$L^p$-norms, log-barriers and Cramer transform in optimization

Jean B. Lasserre(lasserrelaas.fr)
E. Santillan Zeron(eszeronmath.cinvestav.mx)

Abstract: We show that the Laplace approximation of a supremum by $L^p$-norms has interesting consequences in optimization. For instance, the logarithmic barrier functions (LBF) of a primal convex problem $P$ and its dual $P^*$ appear naturally when using this simple approximation technique for the value function $g$ of $P$ or its Legendre-Fenchel conjugate $g^*$. In addition, minimizing the LBF of the dual $P^*$ is just evaluating the Cramer transform of the Laplace approximation of $g$. Finally, this technique permits to sometimes define an explicit dual problem $P^*$ in cases when the Legendre-Fenchel conjugate $g^*$ cannot be derived explicitly from its definition.

Keywords: Logarithmic Barrier Function; Legendre-Fenchel and Cramer transforms

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

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Entry Submitted: 01/13/2010
Entry Accepted: 01/13/2010
Entry Last Modified: 01/13/2010

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