We study time-varying semidefinite programs (TV-SDPs), which are semidefinite programs whose data (and solutions) are functions of time. Our focus is on the setting where the data varies polynomially with time. We show that under a strict feasibility assumption, restricting the solutions to also be polynomial functions of time does not change the optimal value … Read more
We study the representability of sets that admit extended formulations using mixed-integer bilevel programs. We show that feasible regions modeled by continuous bilevel constraints (with no integer variables), complementarity constraints, and polyhedral reverse convex constraints are all finite unions of polyhedra. Conversely, any finite union of polyhedra can be represented using any one of these … Read more
A set of 78 test examples is presented for the trust region method MHT described in J. Thomann, G. Eichfelder, A trust region algorithm for heterogeneous multi-objective optimization, 2018 (available as preprint: http://optimization-online.org/DB_HTML/2018/03/6495.html) . It is designed for multi-objective heterogeneous optimization problems where one of the objective functions is an expensive black-box function, for example … Read more
We consider the problem of minimizing the sum of two convex functions: one is differentiable and relatively smooth with respect to a reference convex function, and the other can be nondifferentiable but simple to optimize. The relatively smooth condition is much weaker than the standard assumption of uniform Lipschitz continuity of the gradients, thus significantly … Read more
In this paper we consider the linear convergence of algorithms for minimizing difference- of-convex functions with convex constraints. We allow nonsmoothness in both of the convex and concave components in the objective function, with a finite max structure in the concave compo- nent. Our focus is on algorithms that compute (weak and standard) d(irectional)-stationary points … Read more
We propose an efficient algorithm to solve positive a semidefinite matrix approximation problem with a trace constraint. Without constraints, it is well known that positive semidefinite matrix approximation problem can be easily solved by one-time eigendecomposition of a symmetric matrix. In this paper, we confirmed that one-time eigendecomposition is also sufficient even if a trace … Read more
We introduce two-stage distributionally robust disjunctive programs (TSDR-DPs) with disjunctive constraints in both stages and a general ambiguity set for the probability distributions. The TSDR-DPs subsume various classes of two-stage distributionally robust programs where the second stage problems are non-convex programs (such as mixed binary programs, semi-continuous program, nonconvex quadratic programs, separable non-linear programs, etc.). … Read more
Split cuts are prominent general-purpose cutting planes in integer programming. The split closure of a rational polyhedron is what is obtained after intersecting the half-spaces defined by all the split cuts for the polyhedron. In this paper, we prove that deciding whether the split closure of a rational polytope is empty is NP-hard, even when … Read more
We study the following problem: given a rational polytope with Chvatal rank 1, does it contain an integer point? Boyd and Pulleyblank observed that this problem is in the complexity class NP ∩ co-NP, an indication that it is probably not NP-complete. It is open whether there is a polynomial time algorithm to solve the … Read more
In this paper, we consider polytopes P that are contained in the unit hypercube. We provide conditions on the set of 0,1 vectors not contained in P that guarantee that P has a small Chvatal rank. Our conditions are in terms of the subgraph induced by these infeasible 0,1 vertices in the skeleton graph of … Read more