In this paper we present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting, that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a “nearly optimal” solution of … Read more
We consider relaxations for nonconvex quadratically constrained quadratic programming (QCQP) based on semidefinite programming (SDP) and the reformulation-linearization technique (RLT). From a theoretical standpoint we show that the addition of a semidefiniteness condition removes a substantial portion of the feasible region corresponding to product terms in the RLT relaxation. On test problems we show that … Read more
In a paper published 1978, McEliece, Rodemich and Rumsey improved Lov\’asz’ bound for the Maximum Clique Problem. This strengthening has become well-known under the name Lov\’asz-Schrijver bound and is usually denoted by $\theta’$. This article now deals with situations where this bound is not exact. To provide instances for which the gap between this bound … Read more
This paper presents a robust branch-cut-and-price algorithm for the Heterogeneous Fleet Vehicle Routing Problem (HFVRP), vehicles may have various capacities and fixed costs. The columns in the formulation are associated to $q$-routes, a relaxation of capacitated elementary routes that makes the pricing problem solvable in pseudo-polynomial time. Powerful new families of cuts are also proposed, … Read more
For a conic linear system of the form Ax \in K, K a convex cone, several condition measures have been extensively studied in the last dozen years. Herein we show that Renegar’s condition number is bounded from above and below by certain purely geometric quantities associated with A and K, and highlights the role of … Read more
We consider semidefinite programming (SDP) formulations of certain truss topology optimization problems, where a lower bound is imposed on the fundamental frequency of vibration of the truss structure. These SDP formulations were introduced in: [M. Ohsaki, K. Fujisawa, N. Katoh and Y. Kanno, Semi-definite programming for topology optimization of trusses under multiple eigenvalue constraints, Comp. … Read more
This paper focuses on monotone L\”{o}wner operators in Euclidean Jordan algebras and their applications to the symmetric cone complementarity problem (SCCP). We prove necessary and sufficient conditions for locally Lipschitz L\”{o}wner operators to be monotone, strictly monotone and strongly monotone. We also study the relationship between monotonicity and operator-monotonicity of L\”{o}wner operators. As a by-product … Read more
In this paper we devise a new version of primal simplex algorithms in which the classical iteration is decomposed two basic operations: the move and the pivot. The move operation decreases the primal objective value and the pivot operation increases the dual objective. We define the condition number of the pivot operation and present a … Read more
Linear Complementarity Problems ($LCP$s) belong to the class of $\mathbb{NP}$-complete problems. Therefore we can not expect a polynomial time solution method for $LCP$s without requiring some special property of the matrix coefficient matrix. Our aim is to construct some interior point algorithms which, according to the duality theorem in EP form, gives a solution of … Read more
The linear complementarity problem (LCP) belongs to the class of NP-complete problems. Therefore we can not expect a polynomial time solution method for LCPs without requiring some special property of the matrix of the problem. We show that the dual LCP can be solved in polynomial time if the matrix is row sufficient, moreover in … Read more