We propose a new approach based DC programming for fnding a solution of the partial facility interdiction problem that belongs to the class of bilevel programming. This model was frst considered in the work of Aksen et al. [1] with a heuristic algorithm named multi-start simplex search (MSS). However, because of the big number of … Read more
In this paper, we consider a partial sparse and partial group sparse optimization problem, where the loss function is a continuously differentiable function (possibly nonconvex), and the penalty term consists of two parts associated with sparsity and group sparsity. The first part is the $\ell_0$ norm of ${\bf x}$, the second part is the $\ell_{2,0}$ … Read more
The minimization of non-lower semicontinuous functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance … Read more
Preconditioning is essential in iterative methods for solving linear systems of equations. We study a nonclassic matrix condition number, the ω-condition number, in the context of optimal conditioning for low rank updating of positive definite matrices. For a positive definite matrix, this condition measure is the ratio of the arithmetic and geometric means of the … Read more
We introduce a notion of self-concordant smoothing for minimizing the sum of two convex functions, one of which is smooth and the other may be nonsmooth. The key highlight of our approach is in a natural property of the resulting problem’s structure which provides us with a variable-metric selection method and a step-length selection rule … Read more
We are concerned with Lipschitzian error bounds and Lipschitzian stability properties for solutions of a complementarity system. For this purpose, we deal with a nonsmooth slack-variable reformulation of the complementarity system, and study conditions under which the reformulation serves as a local error bound for the solution set of the complementarity system. We also discuss … Read more
Bilevel optimization problems are hierarchical problems with a constraint set which is a subset of the graph of the solution set mapping of a second optimization problem. To investigate their properties and derive solution algorithms, their transformation into single-level ones is necessary. For this, various approaches have been developed. The rst and most often used … Read more
We study the convergence of the last iterate in subgradient methods applied to the minimization of a nonsmooth convex function with bounded subgradients. We first introduce a proof technique that generalizes the standard analysis of subgradient methods. It is based on tracking the distance between the current iterate and a different reference point at each … Read more
In this work, we introduce new direct search schemes for the solution of bilevel optimization (BO) problems. Our methods rely on a fixed accuracy black box oracle for the lower-level problem, and deal both with smooth and potentially nonsmooth true objectives. We thus analyze for the first time in the literature direct search schemes in … Read more
Restricted normal cones are of interest, for instance, in the theory of local error bounds, where they have recently been used to characterize the exis- tence of a constrained Lipschitzian error bound. In this paper, we establish rela- tions between two concepts for restricted normals. The first of these concepts was introduced in the late … Read more