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

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

2010-01-08

arxiv.org/abs/1001.1208v2

Mathematics

Optimization and Control

Scientific paper

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

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