Structured Blackbox Optimization

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Optimization, the study of minimizing or maximizing a function, arises naturally in virtually every area of science. In some applications the use of optimization is obvious, such as minimizing the cost when designing a new road. In other applications the use of optimization is more subtle, such as denoising in medical imaging.In optimization, a blackbox is any function that is not analytically available. When evaluated at a point, a blackbox returns an objective function value. In addition, some blackboxes return a (sub)gradient vector. A blackbox optimization problem is any optimization problem where some, or all, of the functions defining the problem are given by blackboxes.One common occurrences of blackbox functions is the output of a computer simulation. Given some input parameters, the simulation executes and returns a function value. If the simulation is implemented in an open source language, then automated differentiation could be employed to further obtain a (sub-)gradient vector. As computer simulations have become ubiquitous in modern research, blackbox optimization represents one of the most important areas of research for solving future real-world applications.In some applications, the blackbox has some visible structure. A structured blackbox optimization problem is any optimization problem where some, or all, of the underlying functions are given by blackboxes, but the problem itself has some visible mathematical structure. A simple example of structured blackbox optimization is minimizing the `worst-case outcome'. In this case, each scenario is provided through a blackbox, and the final objective is to minimize the maximum of all the blackbox functions. This can be (and often has been) approached by considering the maximum of all the blackbox functions as a single blackbox function. However, if we recognize the structure of the max function, we can design algorithms that are faster and more accurate for this problem.My research focuses on structured blackbox optimization. My work includes the development of novel algorithms for structured blackbox optimization, the application of algorithms to solve real-world optimization problems, and the advancement of knowledge in the mathematics behind structured blackbox optimization.HQP interested in working in algorithm design will be trained in developing convergence analysis, implementing algorithms, and numerical testing of algorithms. HQP interested in working in optimization applications will be trained in determining the structures within optimization problems, considering methods to exploit this structure, and selecting the appropriate algorithm to solve problems while considering solution time and quality. HQP interested in working theoretical analysis will be trained in the broad field of optimization theory, including strong foundations in functional analysis and variational analysis.