Maximum length of steepest descent curves for quasi-convex functions

Manselli, Paolo; Pucci, Carlo

doi:10.1007/bf00181220

Cited by 34 publications

(78 citation statements)

References 4 publications

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“…After the initial announcement of this work, Sébastien Bubeck pointed out to us that the greedy algorithm (which moves at each time to the closest feasible point from the current location) is also d O(d) -competitive, and that this can be shown using the results about gradient flows due to Manselli and Pucci [MP91].…”

Section: Discussionmentioning

confidence: 98%

Nested Convex Bodies are Chaseable

Bansal

Böhm

Eliáš

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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In the Convex Body Chasing problem, we are given an initial point v 0 ∈ R d and an online sequence of n convex bodies F 1 , . . . , F n . When we receive F i , we are required to move inside F i . Our goal is to minimize the total distance traveled. This fundamental online problem was first studied by Friedman and Linial (DCG 1993). They proved an Ω( √ d) lower bound on the competitive ratio, and conjectured that a competitive ratio depending only on d is possible. However, despite much interest in the problem, the conjecture remains wide open.We consider the setting in which the convex bodies are nested: F 1 ⊃ . . . ⊃ F n . The nested setting is closely related to extending the online LP framework of Buchbinder and Naor (ESA 2005) to arbitrary linear constraints. Moreover, this setting retains much of the difficulty of the general setting and captures an essential obstacle in resolving Friedman and Linial's conjecture. In this work, we give a f (d)-competitive algorithm for chasing nested convex bodies in R d .

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Section: Discussionmentioning

confidence: 98%

Nested Convex Bodies are Chaseable

Bansal

Böhm

Eliáš

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…The previous corollary shows a significant difference of the isoclines from steepest descent lines for quasi convex functions (see [14]). This difference also appears in L 1 norm as a consequence of the following theorem.…”

Section: Isoclines Of Arbitrary Lengthmentioning

confidence: 90%

“…Plugging (15) into (12) one gets (14). g Let us compare isoclines with level lines and steepest descent lines, parameterized by…”

Section: Isocline Linesmentioning

confidence: 99%

On isocline lines for functions and convex stratifications of two variables

Longinetti

Manselli

Venturi

2010

Applicable Analysis

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Let the isoclines of a function u be the level lines of the function ¼ arg(Du). Formulas for the curvature and the length of isocline lines in terms of the curvatures k, j of the level curves and of the steepest descent lines of u are given. The special case when all isoclines are straight lines is studied: in this case the steepest descent lines bend proportionally to the level lines; the support function of the level lines is linear function on the isoclines parameterized by the level values, possibly changing them. This characterization gives a new proof of a property of the developable surfaces found in [A. Fialkow, Geometric characterization of invariant partial differential equations, Amer. J. Math. 59(4) (1937), pp. 833-844]. When u is in the class of quasi convex functions, the L p norm of the length function I of the isoclines has minimizers with isoclines straight lines; the same occurs for other functionals on u depending on k, j. For a strictly regular quasi convex function isoclines may have arbitrarily large length and arbitrarily large L 1 norm of I .

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“…In [1] (see also [4,5]), it was proved that if w(K ) is bounded a priori, then (γ, 1, co(γ )) has an upper bound depending on n only.…”

Section: Introductionmentioning

confidence: 97%

“…[6] For n > 3, finding the value of I is an open problem. The relation of previous problem with steepest descent curves was hinted in [1] and made more precise in Section 5.…”

Section: Introductionmentioning

confidence: 98%

On variational problems related to steepest descent curves and self-dual convex sets on the sphere

Longinetti

Manselli

Venturi

2014

Applicable Analysis

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Let C be the family of compact convex subsets S of the hemisphere in R n with the property that S contains its dual S * ; let u ∈ S * , and let (S, u) = 2 ω n S θ, u dσ (θ). The problem to study inf (S, u), S ∈ C, u ∈ S * is considered. It is proved that there exists a minimizing couple (S, u) such that S is self-dual and u is on its boundary. More can be said for n = 3: the minimum set is a Reuleaux triangle on the sphere. The previous problem is related to the one to find the maximal length of steepest descent curves for quasi-convex functions, satisfying suitable constraints. For n = 2, let us refer to [Manselli P, Pucci C. Maximum length of steepest descent curves for quasi-convex functions. Geom. Dedicata. 1991]. Here, quite different results are obtained for n ≥ 3.

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Maximum length of steepest descent curves for quasi-convex functions

Cited by 34 publications

References 4 publications

Nested Convex Bodies are Chaseable

Nested Convex Bodies are Chaseable

On isocline lines for functions and convex stratifications of two variables

On variational problems related to steepest descent curves and self-dual convex sets on the sphere

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