We show that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with two integer variables is a crooked cross cut (which we defined in 2010). We extend this result to show that crooked cross cuts give the convex hull of mixed-integer sets with more integer variables if the coefficients of the integer variables form a matrix of rank 2. We also present an alterna...

Global optimization of mixed-integer nonlinear bilevel optimization problems is addressed using a novel technique. For problems where integer variables participate in both inner and outer problems, the outer level may involve general mixed-integer nonlinear functions. The inner level may involve functions that are mixed-integer nonlinear in outer variables, linear, polynomial, or multilinear in...

Abstract We study the existence and construction of very small formulations for disjunctive constraints in optimization problems: that is, formulations that use very few integer variables and extra constraints. To accomplish this, we present a novel mixed-integer branching formulation framework, which preserves many of the favorable algorithmic properties of a traditional mixed-integer programm...

This paper makes a review of interactive methods devoted to multiobjective integer and mixed-integer programming (MOIP/MOMIP) problems. The basic concepts concerning the characterization of the non-dominated solution set are first introduced, followed by a remark about non-interactive methods vs. interactive methods. Then, we focus on interactive MOIP/MOMIP methods, including their characteriza...

This survey presents tools from polyhedral theory that are used in integer programming. It applies them to the study of valid inequalities for mixed integer linear sets, such as Gomory’s mixed integer cuts.

In this paper we investigate the sub-channel assignment and power control to maximize the total sum rate in the uplink of two-cell network. It is assumed that there are some sub-channels in each cell which should be allocated among some users. Also, each user is subjected to a power constraint. The underlying problem is a non-convex mixed integer non-linear optimization problem which does not h...

Clustering parts and machines into part families and machine cells is a major decision in the design of cellular manufacturing systems which is defined as cell formation. This paper presents a non-linear mixed integer programming model to design cellular manufacturing systems which assumes that the arrival rate of parts into cells and machine service rate are stochastic parameters and descri...

We view mixed integer/linear problem formulation as a process of identifying disjunctive and knapsack constraints in a problem and converting them to mixed integer form. We show through a series of examples that following this process can yield mixed integer models that automatically incorporate some of the modeling devices that have been discovered over the years for making the formulation tig...

This article investigates cutting planes for mixed-integer disjunctive programs. In the early 1980s, Balas and Jeroslow presented monoidal disjunctive cuts exploiting the integrality of variables. For disjunctions arising from binary variables, it is known that these cutting planes are essentially the same as Gomory mixed-integer and mixed-integer rounding cuts. In this article, we investigate ...