Extended formulations in mixed integer conic quadratic programming

Vielma, Juan Pablo; Dunning, Iain; Huchette, Joey; Lubin, Miles

doi:10.1007/s12532-016-0113-y

Cited by 32 publications

(27 citation statements)

References 43 publications

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“…CXLPE: CPLEX branch-and-bound algorithm with LP outer approximations, setting the branching rule to maximum infeasibility, the node selection rule to best bound, and disabling cuts and heuristic. Since presolve is activated, CPLEX uses extended formulations described in [40]. Besides presolve, all other parameters are set as in CXLP.…”

Section: Discrete Instancesmentioning

confidence: 99%

Simplex QP-based methods for minimizing a conic quadratic objective over polyhedra

Atamtürk

Gómez

2018

Math. Prog. Comp.

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We consider minimizing a conic quadratic objective over a polyhedron. Such problems arise in parametric value-at-risk minimization, portfolio optimization, and robust optimization with ellipsoidal objective uncertainty; and they can be solved by polynomial interior point algorithms for conic quadratic optimization. However, interior point algorithms are not well-suited for branch-and-bound algorithms for the discrete counterparts of these problems due to the lack of effective warm starts necessary for the efficient solution of convex relaxations repeatedly at the nodes of the search tree.In order to overcome this shortcoming, we reformulate the problem using the perspective of the quadratic function. The perspective reformulation lends itself to simple coordinate descent and bisection algorithms utilizing the simplex method for quadratic programming, which makes the solution methods amenable to warm starts and suitable for branch-and-bound algorithms. We test the simplex-based quadratic programming algorithms to solve convex as well as discrete instances and compare them with the state-of-the-art approaches. The computational experiments indicate that the proposed algorithms scale much better than interior point algorithms and return higher precision solutions. In our experiments, for large convex instances, they provide up to 22x speed-up. For smaller discrete instances, the speed-up is about 13x over a barrier-based branch-and-bound algorithm and 6x over the LPbased branch-and-bound algorithm with extended formulations.

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Section: Discrete Instancesmentioning

confidence: 99%

Simplex QP-based methods for minimizing a conic quadratic objective over polyhedra

Atamtürk

Gómez

2018

Math. Prog. Comp.

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“…Higher is better. CPLEX is the best overall, since notably it already implements the extended formulation proposed by Vielma et al [44].…”

Section: Computational Experimentsmentioning

confidence: 99%

“…Fortunately, [28] also propose a solution based on ideas from [41] that can significantly improve the quality of a polyhedral approximation by constructing the approximation in a higher dimensional space. These constructions are known as extended formulations, which have also been considered by [44,32]. Although Hijazi et al demonstrate impressive computational gains by using extended formulations, implementing these techniques within traditional MICP solvers requires more structural information than provided by "black-box" oracles through which these solvers typically interact with nonlinear functions.…”

Section: Introductionmentioning

confidence: 99%

Polyhedral approximation in mixed-integer convex optimization

et al. 2017

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Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we intend to provide a broadly accessible introduction to our recent work in developing algorithms and software for this problem class. Our approach is based on constructing polyhedral outer approximations of the convex constraints, resulting in a global solution by solving a finite number of mixed-integer linear and continuous convex subproblems. The key advance we present is to strengthen the polyhedral approximations by constructing them in a higher-dimensional space. In order to automate this extended formulation we rely on the algebraic modeling technique of disciplined convex programming (DCP), and for generality and ease of implementation we use conic representations of the convex constraints. Although our framework requires a manual translation of existing models into DCP form, after performing this transformation on the MINLPLIB2 benchmark library we were able to solve a number of unsolved instances and on many other instances achieve superior performance compared with state-of-the-art solvers like Bonmin, SCIP, and Artelys Knitro

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“…Vielma et al [30] discuss the motivation for extended formulations in MICP: many successful MICP algorithms use polyhedral outer approximations of nonlinear constraints, and polyhedral outer approximations in a higher dimensional space can often be much stronger than approximations in the original space. Hijazi et al [20] give an example of an approximation of an 2 ball in R n which requires 2 n tangent hyperplanes in the original space to prove that the intersection of the ball with the integer lattice is in fact empty.…”

Section: Extended Formulations and Conic Representabilitymentioning

confidence: 99%

“…Hijazi et al generated these extended formulations by hand, and no subsequent work has proposed techniques for off-the-shelf MICP solvers to detect and exploit separability. Building on these results, Vielma et al [30] propose extended formulations for second-order cones. These extended formulations improved solution times for mixed-integer second-order cone programming (MISOCP) over state of the art commercial solvers CPLEX and Gurobi quite significantly; both solvers adopted the technique within a few months after its publication.…”

Section: Introductionmentioning

confidence: 99%

Extended Formulations in Mixed-Integer Convex Programming

Lubin

Yamangil

Bent

et al. 2016

Integer Programming and Combinatorial Optimization

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Abstract. We present a unifying framework for generating extended formulations for the polyhedral outer approximations used in algorithms for mixed-integer convex programming (MICP). Extended formulations lead to fewer iterations of outer approximation algorithms and generally faster solution times. First, we observe that all MICP instances from the MINLPLIB2 benchmark library are conic representable with standard symmetric and nonsymmetric cones. Conic reformulations are shown to be effective extended formulations themselves because they encode separability structure. For mixed-integer conic-representable problems, we provide the first outer approximation algorithm with finite-time convergence guarantees, opening a path for the use of conic solvers for continuous relaxations. We then connect the popular modeling framework of disciplined convex programming (DCP) to the existence of extended formulations independent of conic representability. We present evidence that our approach can yield significant gains in practice, with the solution of a number of open instances from the MINLPLIB2 benchmark library.

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Extended formulations in mixed integer conic quadratic programming

Cited by 32 publications

References 43 publications

Simplex QP-based methods for minimizing a conic quadratic objective over polyhedra

Simplex QP-based methods for minimizing a conic quadratic objective over polyhedra

Polyhedral approximation in mixed-integer convex optimization

Extended Formulations in Mixed-Integer Convex Programming

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