Lp-norms, Log-barriers and Cramer transform in Optimization

We show that the Laplace approximation of a supremum by L-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.