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A finite dimensional linear programming approximation of Mather's variational problem

Luca Granieri (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We provide an approximation of Mather variational problem by finite dimensional minimization problems in the framework of Γ-convergence. By a linear programming interpretation as done in [Evans and Gomes, ESAIM: COCV 8 (2002) 693–702] we state a duality theorem for the Mather problem, as well a finite dimensional approximation for the dual problem.

A generalized limited-memory BNS method based on the block BFGS update

Vlček, Jan, Lukšan, Ladislav (2017)

Programs and Algorithms of Numerical Mathematics

A block version of the BFGS variable metric update formula is investigated. It satisfies the quasi-Newton conditions with all used difference vectors and gives the best improvement of convergence in some sense for quadratic objective functions, but it does not guarantee that the direction vectors are descent for general functions. To overcome this difficulty and utilize the advantageous properties of the block BFGS update, a block version of the limited-memory BNS method for large scale unconstrained...

A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two

Alexandre Caboussat, Roland Glowinski, Danny C. Sorensen (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We address in this article the computation of the convex solutions of the Dirichlet problem for the real elliptic Monge − Ampère equation for general convex domains in two dimensions. The method we discuss combines a least-squares formulation with a relaxation method. This approach leads to a sequence of Poisson − Dirichlet problems and another sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite element approximations with a smoothing procedure are used for the...

A minimum effort optimal control problem for elliptic PDEs

Christian Clason, Kazufumi Ito, Karl Kunisch (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation...

A minimum effort optimal control problem for elliptic PDEs

Christian Clason, Kazufumi Ito, Karl Kunisch (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation...

A moving mesh fictitious domain approach for shape optimization problems

Raino A.E. Mäkinen, Tuomo Rossi, Jari Toivanen (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

A new numerical method based on fictitious domain methods for shape optimization problems governed by the Poisson equation is proposed. The basic idea is to combine the boundary variation technique, in which the mesh is moving during the optimization, and efficient fictitious domain preconditioning in the solution of the (adjoint) state equations. Neumann boundary value problems are solved using an algebraic fictitious domain method. A mixed formulation based on boundary Lagrange multipliers is...

A new series of conjectures and open questions in optimization and matrix analysis

Jean-Baptiste Hiriart-Urruty (2009)

ESAIM: Control, Optimisation and Calculus of Variations

We present below a new series of conjectures and open problems in the fields of (global) Optimization and Matrix analysis, in the same spirit as our recently published paper [J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAM Review 49 (2007) 255–273]. With each problem come a succinct presentation, a list of specific references, and a view on the state of the art of the subject.

A new series of conjectures and open questions in optimization and matrix analysis

Jean-Baptiste Hiriart-Urruty (2008)

ESAIM: Control, Optimisation and Calculus of Variations

We present below a new series of conjectures and open problems in the fields of (global) Optimization and Matrix analysis, in the same spirit as our recently published paper [J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAM Review49 (2007) 255–273]. With each problem come a succinct presentation, a list of specific references, and a view on the state of the art of the subject.

A nonsmooth optimisation approach for the stabilisation of time-delay systems

Stefan Vandewalle, Wim Michiels, Koen Verheyden, Joris Vanbiervliet (2008)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is concerned with the stabilisation of linear time-delay systems by tuning a finite number of parameters. Such problems typically arise in the design of fixed-order controllers. As time-delay systems exhibit an infinite amount of characteristic roots, a full assignment of the spectrum is impossible. However, if the system is stabilisable for the given parameter set, stability can in principle always be achieved through minimising the real part of the rightmost characteristic root, or...

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