Given a closed convex cone P with nonempty interior in a locally convex vector space, and a set A ⊂ Y , we provide various equivalences to the implication A ∩ (−int P ) = ∅ =⇒ co(A) ∩ (−int P ) = ∅, among them, to the pointedness of cone(A + int P ). This allows us to establish an optimal alternative theorem, suitable for vector optimization problems. In addition, we characterize the two-dimens...

This paper introduces to constructing problems of convex relaxations for nonconvex polynomial optimization problems. Branch-and-bound algorithms are convex relaxation based. The convex envelopes are of primary importance since they represent the uniformly best convex underestimators for nonconvex polynomials over some region. The reformulationlinearization technique (RLT) generates LP (linear p...

We present the online Newton's method, a single-step second-order method for nonconvex optimization. analyze its performance and obtain dynamic regret bound that is linear in cumulative variation between round optima. show if optima limited, leads to constant bound. In general case, outperforms convex optimization algorithms functions performs similarly specialized algorithm strongly functions....

In this paper, we derive a portfolio optimization model by minimizing upper and lower bounds of loss probability. These bounds are obtained under a nonparametric assumption of underlying return distribution by modifying the so-called generalization error bounds for the support vector machine, which has been developed in the field of statistical learning. Based on the bounds, two fractional prog...

Abstract In this paper, the class of differentiable semi-infinite multiobjective programming problems with vanishing constraints is considered. Both Karush–Kuhn–Tucker necessary optimality conditions and, under appropriate invexity hypotheses, sufficient are proved for such nonconvex smooth vector optimization problems. Further, duals in sense Mond–Weir defined considered and several duality re...

  Semidefinite optimization relaxations are among the widely used approaches to find global optimal or approximate solutions for many nonconvex problems. Here, we consider a specific quadratically constrained quadratic problem with an additional linear constraint. We prove that under certain conditions the semidefinite relaxation approach enables us to find a global optimal solution of the unde...