Duality in Convex Optimisation

2 Maio 2019, 14:30 • José Francisco da Silva Costa Rodrigues

Motivation: formulation of the Primal and the Dual problem in the example of de Dirichlet integral associated with the Dirichlet problem for the Poisson equation—example of the conjugate function; minimisation of the solution versus maximising the symmetric of its gradient.Formulation of the abstract Primal and the Dual problem in the duality framework of convex functionals and sufficient conditions for the close relations between the two problems, for the existence of solutions and the respective extremality equations in general reflexive Banach spaces. Concretisation in important special cases in the Calculus of Variations. Examples: the linear Neumann problem; the nonlinear Dirichlet problem of p-Laplacian type; the Stokes problem; and the linear problem in elastostatics.References: Chap. 3 and 4 of Ekeland-Teman and the Duvaut-Lions' book for the Dual formulation in linear elasticity in page 119 and its relation with "Lagrange multipliers" in page 120 (see the connections of page 67 of Ekeland-Teman book). — The challenge is to put this system of linear elasticity in the duality framework, i.e., a special case of the general abstract theory of duality in convex analysis. (suggestion: start with the full Dirichlet homogeneous boundary data).