# Convex Approximation of Chance Constrained Optimization Problems: Application in System and Control

Open Access

- Graduate Program:
- Electrical Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- August 26, 2016
- Committee Members:
- Constantino Manuel Lagoa, Dissertation Advisor
- Necdet S Aybat, Committee Chair
- Vishal Monga, Committee Member
- Minghui Zhu, Committee Member
- Antonios Armaou, Outside Member
- Alexei Novikov, Committee Member

- Keywords:
- Chance Constrained
- Convex Optimization
- Semidefinite Programming
- Measure and Moments
- Polynomials
- Sum of Squares Optimization
- Duality

- Abstract:
- This dissertation concentrates on chance constrained optimization problems and their application in systems and control area. In “chance optimization” problems, we aim at maximizing the probability of a set defined by polynomial inequalities involving decision and uncertain parameters. 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. In addition, we apply obtained results on chance optimization problems to challenging problems in the area of systems, control and data science. We consider the problem of probabilistic control of uncertain systems to ensure that the probability of defined failure/success is minimized/maximized. In particular, we consider the probabilistic robust control and chance constrained model predictive control problems. We also use the obtained results to analysis of stochastic and deterministic systems. More precisely, we address the problem of uncertainty set propagation and computing invariant robust set for uncertain systems and problem of computing region of attraction set for deterministic systems. In the problem of uncertainty propagation, we propagate the set of initial sets through uncertain dynamical systems and find the uncertainty set of states of the system for given time step. In the problem of region of attraction and invariant robust set, we aim at finding the largest set of all initial states whose trajectories converge to the origin. Moreover, we present the problem of corrupted and sparse data reconstruction where we want to complete the data with least possible complexity. In this thesis, to be able to efficiently solve the resulting large-scale problems, a first-order augmented Lagrangian algorithm is also implemented. Numerical examples are presented to illustrate the computational performance of the proposed approach.