33 research outputs found
Convex Hull Formulations for Mixed-Integer Multilinear Functions
In this paper, we present convex hull formulations for a mixed-integer,
multilinear term/function (MIMF) that features products of multiple continuous
and binary variables. We develop two equivalent convex relaxations of an MIMF
and study their polyhedral properties in their corresponding higher-dimensional
spaces. We numerically observe that the proposed formulations consistently
perform better than state-of-the-art relaxation approaches
A New Mathematical Programming Framework for Facility Layout Design
We present a new framework for efficiently finding competitive solutions for the facility layout problem. This framework is based on the combination of two new mathematical programming models. The first model is a relaxation of the layout problem and is intended to find good starting points for the iterative algorithm used to solve the second model. The second model is an exact formulation of the facility layout problem as a non-convex mathematical program with equilibrium constraints (MPEC). Aspect ratio constraints, which are frequently used in facility layout methods to restrict the occurrence of overly long and narrow departments in the computed layouts, are easily incorporated into this new framework. Finally, we present computational results showing that both models, and hence the complete framework, can be solved efficiently using widely available optimization software. This important feature of the new framework implies that it can be used to find competitive layouts with relatively little computational effort. This is advantageous for a user who wishes to consider several competitive layouts rather than simply using the mathematically optimal layout
Scanning integer points with lex-inequalities: A finite cutting plane algorithm for integer programming with linear objective
We consider the integer points in a unimodular cone K ordered by a
lexicographic rule defined by a lattice basis. To each integer point x in K we
associate a family of inequalities (lex-cuts) that defines the convex hull of
the integer points in K that are not lexicographically smaller than x. The
family of lex-cuts contains the Chvatal-Gomory cuts, but does not contain and
is not contained in the family of split cuts. This provides a finite cutting
plane method to solve the integer program min{cx : x \in S \cap Z^n }, where S
\subset R^n is a compact set and c \in Z^n . We analyze the number of
iterations of our algorithm.Comment: 16 pages, 1 figur
Mixed-Integer Convex Representability
We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer assignments is finite. We develop a characterization for the more general case of unbounded integer variables together with a simple necessary condition for representability which we use to prove the first known negative results. Finally, we study representability of subsets of the natural numbers, developing insight towards a more complete understanding of what modeling power can be gained by using convex sets instead of polyhedral sets; the latter case has been completely characterized in the context of mixed-integer linear optimization.United States. National Science Foundation. (Grant CMMI-1351619