Universal Convexification via Risk-Aversion

We develop a framework for convexifying a fairly general class of optimization problems. Under additional assumptions, we analyze the suboptimality of the solution to the con-vexified problem relative to the original non-convex problem and prove additive approx-imation guarantees. We then develop algo-rithms based on stochastic gradient methods to solve the resulting optimization problems and show bounds on convergence rates. We then extend this framework to apply to a gen-eral class of discrete-time dynamical systems. In this context, our convexification approach falls under the well-studied paradigm of risk-sensitive Markov Decision Processes. We de-rive the first known model-based and model-free policy gradient optimization algorithms with guaranteed convergence to the optimal solution. Finally, we present numerical re-sults validating our formulation in different applications.