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...

We consider the following basic geometric problem: Given ∈ (0, 1/2), a 2-dimensional figure that consists of a black object and a white background is -far from convex if it differs in at least an fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is -far from convex are needed to detect a violation of convexity wi...

The purpose of this paper is to consider the set-valued optimization problem in Asplund spaces without convexity assumption. By a scalarization function introduced by Tammer and Weidner (J Optim Theory Appl 67:297–320, 1990), we obtain the Lagrangian condition for approximate solutions on set-valued optimization problems in terms of the Mordukhovich coderivative.

In the area of sparse recovery, numerous researches hint that non-convex penalties might induce better sparsity than convex ones, but up until now the non-convex algorithms lack convergence guarantee from the initial solution to the global optimum. This paper aims to provide theoretical guarantee for sparse recovery via non-convex optimization. The concept of weak convexity is incorporated into...

This article deals with the different classes of convexity and generalizations. Firstly, we reveal the new generalization of the definition of convexity that can reduce many order of convexity. We have showed features of algebra for this new convex function. Then after we have constituted Hermite-Hadamard type inequalities for this class of functions. Finally the identity has been revealed for ...

We study the problem of reconstructing finite subsets of the integer lattice Z from their approximate X-rays in a finite number of prescribed lattice directions. We provide a polynomial-time algorithm for reconstructing Q-convex sets from their “approximate” X-rays. A Qconvex set is a special subset of Z having some convexity properties. This algorithm can be used for reconstructing convex subs...

Maximum a-posteriori (MAP) estimation is an important task in many applications of probabilistic graphical models. Although finding an exact solution is generally intractable, approximations based on linear programming (LP) relaxation often provide good approximate solutions. In this paper we present an algorithm for solving the LP relaxation optimization problem. In order to overcome the lack ...