Minimum Problems over Sets of Concave Functions and Related Questions

Buttazzo, Giuseppe; Ferone, Vincenzo; Kawohl, Bernd

doi:10.1002/mana.19951730106

Cited by 122 publications

(155 citation statements)

References 12 publications

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“…Variational problems subject to a convexity constraint arise in different applied settings: economics with Rochet-Choné model [30] (which corresponds to the principal-agent problem with a bilinear b), but also Newton's least resistance problem (see [7], [8], [20]). We shall also see in section 5 how to relate such variational problems with the computations of convex envelopes, a problem with its own interest.…”

Section: The Case Of the Convexity Constraintmentioning

confidence: 99%

“…Variational problems subject to a convexity constraint arise in several different contexts such as mathematical economics [30], Newton's least resistance problem [7], [8], [20], optimal transport for the quadratic cost [6] or shape optimization [9], [21]. Existence of minimizers is generally not an issue since the set of convex functions have good local compactness properties in most reasonable functional spaces.…”

Section: Introductionmentioning

confidence: 99%

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An Iterated Projection Approach to Variational Problems Under Generalized Convexity Constraints

Carlier

Dupuis

2016

Appl Math Optim

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The principal-agent problem in economics leads to variational problems subject to global constraints of b-convexity on the admissible functions, capturing the so-called incentive-compatibility constraints. Typical examples are minimization problems subject to a convexity constraint. In a recent pathbreaking article, Figalli, Kim and McCann [19] identified conditions which ensure convexity of the principal-agent problem and thus raised hope on the development of numerical methods. We consider special instances of projections problems over b-convex functions and show how they can be solved numerically using Dykstra's iterated projection algorithm to handle the b-convexity constraint in the framework of [19]. Our method also turns out to be simple for convex envelope computations.

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Section: The Case Of the Convexity Constraintmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An Iterated Projection Approach to Variational Problems Under Generalized Convexity Constraints

Carlier

Dupuis

2016

Appl Math Optim

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“…Therefore, the usual direct methods of the calculus of variations for obtaining the existence of an optimal solution may fail. However, due to the concavity constraint, the existence of a minimizer u still holds, as it has been proved in [59]. A complete discussion on the problem above can be found in [37] where all the concerning references are quoted.…”

Section: The Newton's Problem Of Optimal Aerodynamical Profilesmentioning

confidence: 99%

Optimal Shapes and Masses, and Optimal Transportation Problems

Buttazzo

Pascale

2003

Optimal Transportation and Applications

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Summary. We present here a survey on some shape optimization problems that received particular attention in the last years. In particular, we discuss a class of problems, that we call mass optimization problems, where one wants to find the distribution of a given amount of mass which optimizes a given cost functional. The relation with mass transportation problems will be discussed, and several open problems will be presented.

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“…Since the early 1990s, there have been obtained new interesting results related to the problem of minimal resistance in various classes of admissible bodies [1][2][3][4][5][6][7][8][9]11,12]. In particular, there has been considered the wider class of convex (generally non-symmetric) bodies inscribed in a given cylinder [1,3,4,7,9].…”

Section: Introductionmentioning

confidence: 99%

“…In particular, there has been considered the wider class of convex (generally non-symmetric) bodies inscribed in a given cylinder [1,3,4,7,9]. It was shown that the solution in this class exists and does not coincide with the Newton one.…”

Section: Introductionmentioning

confidence: 99%

The problem of the body of revolution of minimal resistance

Plakhov

Aleksenko

2008

ESAIM: COCV

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Abstract. Newton's problem of the body of minimal aerodynamic resistance is traditionally stated in the class of convex axially symmetric bodies with fixed length and width. We state and solve the minimal resistance problem in the wider class of axially symmetric but generally nonconvex bodies. The infimum in this problem is not attained. We construct a sequence of bodies minimizing the resistance. This sequence approximates a convex body with smooth front surface, while the surface of approximating bodies becomes more and more complicated. The shape of the resulting convex body and the value of minimal resistance are compared with the corresponding results for Newton's problem and for the problem in the intermediate class of axisymmetric bodies satisfying the single impact assumption [Comte and Lachand-Robert, J. Anal. Math. 83 (2001) 313-335]. In particular, the minimal resistance in our class is smaller than in Newton's problem; the ratio goes to 1/2 as (length)/(width of the body) → 0, and to 1/4 as (length)/(width) → +∞.Mathematics Subject Classification. 49K30, 49Q10.

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Minimum Problems over Sets of Concave Functions and Related Questions

Cited by 122 publications

References 12 publications

An Iterated Projection Approach to Variational Problems Under Generalized Convexity Constraints

An Iterated Projection Approach to Variational Problems Under Generalized Convexity Constraints

Optimal Shapes and Masses, and Optimal Transportation Problems

The problem of the body of revolution of minimal resistance

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