Approximation of convex bodies by polytopes

Gruber, Peter M.; Kenderov, P. S.

doi:10.1007/bf02844354

Cited by 56 publications

(58 citation statements)

References 14 publications

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“…The problem of outer-approximating convex functions is closely related to that of polyhedral approximations to convex sets [15,14]. It is known that the distance between a planar convex figure and its best approximating n − gon is O(1/n 2 ) under various error measures like Hausdorff, and area of the symmetric difference.…”

Section: Outer-approximationmentioning

confidence: 99%

Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

Tawarmalani

Sahinidis²

2004

Mathematical Programming

477

285

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This work addresses the development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixed-integer nonlinear programs. Towards this end, we develop novel relaxation schemes, range reduction tests, and branching strategies which we incorporate into the prototypical branch-and-bound algorithm.In the theoretical/algorithmic part of the paper, we begin by developing novel strategies for constructing linear relaxations of mixed-integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties. We then use Lagrangian/linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming. This theory leads to new range reduction schemes, including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch-and-bound tree. Finally, we incorporate these relaxation and reduction strategies in a branch-and-bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems.In the computational part of the paper, we describe our implementation discussing, wherever appropriate, the use of suitable data structures and associated algorithms. We present computational experience with benchmark separable concave quadratic programs, fractional 0 − 1 programs, and mixed-integer nonlinear programs from applications in synthesis of chemical processes, engineering design, just-in-time manufacturing, and molecular design.

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Section: Outer-approximationmentioning

confidence: 99%

Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

Tawarmalani

Sahinidis²

2004

Mathematical Programming

477

285

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“…A new understanding of the problem came by the method developed by Gruber [7]. This method could handle all related questions (say for the Banach-Mazur distance), and the the assumption on the boundary could be relaxed to C. See Gruber's comprehensive surveys [8] and [9] for a detailed history of this field. The central problem of this paper is to estimate the error of these asymptotic formulae as n tends to infinity.…”

Section: (M P(") )mentioning

confidence: 99%

“…In this paper a bound of order n- 5/(2(d-1)) is given for the error of the asymptotic formulae. This bound is clearly not the best possible, and Gruber [9] conjectured that if the boundary of M is sufficiently smooth, then there exist asymptotic expansions for SH(P", M) and 8H (P(5) , M). With the help of quasiconformal mappings, we show for the three-dimensional unit ball that the error is at least f (n) • n -2 where f (n) tends to infinity.…”

mentioning

confidence: 99%

About the error term for best approximation with respect to the Hausdorff related metrics

Böröczky

2001

Discrete Comput Geom

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Abstract. Let M be a convex body with C+ boundary in R°, d > 3, and consider a polytope P. (or P(" ) ) with at most n vertices (at most n facets) minimizing the Hausdorff distance from M. It has long been known that as n tends to infinity, there exist asymptotic formulae of order n-2/(d-1) for the Hausdorff distances 8H(Pa , M) and 8H(P(5), M). In this paper a bound of order n- 5/(2(d-1)) is given for the error of the asymptotic formulae. This bound is clearly not the best possible, and Gruber [9] conjectured that if the boundary of M is sufficiently smooth, then there exist asymptotic expansions for SH(P", M) and 8H (P(5) , M). With the help of quasiconformal mappings, we show for the three-dimensional unit ball that the error is at least f (n) • n -2 where f (n) tends to infinity. Therefore in this case, no asymptotic expansion exists in terms of n -2/ (d-l) = n -l .

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“…Generalizing, Similarly, we replace each convex inequality with a valid linear outer-approximation. (Tawarmalani & Sahinidis , 2002, Gruber & Kenderov, 1982.…”

Section: Gdp B and Gdp Co )mentioning

confidence: 99%

Strengthening of lower bounds in the global optimization of Bilinear and Concave Generalized Disjunctive Programs

Ruiz

Grossmann

2010

Computers & Chemical Engineering

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This paper is concerned with global optimization of Bilinear and Concave Generalized Disjunctive Programs. A major objective is to propose a procedure to find relaxations that yield strong lower bounds. We first present a general framework for obtaining a hierarchy of linear relaxations for nonconvex Generalized Disjunctive Programs (GDP).This framework combines linear relaxation strategies proposed in the literature for nonconvex MINLPs with the results of the work by Sawaya & Grossmann (2008) for Linear GDPs. We further exploit the theory behind Disjunctive Programming to guide more efficiently the generation of relaxations by considering the particular structure of the problems. Finally, we show through a set of numerical examples how these new relaxations can strenghten substantially the lower bounds for the global optimum, often leading to a significant reduction of the number of nodes when used within a spatial branch and bound framework.

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Approximation of convex bodies by polytopes

Cited by 56 publications

References 14 publications

Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

About the error term for best approximation with respect to the Hausdorff related metrics

Strengthening of lower bounds in the global optimization of Bilinear and Concave Generalized Disjunctive Programs

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