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Semidefinite Programming For Chance Constrained Optimization Over Semialgebraic Sets


In this work, “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. 

The 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


 Main Problem


Application


Motivating Examples


Convex Equivalent Problem


Semidefinite Relaxations


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