipschitz continuous functions. This algorithm is applied to design a minimization method, called a secant met...

We show that local minimizers of functionals of the form Z Ω [f(Du(x)) + g(x , u(x))] dx, u ∈ u0 + W 1,p 0 (Ω), are locally Lipschitz continuous provided f is a convex function with p − q growth satisfying a condition of qualified convexity at infinity and g is Lipschitz continuous in u. As a consequence of this, we obtain an existence result for a related nonconvex functional.

We prove the existence of solutions to the differential inclusion ẍ(t) ∈ F (x(t), ẋ(t)) + f(t, x(t), ẋ(t)), x(0) = x0, ẋ(0) = y0, where f is a Carathéodory function and F with nonconvex values in a Hilbert space such that F (x, y) ⊂ γ(∂g(y)), with g a regular locally Lipschitz function and γ a linear operator.

We establish optimal local regularity results for vector-valued extremals and minimizers of variational integrals whose integrand is the squared distance function to a compact setK in matrix space MN×n. The optimality is illustrated by explicit examples showing that, in the nonconvex case, minimizers need not be locally Lipschitz. This is in contrast to the case when the set K is suitably conve...

‎By p-power (or partial p-power) transformation‎, ‎the Lagrangian function in nonconvex optimization problem becomes locally convex‎. ‎In this paper‎, ‎we present a neural network based on an NCP function for solving the nonconvex optimization problem‎. An important feature of this neural network is the one-to-one correspondence between its equilibria and KKT points of the nonconvex optimizatio...

We present a random perturbation of the projected variable metric method for solving linearly constrained nonsmooth (i.e., nondifferentiable) nonconvex optimization problems, and we establish the convergence to a global minimum for a locally Lipschitz continuous objective function which may be nondifferentiable on a countable set of points. Numerical results show the effectiveness of the propos...

We consider vector optimization problems on Banach spaces without convexity assumptions. Under the assumption that the objective function is locally Lipschitz we derive Lagrangian necessary conditions on the basis of Mordukhovich subdifferential and the approximate subdifferential by Ioffe using a non-convex scalarization scheme. Finally, we apply the results for deriving necessary conditions f...