We study a generalized job-shop problem called the body shop scheduling problem (bssp). This problem arises from the industrial application of welding in a car body production line, where possible collisions between industrial robots have to be taken into account. bssp corresponds to a job-shop problem where the operations of a job have to follow … Read more
Matrix rank minimization problems are gaining a plenty of recent attention in both mathematical and engineering fields. This class of problems, arising in various and across-discipline applications, is known to be NP-hard in general. In this paper, we aim at providing an approximation theory for the rank minimization problem, and prove that a rank minimization … Read more
The behavior of enumeration-based programs for solving MINLPs depends at least in part on the quality of the bounds on the optimal value and of the feasible solutions found. We consider MINLP problems with linear constraints. The convex hull relaxation (CHR) is a special case of the primal relaxation (Guignard 1994, 2007) that is very … Read more
The paper describes a new primal relaxation (PR) for computing bounds on nonlinear integer programming (NLIP) problems. It is a natural extension to NLIP problems of the geometric interpretation of Lagrangean relaxation presented by Geoffrion (1974) for linear problems, and it is based on the same assumption that some constraints are complicating and are treated … Read more
The convex hull relaxation (CHR) method (Albornoz 1998, Ahlatçıoğlu 2007, Ahlatçıoğlu and Guignard 2010) provides lower bounds and feasible solutions (thus upper bounds) on convex 0-1 nonlinear programming problems with linear constraints. In the quadratic case, these bounds may often be improved by a preprocessing step that adds to the quadratic objective function terms which … Read more
Minimization with orthogonality constraints (e.g., $X^\top X = I$) and/or spherical constraints (e.g., $\|x\|_2 = 1$) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. … Read more
Burer has shown that completely positive relaxations of nonconvex quadratic programs with nonnegative and binary variables are exact when the binary variables satisfy a so-called key assumption. Here we show that introducing binary variables to obtain an equivalent problem that satisfies the key assumption will not improve the semidefinite relaxation, and only marginally improve the … Read more
Min-$k$-cut is the problem of partitioning vertices of a given graph or hypergraph into $k$ subsets such that the total weight of edges or hyperedges crossing different subsets is minimized. For the capacitated min-$k$-cut problem, each edge has a non-negative weight, and each subset has a possibly different capacity that imposes an upper bound on … Read more
In this paper we describe an algorithm to approximately solve a class of semidefinite programs called covering semidefinite programs. This class includes many semidefinite programs that arise in the context of developing algorithms for important optimization problems such as sparsest cut, wireless multicasting, and pattern classification. We give algorithms for covering SDPs whose dependence on … Read more
We study the approximation of the least core value and the least core of supermodular cost cooperative games. We provide a framework for approximation based on oracles that approximately determine maximally violated constraints. This framework yields a (3 + \epsilon)-approximation algorithm for computing the least core value of supermodular cost cooperative games, and a polynomial-time … Read more