Concave envelopes of monomial functions over rectangles

Benson, Harold P.

doi:10.1002/nav.20011

Cited by 9 publications

(6 citation statements)

References 30 publications

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“…), for every affine function g. (5) Proof (⇒) First note that the assumptions on S and T imply conv(S) = conv(T ). Let g(x) be an affine function.…”

Section: Theorem 1 Let F Be a Lower Semicontinuous Function On A Compmentioning

confidence: 97%

“…where the fifth equality follows from assumption (5). In order to complete the proof, we apply Proposition 3 to the functions f and conv T ( f ) to obtain…”

Section: Theorem 1 Let F Be a Lower Semicontinuous Function On A Compmentioning

confidence: 97%

“…Knowing that a function has a vertex polyhedral convex envelope is important because such envelope can be always be expressed as the maximum of a finite number of affine functions and such expression can be explicitely computed (possibly in exponential time) (see [3][4][5][6]8,12,17,[20][21][22][24][25][26][27][30][31][32][33][34]). …”

Section: Theorem 2 a Function F Has A Vertex Polyhedral Convex Envelomentioning

confidence: 99%

“…In this paper we mainly focus on functions that admit a vertex polyhedral convex envelope on a polytope P, i.e., on those functions whose convex envelope on P coincides with the convex envelope of their restrictions to the vertices of P. This property is known to hold for concave functions [11], for multilinear functions on boxes and other special polytopes [5,24,26], and for some other rather restricted classes of functions [4,6,26,27,32]. Note that if a function admits a vertex polyhedral convex envelope on P ⊆ R n , then such convex envelope can be computed exactly, and it can be expressed as the maximum of a finite family of affine functions.…”

Section: Introductionmentioning

confidence: 98%

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Existence and sum decomposition of vertex polyhedral convex envelopes

Tardella

2007

Optimization Letters

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Convex envelopes are a very useful tool in global optimization. However finding the exact convex envelope of a function is a difficult task in general. This task becomes considerably simpler in the case where the domain is a polyhedron and the convex envelope is vertex polyhedral, i.e., has a polyhedral epigraph whose vertices correspond to the vertices of the domain. A further simplification is possible when the convex envelope is sum decomposable, i.e., the convex envelope of a sum of functions coincides with the sum of the convex envelopes of the summands. In this paper we provide characterizations and sufficient conditions for the existence of a vertex polyhedral convex envelope. Our results extend and unify several results previously obtained for special cases of this problem. We then characterize sum decomposability of vertex polyhedral convex envelopes, and we show, among else, that the vertex polyhedral convex envelope of a sum of functions coincides with the sum of the vertex polyhedral convex envelopes of the summands if and only if the latter sum is vertex polyhedral

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“…), for every affine function g. (5) Proof (⇒) First note that the assumptions on S and T imply conv(S) = conv(T ). Let g(x) be an affine function.…”

Section: Theorem 1 Let F Be a Lower Semicontinuous Function On A Compmentioning

confidence: 97%

“…where the fifth equality follows from assumption (5). In order to complete the proof, we apply Proposition 3 to the functions f and conv T ( f ) to obtain…”

Section: Theorem 1 Let F Be a Lower Semicontinuous Function On A Compmentioning

confidence: 97%

Section: Theorem 2 a Function F Has A Vertex Polyhedral Convex Envelomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Existence and sum decomposition of vertex polyhedral convex envelopes

Tardella

2007

Optimization Letters

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“…An important area of study regarding convexification schemes for QCQPs is to convexify commonly occurring substructures, like in the case of integer programming. However, most of the work in this direction in the global optimization area has been focused on convexification of functions (i.e., finding convex and concave envelopes), see for example [4,46,33,10,36,9,8,38,21,48,45,37,52,53,16,19,1,26]. There are relatively lesser number of results on convexification of sets [51,41,42,50,24,30,44,20,32,17,40,23].…”

Section: Introduction 1motivationmentioning

confidence: 99%

A study of rank-one sets with linear side constraints and application to the pooling problem

Dey,

Kocuk,

Santana

2019

Preprint

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We study sets defined as the intersection of a rank-1 constraint with different choices of linear side constraints. We identify different conditions on the linear side constraints, under which the convex hull of the rank-1 set is polyhedral or second-order cone representable. In all these cases, we also show that a linear objective can be optimized in polynomial time over these sets. Towards the application side, we show how these sets relate to commonly occurring substructures of a general quadratically constrained quadratic program. To further illustrate the benefit of studying quadratically constrained quadratic programs from a rank-1 perspective, we propose new rank-1 formulations for the generalized pooling problem and use our convexification results to obtain several new convex relaxations for the pooling problem. Finally, we run a comprehensive set of computational experiments and show that our convexification results together with discretization significantly help in improving dual bounds for the generalized pooling problem.

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