The goal of this paper is an easy and self-contained presentation of optimality conditions for nonlinear semidefinite programs concentrating on the differences between nonlinear semidefinite programs and nonlinear programs. Citation Technical Report, Department of Mathematics, Universit\”at D\”usseldorf. Article Download View Elementary optimality conditions for nonlinear SDPs
We consider convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and (possibly nonconvex) quadratic constraints. Let F denote the feasible region for the linear constraints. We first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on F is dominated by an alternative … Read more
In this paper, we propose an efficient technique for solving some infinite-dimensional problems over the sets of functions of time. In our problem, besides the convex point-wise constraints on state variables, we have convex coupling constraints with finite-dimensional image. Hence, we can formulate a finite-dimensional dual problem, which can be solved by efficient gradient methods. … Read more
We solve a class of convex infinite-dimensional optimization problems using a numerical approximation method that does not rely on discretization. Instead, we restrict the decision variable to a sequence of finite-dimensional linear subspaces of the original infinite-dimensional space and solve the corresponding finite-dimensional problems in a efficient way using structured convex optimization techniques. We prove … Read more
Ordering problems assign weights to each ordering and ask to find an ordering of maximum weight. We consider problems where the cost function is either linear or quadratic. In the first case, there is a given profit if the element u is before v in the ordering. In the second case, the profit depends on … Read more
We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in … Read more
This work gives an overview over the major codes available for the solution of linear semidefinite (SDP) and second-order cone (SOCP) programs. Some developments since the 7th DIMACS Challenge [9, 17] are pointed out as well as some currently under way. Instead of presenting per- formance tables, reference is made to the ongoing benchmark [19] … Read more