Integer Optimization on Convex Semialgebraic Sets

Khachiyan, Leonid; Porkolab, Lorant

doi:10.1007/pl00009496

Cited by 46 publications

(41 citation statements)

References 17 publications

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“…If the quadratic function is not convex, the problem is easily shown to be unbounded. If, on the other hand, the quadratic function is convex, then the problem can be solved for fixed n with the algorithm described in [20].…”

Section: Complexitymentioning

confidence: 99%

Unbounded convex sets for non-convex mixed-integer quadratic programming

Burer

Letchford

2012

Math. Program.

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This paper introduces a fundamental family of unbounded convex sets that arises in the context of non-convex mixed-integer quadratic programming. It is shown that any mixed-integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a set in the family. Some fundamental properties of the convex sets are derived, along with connections to some other well-studied convex sets. Several classes of valid and facet-inducing inequalities are also derived.

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Section: Complexitymentioning

confidence: 99%

Unbounded convex sets for non-convex mixed-integer quadratic programming

Burer

Letchford

2012

Math. Program.

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“…Case 3a1: Suppose g * 2 < 0. Then let λ * 2 be a minimizer to the minimization problem (14). Letx ∈ Z 2 be such thatx =x + λ * 2 r 2 + λ 1 r 1 ∈ Z 2 for some λ 1 ≥ 0, which can be found using a linear integer program.…”

Section: Cubic Polynomials and Unbounded Polyhedramentioning

confidence: 99%

“…This technique is based on an operator that determines integer feasibility on sets P ∩ C and P \ C, where P is a polyhedron, C is a convex set, and the dimension is fixed. It relies on two important previous results, namely that in fixed dimension the feasibility problem over semialgebraic sets can be solved in polynomial time [14], and the vertices of the integer hull of a polyhedron can be computed in polynomial time [5,10]. We employ this operator to solve the feasibility problem by dividing the domain into regions where this operator can be applied.…”

Section: Introductionmentioning

confidence: 99%

“…Otherwise, since vertices are integral, we have found an integer point in (P \ C) ∩ Z n . Next, since P ∩ C is a convex, semialgebraic set, by [14] we can determine in polynomial time whether P ∩ C ∩ Z n is empty, and if it is not, compute a point contained in it. If we can appropriately divide up the feasible domain into regions of the type that Lemma 2.2 applies to, then we are able to solve Problem (1) in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

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Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane

et al. 2016

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We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in R 2 . Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in R 2 can be done in polynomial time, while optimizing a quartic polynomial in the same type of region is NP-hard. We close the gap by showing that this problem can be solved in polynomial time for cubic polynomials. Furthermore, we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a bounded polyhedron in R 2 is solvable in polynomial time. We show that this holds for polynomials that can be translated into homogeneous polynomials, even when the translation vector is unknown. We demonstrate that such problems in the unbounded case can have smallest optimal solutions of exponential size in the size of the input, thus requiring a compact representation of solutions for a general polynomial time algorithm for the unbounded case.

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“…The complexity result is the following. A complexity result of greater generality was presented by Khachiyan and Porkolab [79]. It covers the case of minimization of convex polynomials over the integer points in convex semialgebraic sets given by arbitrary (not necessarily quasiconvex) polynomials.…”

Section: Fixed Dimensionmentioning

confidence: 99%

Nonlinear Integer Programming

Li¹,

Sun²

2006

International Series in Operations Research &Amp; Management Science

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Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic.The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms.We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems.It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.

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Integer Optimization on Convex Semialgebraic Sets

Cited by 46 publications

References 17 publications

Unbounded convex sets for non-convex mixed-integer quadratic programming

Unbounded convex sets for non-convex mixed-integer quadratic programming

Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane

Nonlinear Integer Programming

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