Exploiting Structure in Nonsmooth Optimization Problems

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Optimization, the study of minimizing or maximizing a function, is essential in modern society. Applications are extremely diverse and include problems like: minimizing the construction costs when designing new roads, maximizing the efficiency (speed) of a new microchip design, minimizing expected hospital bed congestion, and maximizing the effectiveness of seismic dampers in building retrofitting. Despite the exponential growth in computing power and significant improvements in algorithm designs, modern applications of optimization are more complex than ever. In this proposal, we advance research and knowledge in Optimization Applications, Algorithm Design, and Optimization Theory. In Optimization Applications, we focus on real-world problems. We develop methods to detect structures within a problem's mathematical formulation. Some structures arise naturally, such as in finite minimax problems, while other structures can be created by the application of certain modelling and optimization techniques, such as regularization. Once detected, we seek to exploit the structures to improve solution time and quality. In Algorithm Design, we develop new methods for solving optimization problems that arise from simulation models. Optimization of simulations is particularly challenging, as the objective function is a "black-box" that cannot be explored analytically. We seek methods to detect and exploit structures within the black-box to improve convergence rates and solution quality. As simulation software becomes an increasingly prevalent tool for innovation, this research will provide valuable assets to researchers in disciplines where optimization has not traditionally been involved. In Optimization Theory, we explore the notions of "prox-regular" and "parametrically prox-regular" functions. These notions provide an extension of convexity that make it possible to work effectively with a much broader class of functions.