Ellipsoidal mixed-integer representability

Pia, Alberto Del; Poskin, Jeffrey

doi:10.1007/s10107-017-1196-6

Cited by 5 publications

(2 citation statements)

References 15 publications

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“…For instance, Theorem 6 in [5] can be used to show that for any α > 0, S α := x ∈ Z 2 : x 1 x 2 ≥ α satisfies (3) with S i containing a single integer vector for each i ∈ k . The only mixed-integer and non-polyhedral result we are aware of is a characterization of the form (3) when M is the intersection of a rational polyhedron with an ellipsoidal cylinder having a rational recession cone [4]. An identical proof also holds when the recession cone of M is a rational subspace and M is contained in a rational polyhedron with the same recession cone as M .…”

Section: Bounded and Other Restricted Micp Representability Resultsmentioning

confidence: 95%

“…To our knowledge, we are the first authors to consider this general case. Related but more specific analysis has been developed by Del Pia and Poskin [4] where they characterized the case where the convex set is an intersection of a polyhedron with an ellipsoidal region and by Dey and Morán [5] where they studied the structure of integer points within convex sets but without allowing a mix of continuous and discrete variables.…”

Section: Introductionmentioning

confidence: 99%

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Mixed-Integer Convex Representability

2022

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Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible regions of MICP problems. We establish several results in this direction, including the first complete characterization for the mixed-binary case and a simple necessary condition for the general case. We use the latter to derive the first nonrepresentability results for various nonconvex sets, such as the set of rank-1 matrices and the set of prime numbers. Finally, in correspondence with the seminal work on mixed-integer linear representability by Jeroslow and Lowe, we study the representability question under rationality assumptions. Under these rationality assumptions, we establish that representable sets obey strong regularity properties, such as periodicity, and we provide a complete characterization of representable subsets of the natural numbers and of representable compact sets. Interestingly, in the case of subsets of natural numbers, our results provide a clear separation between the mathematical modeling power of mixed-integer linear and mixed-integer convex optimization. In the case of compact sets, our results imply that using unbounded integer variables is necessary only for modeling unbounded sets.

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Section: Bounded and Other Restricted Micp Representability Resultsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Mixed-Integer Convex Representability

2022

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Mixed-Integer Convex Representability

Lubin

Vielma

Zadik

2017

Integer Programming and Combinatorial Optimization

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Abstract. We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer assignments is finite. We develop a characterization for the more general case of unbounded integer variables together with a simple necessary condition for representability which we use to prove the first known negative results. Finally, we study representability of subsets of the natural numbers, developing insight towards a more complete understanding of what modeling power can be gained by using convex sets instead of polyhedral sets; the latter case has been completely characterized in the context of mixed-integer linear optimization.

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Characterizations of mixed binary convex quadratic representable sets

Pia

Poskin

2018

Math. Program.

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In pure integer linear programming it is often desirable to work with polyhedra that are full-dimensional, and it is well known that it is possible to reduce any polyhedron to a fulldimensional one in polynomial time. More precisely, using the Hermite normal form, it is possible to map a non full-dimensional polyhedron to a full-dimensional isomorphic one in a lower-dimensional space, while preserving integer vectors. In this paper, we extend the above result simultaneously in two directions. First, we consider mixed integer vectors instead of integer vectors, by leveraging on the concept of "integer reflexive generalized inverse." Second, we replace polyhedra with convex quadratic sets, which are sets obtained from polyhedra by enforcing one additional convex quadratic inequality. We study structural properties of convex quadratic sets, and utilize them to obtain polynomial time algorithms to recognize full-dimensional convex quadratic sets, and to find an affine function that maps a non full-dimensional convex quadratic set to a full-dimensional isomorphic one in a lower-dimensional space, while preserving mixed integer vectors. We showcase the applicability and the potential impact of these results by showing that they can be used to prove that mixed integer convex quadratic programming is fixed parameter tractable with parameter the number of integer variables. Our algorithm unifies and extends the known polynomial time solvability of pure integer convex quadratic programming in fixed dimension and of convex quadratic programming.

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Ellipsoidal mixed-integer representability

Cited by 5 publications

References 15 publications

Mixed-Integer Convex Representability

Mixed-Integer Convex Representability

Mixed-Integer Convex Representability

Characterizations of mixed binary convex quadratic representable sets

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