We present a faster exponential-time algorithm for integer optimization over quasi-convex polynomials. We study the minimization of a quasi-convex polynomial subject to s quasi-convex polynomial constraints and integrality constraints for all variables. The new algorithm is an improvement upon the best known algorithm due to Heinz (Journal of Complexity, 2005). A lower time complexity is … Read more
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 recently in another paper). We then extend this observation to show that crooked cross cuts give the convex hull of mixed-integer sets with more integer variables provided that … Read more
We will analyze mixed 0/1 second order cone programs where the fractional and binary variables are solely coupled via the conic constraints. For this special type of mixed-integer second order cone programs we devise a cutting-plane framework based on the generalized Benders cut and an implicit Sherali-Adams reformulation. We show that the resulting cuts are … Read more
We provide a probabilistic comparison of split and type 1 triangle cuts for mixed-integer programs with two rows and two integer variables. Under a simple probabilistic model of the problem parameters, we show that a simple split cut, i.e. a Gomory cut, is more likely to be better than a type 1 triangle cut in … Read more
In this paper, we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be constructed easily. In particular, we use these techniques to derive, in closed-form, the concave envelopes of … Read more
Let S be a subset of integer points that satisfy the property that $conv(S) \cap Z^n = S$. Then a convex set K is called an S-free convex set if $int(K) \cap S = \emptyset$. A maximal S-free convex set is an S-free convex set that is not properly contained in any S-free convex set. … Read more
CitationIBM Research Report RC24999ArticleDownload View PDF
The Quadratic Assignment Problem (QAP) can be solved by linearization, where one formulates the QAP as a mixed integer linear programming (MILP) problem. On the one hand, most of these linearization are tight, but hardly exploited within a reasonable computing time because of their size. On the other hand, Kaufman and Broeckx formulation [1] is … Read more
The paper studies large scale mixed integer reformulation approach to stochastic programming problems containing probability and quantile functions, under assumption of discreteness of the probability distribution involved. Jointly with general sample approximation technique and contemporary mixed integer programming solvers the approach gives a regular framework to solution of practical probabilistic programming problems. In the literature … Read more
We propose a new look at the problem of truss topology optimization with integer or binary variables. We show that the problem can be equivalently formulated as an integer \emph{linear} semidefinite optimization problem. This makes its numerical solution much easier, compared to existing approaches. We demonstrate that one can use an off-the-shelf solver with default … Read more