Introduction

Following the theory of idempotent Maslov measures, a formalism analogous to probability calculus is obtained for optimization by replacing the classical structure of real numbers (ℝ, +, ×) by the idempotent semifield obtained by endowing the set ℝ ∪ {+∞} with the “min” and “+” operations. To the probability of an event corresponds the cost of a set of decisions. To random variables correspond decision variables.

Weak convergence, tightness and limit theorems of probability have an optimization counterpart which is useful for approximating the Hamilton–;Jacobi–Bellman (HJB) equation and obtaining asymptotics for this equation. The introduction of tightness for cost measures and its consequences is the main contribution of this paper. A link is established between weak convergence and the epigraph convergence used in convex analysis.

The Cramér transform used in the large deviation literature is defined as the composition of the Laplace transform by the logarithm by the Fenchel transform. It transforms convolution into inf-convolution. Probabilistic results about processes with independent increments are then transformed into similar results on dynamic programming equations. The Cramér transform gives new insight into the Hopf method used to compute explicit solutions of some HJB equations. It also explains the limit theorems obtained directly as the image of the classic limit theorems of probability.

Cost Measures and Decision Variables

Let us denote by ℝmin the idempotent semifield (ℝ ∪ {+∞}, min, +) and by extension the metric space ℝ ∪ {+∞} endowed with the exponential distance d(x, y) = I exp(–x) – exp(–y).