“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,
a semidefinite relaxations is implemented. first-order augmented Lagrangian algorithmThe outline of this work is as follows: I) Chance Optimization over a Semialgebraic Set - An Equivalent Problem - Semidefinite Relaxations - Discussion on Improving Estimates of Probability - Orthogonal Basis II) Chance Optimization over a Union of Sets - An Equivalent Problem - Semidefinite Relaxations III) Regularized Chance Optimization Using Trace Norm IV) First-Order Augmented Lagrangian Algorithm V) Numerical Examples ## |

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