Mixed-Integer Convex Representability

Lubin, Miles; Vielma, Juan Pablo; Zadik, Ilias

doi:10.1007/978-3-319-59250-3_32

Cited by 14 publications

(10 citation statements)

References 23 publications

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“…When 0 ă L ď U , A is a nonconvex set. Moreover, the annulus is not mixedinteger convex representable [23,24]: that is, there do not exist mixed-integer formulations for the annulus even if we allow the relaxation R to be an arbitrary convex set. Foster [20] proposes a disjunctive relaxation for the annulus given aŝ A def "…”

Section: Relaxations Of the Annulusmentioning

confidence: 99%

A geometric way to build strong mixed-integer programming formulations

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Vielma²

2019

Operations Research Letters

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We give an explicit geometric way to build mixed-integer programming (MIP) formulations for unions of polyhedra. The construction is simply described in terms of spanning hyperplanes in an r-dimensional linear space. The resulting MIP formulation is ideal, and uses exactly r integer variables and 2ˆp# of spanning hyperplanesq general inequality constraints. We use this result to derive novel logarithmicsized ideal MIP formulations for discontinuous piecewise linear functions and structures appearing in robotics and power systems problems.

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Section: Relaxations Of the Annulusmentioning

confidence: 99%

A geometric way to build strong mixed-integer programming formulations

Huchette

Vielma²

2019

Operations Research Letters

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“…We note that conic representations are useful for constructing tight formulations for disjunctions or unions of convex sets [Lubin et al, 2017b,c, Vielma, 2018].…”

Section: Mixed-integer Conic Formmentioning

confidence: 99%

Outer approximation with conic certificates for mixed-integer convex problems

2020

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A mixed-integer convex (MI-convex) optimization problem is one that becomes convex when all integrality constraints are relaxed. We present a branch-and-bound LP outer approximation algorithm for an MI-convex problem transformed to MI-conic form. The polyhedral relaxations are refined with K * cuts derived from conic certificates for continuous primaldual conic subproblems. Under the assumption that all subproblems are well-posed, the algorithm detects infeasibility or unboundedness or returns an optimal solution in finite time. Using properties of the conic certificates, we show that the K * cuts imply certain practically-relevant guarantees about the quality of the polyhedral relaxations, and demonstrate how to maintain helpful guarantees when the LP solver uses a positive feasibility tolerance. We discuss how to disaggregate K * cuts in order to tighten the polyhedral relaxations and thereby improve the speed of convergence, and propose fast heuristic methods of obtaining useful K * cuts. Our new open source MI-conic solver Pajarito (github.com/JuliaOpt/Pajarito.jl) uses an external mixed-integer linear (MILP) solver to manage the search tree and an external continuous conic solver for subproblems. Benchmarking on a library of mixed-integer second-order cone (MISOCP) problems, we find that Pajarito greatly outperforms Bonmin (the leading open source alternative) and is competitive with CPLEX's specialized MISOCP algorithm. We demonstrate the robustness of Pajarito by solving diverse MI-conic problems involving mixtures of positive semidefinite, second-order, and exponential cones, and provide evidence for the practical value of our analyses and enhancements of K * cuts. * M. Lubin is now at Google. The authors thank MOSEK for access to alpha releases of MOSEK 9 and Russell Bent and Emre Yamangil for support for and early contributions to Pajarito.

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“…Jeroslow and Lowe's result also serves as inspiration for recent interest in the representability of a variety of problems. See Vielma [35] for a review of the literature until 2015 and [5,13,14,[26][27][28] for examples of more recent work.…”

Section: Introductionmentioning

confidence: 99%

“…This throws a wrench in the classical narrative of representability that has largely focused on closed sets. Indeed, the recent work of Lubin et al [27] is careful to study representability by closed convex sets. This focus is entirely justified.…”

Section: Introductionmentioning

confidence: 99%

Mixed-integer bilevel representability

2019

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We study the representability of sets that admit extended formulations using mixed-integer bilevel programs. We show that feasible regions modeled by continuous bilevel constraints (with no integer variables), complementarity constraints, and polyhedral reverse convex constraints are all finite unions of polyhedra. Conversely, any finite union of polyhedra can be represented using any one of these three paradigms. We then prove that the feasible region of bilevel problems with integer constraints exclusively in the upper level is a finite union of sets representable by mixed-integer programs and vice versa. Further, we prove that, up to topological closures, we do not get additional modeling power by allowing integer variables in the lower level as well. To establish the last statement, we prove that the family of sets that are finite unions of mixed-integer representable sets forms an algebra of sets (up to topological closures). *

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Mixed-Integer Convex Representability

Cited by 14 publications

References 23 publications

A geometric way to build strong mixed-integer programming formulations

A geometric way to build strong mixed-integer programming formulations

Outer approximation with conic certificates for mixed-integer convex problems

Mixed-integer bilevel representability

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