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Facets of a mixed-integer bilinear covering set with bounds on variables [article]

Hamidur Rahman, Ashutosh Mahajan
2019 arXiv   pre-print
We derive a closed form description of the convex hull of mixed-integer bilinear covering set with bounds on the integer variables.  ...  We also derive a linear time separation algorithm for finding the facet defining inequalities of this convex hull.  ...  Acknowledgments We are grateful to three anonymous reviewers for giving insightful comments and suggestions, specially the idea of the extended formulation and simplification of proofs.  ... 
arXiv:1707.06712v2 fatcat:gvkqm5ev7bahpa6dm4q5a6arhy

Lifting inequalities: a framework for generating strong cuts for nonlinear programs

Jean-Philippe P. Richard, Mohit Tawarmalani
2008 Mathematical programming  
We specialize our lifting results to derive facet-defining inequalities for mixed-integer bilinear knapsack sets.  ...  on a tight reformulation of the problem.  ...  For the mixedinteger bilinear set, we use the proposed lifting theory to generate, in closed-form, a large family of facet-defining lifted cover inequalities.  ... 
doi:10.1007/s10107-008-0226-9 fatcat:u3ozvfeflfcfdpcxc3wd5fv7fa

Solving Mixed Integer Bilinear Problems Using MILP Formulations

Akshay Gupte, Shabbir Ahmed, Myun Seok Cheon, Santanu Dey
2013 SIAM Journal on Optimization  
The effectiveness of this reformulation and associated facet-defining inequalities are computationally evaluated on five classes of instances.  ...  We present the convex hull of the underlying mixed integer linear set.  ...  Particular thanks for encouraging us to experiment on more instances, helping us simplify the proof of Proposition 3.4, and for correcting an earlier version of Proposition 2.3.  ... 
doi:10.1137/110836183 fatcat:72ipukixhzbpzfz4lbnwfwr66u

Traces of the XII Aussois Workshop on Combinatorial Optimization

Michael Jünger, Thomas M. Liebling, Denis Naddef, William R. Pulleyblank, Gerhard Reinelt, Giovanni Rinaldi, Laurence A. Wolsey
2010 Mathematical programming  
"Strong Valid Inequalities for Orthogonal Disjunctions and Bilinear Covering Sets" by Mohit Tawarmalani, Jean-Philippe Richard, and Kwanghun Chung deals with the problem of finding tight convex envelopes  ...  Wolsey studies the recently characterized extreme valid inequalities of a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables.  ... 
doi:10.1007/s10107-010-0369-3 fatcat:b7slvsqv4ncy7j45l3u4dcilz4

Strong valid inequalities for orthogonal disjunctions and bilinear covering sets

Mohit Tawarmalani, Jean-Philippe P. Richard, Kwanghun Chung
2010 Mathematical programming  
We demonstrate its applicability in integer programming by deriving the intersection cut for mixed-integer polyhedral sets and the convex hull of certain mixed/pure-integer bilinear sets.  ...  We illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non-negative orthant.  ...  Such a technique will be useful in deriving a separating hyperplane for mixed-integer and pure-integer bilinear covering sets.  ... 
doi:10.1007/s10107-010-0374-6 fatcat:d37g7jyhvjaqfcnykbk5g32cua

The Bipartite Boolean Quadric Polytope with Multiple-Choice Constraints [article]

Andreas Bärmann, Alexander Martin, Oskar Schneider
2020 arXiv   pre-print
In this work, we study the case where there is a partition on one of the two bipartite node sets such that at most one node per subset of the partition can be chosen.  ...  The well-studied BQP is defined as the convex hull of all quadric incidence vectors over a bipartite graph.  ...  Futhermore, we acknowledge financial support by the Bavarian Ministry of Economic Affairs, Regional Development and Energy through the Center for Analytics -Data -Applications (ADA-Center) within the framework  ... 
arXiv:2009.11674v1 fatcat:zmatqjydtnaafdqssjem2qp2ou

Convex hulls of superincreasing knapsacks and lexicographic orderings

Akshay Gupte
2016 Discrete Applied Mathematics  
We describe the convex hull of this n-dimensional set with O(n) facets.  ...  We also establish a distributive property by proving that the convex hull of <- and >-type superincreasing knapsacks can be obtained by intersecting the convex hulls of <- and >-sets taken individually  ...  Gupte et al. demonstrated the practical usefulness of facets to binary expansion knapsacks as cutting planes in a branch-and-cut algorithm for solving mixed integer bilinear programs. Remark 1.  ... 
doi:10.1016/j.dam.2015.08.010 fatcat:lzpvcljrc5bjfgy7pn7fjzbwjm

Relaxations and discretizations for the pooling problem

Akshay Gupte, Shabbir Ahmed, Santanu S. Dey, Myun Seok Cheon
2016 Journal of Global Optimization  
Valid inequalities are derived for the discretized models, which are formulated as mixed integer linear programs.  ...  The classical minimum cost network flow problem seeks to find the optimal way of sending raw materials from a set of suppliers to a set of customers via certain transshipment nodes in a directed capacitated  ...  set of combinatorial constraints that make this optimization model a mixed integer bilinear program (MIBLP).  ... 
doi:10.1007/s10898-016-0434-4 fatcat:x4svzvfpbbg2lbyf6m6h6lxpwe

A hierarchy of relaxations and convex hull characterizations for mixed-integer zero—one programming problems

Hanif D. Sherali, Warren P. Adams
1994 Discrete Applied Mathematics  
The methodology readily extends to multilinear mixed-integer zero-one polynomial programming problems in which the continuous variables appear linearly in the problem.  ...  For the linear case, we propose a technique which first converts the problem into a nonlinear, polynomial mixed-integer zero-one problem by multiplying the constraints with some suitable d-degree polynomial  ...  to the multilinear mixed-integer zero-one polynomial programming problem.  ... 
doi:10.1016/0166-218x(92)00190-w fatcat:uujo7d4r6vak7d23t7v4mf6k6e

Solving Chance-Constrained Optimization Problems with Stochastic Quadratic Inequalities

Miguel A. Lejeune, François Margot
2016 Operations Research  
The test instances are epidemiology and disaster management facility location models and cover the three types of stochastic quadratic inequalities, namely product of two variables that are (i) both binary  ...  We present detailed empirical results comparing the various reformulations and several easy to implement algorithmic ideas that improve performances of the mixed-integer nonlinear solver Couenne for solving  ...  Properties of Threshold Boolean Function The following two definitions are useful to derive a set of mixed-integer nonlinear inequalities defining the feasible set of the chance constraint (3).  ... 
doi:10.1287/opre.2016.1493 fatcat:o47cqe2xmzhlrhf6jr7xtrwpui

Global optimization in the 21st century: Advances and challenges

C.A. Floudas, I.G. Akrotirianakis, S. Caratzoulas, C.A. Meyer, J. Kallrath
2005 Computers and Chemical Engineering  
The review part covers the areas of (a) twice continuously differentiable nonlinear optimization, (b) mixedinteger nonlinear optimization, (c) optimization with differential-algebraic models, (d) optimization  ...  Computational studies will illustrate the potential of these advances.  ...  Floudas gratefully acknowledges support from the National Science Foundation, the National Institutes of Health, AspenTech Corporation and Atofina Chemicals.  ... 
doi:10.1016/j.compchemeng.2005.02.006 fatcat:egaic2jncfc6zfsbfzcvqnbtk4

Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations

E. Cattani
2010 International mathematics research notices  
The Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in various contexts: they impose restrictions on the cohomology algebra of a smooth compact Kähler manifold; they restrict  ...  the local monodromy of a polarized variation of Hodge structure; they impose conditions on the f-vectors of convex polytopes.  ...  of HLT and HRR for toric varieties and to deduce Hodge-Riemann Bilinear Relations 3 the Alexandrov-Fenchel inequality for the mixed volume of polytopes, as well as other similar properties, from the  ... 
doi:10.1093/imrn/rnn025 fatcat:ewsx3lamwrbrdpb3birnw7ybzy

Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations [article]

Eduardo Cattani
2008 arXiv   pre-print
Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose restrictions on the cohomology algebra of a smooth compact  ...  Kähler manifold or on the intersection cohomology of a projective toric variety; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions on the possible f-vectors  ...  There is also a distinguished set of linear coordinates x 1 , . . . , x r on A(∆) defined in the following way: let ξ 1 , . . . , ξ r be a choice of outward normal vectors for ∆, we then set for each P  ... 
arXiv:0707.1352v2 fatcat:yej3uisnebbgpplm43bq7xzjbe

A review of recent advances in global optimization

C. A. Floudas, C. E. Gounaris
2008 Journal of Global Optimization  
It covers the areas of twice continuously differentiable nonlinear optimization, mixed-integer nonlinear optimization, optimization with differential-algebraic models, semi-infinite programming, optimization  ...  This paper presents an overview of the research progress in deterministic global optimization during the last decade (1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006) (2007) (2008).  ...  Floudas gratefully acknowledges support from the National Science Foundation, the National Institutes of Health (R01 GM52032 and R24 GM069736), the Environmental Protection Agency (GAD R 832721-010), AspenTech  ... 
doi:10.1007/s10898-008-9332-8 fatcat:72fpfq72hrdzhf6mxqyc6ssezm

Branch-and-Cut for the Maximum Feasible Subsystem Problem

Marc E. Pfetsch
2008 SIAM Journal on Optimization  
The complementary problem, where one has to remove as few inequalities as possible in order to make the system feasible, can be formulated as a set covering problem.  ...  We present three heuristics for the corresponding NP-hard separation problem and discuss cutting planes from the literature, such as set covering cuts of Balas and Ng, Gomory cuts, and {0, 1 2 }-cuts.  ...  Furthermore, he thanks Edoardo Amaldi and Pietro Belotti for providing the DVB instances of Section 4.3, and Gianni Codato and Matteo Fischetti for the data used in Section 4.4.  ... 
doi:10.1137/050645828 fatcat:w7syhnzs75fm7dgd2tpbn3pyae
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