  In this paper, "chance optimization" problems are introduced, where one aims
at maximizing the probability of a set defined by polynomial inequalities.
These problems are, in general, nonconvex and computationally hard. With the
objective of developing systematic numerical procedures to solve such problems,
a sequence of convex relaxations based on the theory of measures and moments is
provided, whose sequence of optimal values is shown to converge to the optimal
value of the original problem. Indeed, we provide a sequence of semidefinite
programs of increasing dimension which can arbitrarily approximate the solution
of the original problem. To be able to efficiently solve the resulting
large-scale semidefinite relaxations, a first-order augmented Lagrangian
algorithm is implemented. Numerical examples are presented to illustrate the
computational performance of the proposed approach.
