Laplace Method and Applications to Optimization Problems

Background

Laplace's method is based on an ingenious trick used by Laplace in one his papers [19]. The technique is most frequently used to perform asymptotic evaluations to integrals that depend on a scalar parameter t, as t tends to infinity. Its use can be theoretically justified for integrals in the following form:

$$ I(t)=\int_\mathcal A \exp\left\{\frac{-f(x)}{T(t)}\right\} \text{d}\Lambda(x)\:. $$

Where $f:\mathbb R^n\rightarrow \mathbb R$, $T:\mathbb R\rightarrow \mathbb R$, are assumed to be smooth, and $T(t)\rightarrow 0$ as t tends to ∞. $\mathcal A$ is some compact set, and Λ is some measure on $\mathcal B$ (the $\sigma-$field generated by $\mathcal A$). We know that since $\mathcal A$ is compact, the continuous function f will have a global minimum in $\mathcal A$. For simplicity, assume that the global minimum x ^* is unique, and that it occurs in the interior $\mathcal A$. Under these conditions, and as ttends to infinity, only...