On the relationship between standard intersection cuts, lift-and-project cuts, and generalized intersection cuts

We examine the connections between the classes of cuts in the title. We show that lift-and-project (L&P) cuts from a given disjunction are equivalent to generalized intersection cuts from the family of polyhedra obtained by taking positive combinations of the complements of the inequalities of each term of the disjunction. While L&P cuts from split disjunctions are known to be equivalent to standard intersection cuts (SICs) from the strip obtained by complementing the terms of the split, we show that L&P cuts from more general disjunctions may not be equivalent to any SIC. In particular, we give easily verifiable necessary and sufficient conditions for a L&P cut from a given disjunction D to be equivalent to a SIC from the polyhedral counterpart of D. Irregular L&P cuts, i.e. those that violate these conditions, have interesting properties. For instance, unlike the regular ones, they may cut off part of the corner polyhedron associated with the LP solution from which they are derived. Furthermore, they are not exceptional: their frequency exceeds that of regular cuts. A numerical example illustrates some of the above properties. © 2016 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Societ

When Lift-and-Project Cuts are Different

In this paper, we present a method to determine if a lift-and-project cut for a mixed-integer linear program is irregular, in which case the cut is not equivalent to any intersection cut from the bases of the linear relaxation. This is an important question due to the intense research activity for the past decade on cuts from multiple rows of simplex tableau as well as on lift-and-project cuts from non-split disjunctions. While it is known since Balas and Perregaard (2003) that lift-and-project cuts from split disjunctions are always equivalent to intersection cuts and consequently to such multi-row cuts, Balas and Kis (2016) have recently shown that there is a necessary and sufficient condition in the case of arbitrary disjunctions: a lift-and-project cut is regular if, and only if, it corresponds to a regular basic solution of the Cut Generating Linear Program (CGLP). This paper has four contributions. First, we state a result that simplifies the verification of regularity for basic CGLP solutions from Balas and Kis (2016). Second, we provide a mixed-integer formulation that checks whether there is a regular CGLP solution for a given cut that is regular in a broader sense, which also encompasses irregular cuts that are implied by the regular cut closure. Third, we describe a numerical procedure based on such formulation that identifies irregular lift-and-project cuts. Finally, we use this method to evaluate how often lift-and-project cuts from simple $t$-branch split disjunctions are irregular, and thus not equivalent to multi-row cuts, on 74 instances of the MIPLIB benchmarks.Comment: INFORMS Journal on Computing (to appear

ADE Spectral Networks

We introduce a new perspective and a generalization of spectral networks for 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ associated to Lie algebras $\mathfrak{g} = \textrm{A}_n$, $\textrm{D}_n$, $\textrm{E}_{6}$, and $\textrm{E}_{7}$. Spectral networks directly compute the BPS spectra of 2d theories on surface defects coupled to the 4d theories. A Lie algebraic interpretation of these spectra emerges naturally from our construction, leading to a new description of 2d-4d wall-crossing phenomena. Our construction also provides an efficient framework for the study of BPS spectra of the 4d theories. In addition, we consider novel types of surface defects associated with minuscule representations of $\mathfrak{g}$.Comment: 68 pages plus appendices; visit http://het-math2.physics.rutgers.edu/loom/ to use 'loom,' a program that generates spectral networks; v2: version published in JHEP plus minor correction

Symmetries of stochastic colored vertex models

We discover a new property of the stochastic colored six-vertex model called flip-invariance. We use it to show that for a given collection of observables of the model, any transformation that preserves the distribution of each individual observable also preserves their joint distribution. This generalizes recent shift-invariance results of Borodin-Gorin-Wheeler. As limiting cases, we obtain similar statements for the Brownian last passage percolation, the Kardar-Parisi-Zhang equation, the Airy sheet, and directed polymers. Our proof relies on an equivalence between the stochastic colored six-vertex model and the Yang-Baxter basis of the Hecke algebra. We conclude by discussing the relationship of the model with Kazhdan-Lusztig polynomials and positroid varieties in the Grassmannian

Balanced partitions of 3-colored geometric sets in the plane

Let SS be a finite set of geometric objects partitioned into classes or colors . A subset S'¿SS'¿S is said to be balanced if S'S' contains the same amount of elements of SS from each of the colors. We study several problems on partitioning 33-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are 2m2m lines of each color, there is a segment intercepting mm lines of each color. (b) Given nn red points, nn blue points and nn green points on any closed Jordan curve ¿¿, we show that for every integer kk with 0=k=n0=k=n there is a pair of disjoint intervals on ¿¿ whose union contains exactly kk points of each color. (c) Given a set SS of nn red points, nn blue points and nn green points in the integer lattice satisfying certain constraints, there exist two rays with common apex, one vertical and one horizontal, whose union splits the plane into two regions, each one containing a balanced subset of SS.Peer ReviewedPostprint (published version

Adaptive Methods for Linear Programming Decoding

Detectability of failures of linear programming (LP) decoding and the potential for improvement by adding new constraints motivate the use of an adaptive approach in selecting the constraints for the underlying LP problem. In this paper, we make a first step in studying this method, and show that it can significantly reduce the complexity of the problem, which was originally exponential in the maximum check-node degree. We further show that adaptively adding new constraints, e.g. by combining parity checks, can provide large gains in the performance.Comment: 22 pages, 8 figures. Submitted to IEEE Transactions on Information Theor

A branch, price, and cut approach to solving the maximum weighted independent set problem

The maximum weight-independent set problem (MWISP) is one of the most well-known and well-studied NP-hard problems in the field of combinatorial optimization. In the first part of the dissertation, I explore efficient branch-and-price (B&P) approaches to solve MWISP exactly. B&P is a useful integer-programming tool for solving NP-hard optimization problems. Specifically, I look at vertex- and edge-disjoint decompositions of the underlying graph. MWISPâÂÂs on the resulting subgraphs are less challenging, on average, to solve. I use the B&P framework to solve MWISP on the original graph G using these specially constructed subproblems to generate columns. I demonstrate that vertex-disjoint partitioning scheme gives an effective approach for relatively sparse graphs. I also show that the edge-disjoint approach is less effective than the vertex-disjoint scheme because the associated DWD reformulation of the latter entails a slow rate of convergence. In the second part of the dissertation, I address convergence properties associated with Dantzig-Wolfe Decomposition (DWD). I discuss prevalent methods for improving the rate of convergence of DWD. I also implement specific methods in application to the edge-disjoint B&P scheme and show that these methods improve the rate of convergence. In the third part of the dissertation, I focus on identifying new cut-generation methods within the B&P framework. Such methods have not been explored in the literature. I present two new methodologies for generating generic cutting planes within the B&P framework. These techniques are not limited to MWISP and can be used in general applications of B&P. The first methodology generates cuts by identifying faces (facets) of subproblem polytopes and lifting associated inequalities; the second methodology computes Lift-and-Project (L&P) cuts within B&P. I successfully demonstrate the feasibility of both approaches and present preliminary computational tests of each

On the interplay of Mixed Integer Linear, Mixed Integer Nonlinear and Constraint Programming

In this thesis we study selected topics in the field of Mixed Integer Programming (MIP), in particular Mixed Integer Linear and Nonlinear Programming (MI(N)LP). We set a focus on the influences of Constraint Programming (CP). First, we analyze Mathematical Programming approaches to water network optimization, a set of challenging optimization problems frequently modeled as non-convex MINLPs. We give detailed descriptions of many variants and survey solution approaches from the literature. We are particularly interested in MILP approximations and present a respective computational study for water network design problems. We analyze this approach by algorithmic considerations and highlight the importance of certain convex substructures in these non-convex MINLPs. We further derive valid inequalities for water network design problems exploiting these substructures. Then, we treat Mathematical Programming problems with indicator constraints, recalling their most popular reformulation techniques in MIP, leading to either big-M constraints or disjunctive programming techniques. The latter give rise to reformulations in higher-dimensional spaces, and we review special cases from the literature that allow to describe the projection on the original space of variables explicitly. We theoretically extend the respective results in two directions and conduct computational experiments. We then present an algorithm for MILPs with indicator constraints that incorporates elements of CP into MIP techniques, including computational results for the JobShopScheduling problem. Finally, we introduce an extension of the class of MILPs so that linear expressions are allowed to have non-contiguous domains. Inspired by CP, this permits to model holes in the domains of variables as a special case. For such problems, we extend the theory of split cuts and show two ways of separating them, namely as intersection and lift-and-project cuts, and present computational results. We further experiment with an exact algorithm for such problems, applied to the Traveling Salesman Problem with multiple time windows