L p -Norms, Log-Barriers and Cramer Transform in Optimization

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.