Extremum problems for zonotopes

Filliman, P.

doi:10.1007/bf00181491

Cited by 13 publications

(21 citation statements)

References 12 publications

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“…Then, following the proof of[16, Theorem 1], D A 1 2 , hence A 1 = d/D. As Z is enclosed in the d-sphere of radius √ D and the d-sphere of radius D/d is enclosed in I, the result follows from the formula of the volume of a d-sphere of radius ρ:π d/2 Γ(d/2+1) ρ d .…”

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confidence: 85%

On the choice of the low-dimensional domain for global optimization via random embeddings

2019

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The challenge of taking many variables into account in optimization problems may be overcome under the hypothesis of low effective dimensionality. Then, the search of solutions can be reduced to the random embedding of a low dimensional space into the original one, resulting in a more manageable optimization problem. Specifically, in the case of time consuming black-box functions and when the budget of evaluations is severely limited, global optimization with random embeddings appears as a sound alternative to random search. Yet, in the case of box constraints on the native variables, defining suitable bounds on a low dimensional domain appears to be complex. Indeed, a small search domain does not guarantee to find a solution even under restrictive hypotheses about the function, while a larger one may slow down convergence dramatically. Here we tackle the issue of low-dimensional domain selection based on a detailed study of the properties of the random embedding, giving insight on the aforementioned difficulties. In particular, we describe a minimal low-dimensional set in correspondence with the embedded search space. We additionally show that an alternative equivalent embedding procedure yields simultaneously a simpler definition of the low-dimensional minimal set and better properties in practice. Finally, the performance and robustness gains of the proposed enhancements for Bayesian optimization are illustrated on numerical examples.

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confidence: 85%

On the choice of the low-dimensional domain for global optimization via random embeddings

2019

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“…Using the notion of zonotopes, it can be shown that the convex hull of these projections is the triacontahedron, and that it is the largest projection of C 6 into R 3 (see [6,Th. 81).…”

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confidence: 99%

A metrical characterization of symmetric polytopes

Filliman

1989

Geom Dedicata

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“…Section 4 relates the largest projections of the regular simplex, cross polytope, and cube to the solutions of other extremum problems and gives some necessary conditions. One interesting result is that if the vertices of a maximal projection of T "-~ lie on a sphere, then they also solve the problem of finding the largest volume of the convex hull of n points on a sphere in R k. Some bounds and solutions for the projections of a cube can be found in [6], [11], and [19].…”

Section: Theorem 1 the Function V( Q : L) Is Piecewise Linear On G( mentioning

confidence: 99%

“…Henceforth the results shall be stated only for cony S, with the understanding that similar results also hold for conv{±S} and Y, [si]. In particular, the theorems for zonotopes can all be found in Sections 2 and 3 of [11].…”

Section: (43)mentioning

confidence: 99%

Exterior algebra and projections of polytopes

Filliman

1990

Discrete Comput Geom

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This paper explores the metrical properties of convex polytopes by means of the classical Pliicker embedding of the Grassmannian G(k, n) of k-planes in R n into the exterior algebra AkR ~. The results follow from the description of the volume of the projection of a polytope into a k-plane by a piecewise linear function on G(k, n). For example, the Hodge-star operator is used to obtain the volume of a polytope from its Gale transform. Also, the classification of the faces of G(2, n) (or G(n-2, n)) imply that the largest projection within a particular combinatorial type is unique if k = 2 or n-2.

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Extremum problems for zonotopes

Cited by 13 publications

References 12 publications

On the choice of the low-dimensional domain for global optimization via random embeddings

On the choice of the low-dimensional domain for global optimization via random embeddings

A metrical characterization of symmetric polytopes

Exterior algebra and projections of polytopes

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