Displaying similar documents to “The Lazy Travelling Salesman Problem in ℝ 2 ”

Linear convergence in the approximation of rank-one convex envelopes

Sören Bartels (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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A linearly convergent iterative algorithm that approximates the rank-1 convex envelope  f r c of a given function f : ℝ n × m → ℝ , the largest function below which is convex along all rank-1 lines, is established. The proposed algorithm is a modified version of an approximation scheme due to Dolzmann and Walkington.

An interior point algorithm for convex quadratic programming with strict equilibrium constraints

Rachid Benouahboun, Abdelatif Mansouri (2010)

RAIRO - Operations Research

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We describe an interior point algorithm for convex quadratic problem with a strict complementarity constraints. We show that under some assumptions the approach requires a total of O ( n L ) number of iterations, where is the input size of the problem. The algorithm generates a sequence of problems, each of which is approximately solved by Newton's method.

Geometric constraints on the domain for a class of minimum problems

Graziano Crasta, Annalisa Malusa (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider minimization problems of the form min u ∈ ϕ + W 0 1 , 1 ( Ω ) ∫ Ω [ f ( D u ( x ) ) - u ( x ) ] d x where Ω ⊆ ℝ N is a bounded convex open set, and the Borel function f : ℝ N → [ 0 , + ∞ ] is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of and the zero level set of , we prove that the viscosity solution of a related Hamilton–Jacobi equation provides a minimizer for the integral functional.

Existence of optimal maps in the reflector-type problems

Wilfrid Gangbo, Vladimir Oliker (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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In this paper, we consider probability measures and on a -dimensional sphere in 𝐑 d + 1 , d ≥ 1 , and cost functions of the form c ( 𝐱 , 𝐲 ) = l ( | 𝐱 - 𝐲 | 2 2 ) that generalize those arising in geometric optics where l ( t ) = - log t . We prove that if and vanish on ( d - 1 ) -rectifiable sets, if lim t → 0 + l ( t ) = + ∞ , and g ( t ) : = t ( 2 - t ) ( l ' ( t ) ) 2 is monotone then there exists a unique optimal map that transports onto ν , where optimality is measured against Furthermore, inf 𝐱 | T o 𝐱 - 𝐱 | > 0 . Our approach is based on direct variational arguments. In the special case when l ( t ) = - log t , existence...

Generalized Characterization of the Convex Envelope of a Function

Fethi Kadhi (2010)

RAIRO - Operations Research

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We investigate the minima of functionals of the form ∫ [ a , b ] g ( u ˙ ( s ) ) d s where is strictly convex. The admissible functions u : [ a , b ] ⟶ ℝ are not necessarily convex and satisfy u ≤ f on , , , is a fixed function on . We show that the minimum is attained by f ¯ , the convex envelope of .