Theory and Efficient Algorithms for Hard, Large Scale, Numerical Optimization

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The focus of my research will be the proper modelling of hard problems, and the design and implementation of efficient and robust numerical algorithms for large scale, hard, optimization problems. The problems I will deal with arise in many important applications, e.g. molecular conformation (MC), sensor network localization (SNL), inverse imaging and machine learning. In particular, many of these problems arise in the relaxations of hard combinatorial optimization problems. In many instances, the usual modelling approaches result in problems that are both large scale and ill-posed. Therefore, they are hard to solve numerically. Rather than being a disadvantage, one can often take advantage of the ill-posedness to get both a stable problem and one that is smaller in size. In particular, for problems such as SNL one can solve huge problems to high accuracy by exploiting the hidden degeneracy. I plan on applying this technique to MC problems with noisy data as well as to protein design problems. The techniques that I use involve continuous optimization, nonlinear programming (NLP) and in particular, semidefinite programming (SDP). For SDP, my research contributions involve theory, algorithms, and applications, i.e., they include strong duality results, stable algorithms, and the study of relaxations for various applications. Some of the codes that I have designed for the standard form Linear Programming (LP) model have already been implemented in MAPLE. I plan to implement algorithms for more general LP models that include both upper and lower bounds on the variables. As well I plan on implementing codes that solve more general NLP problems. For the NLP problems, I use algorithms for generalized trust region problems to solve large scale unconstrained minimization, as well as solve general NLP using stable sequential quadratic programming methods. The success of these implementations means that researchers in my field will have access to stable, high accuracy, algorithms.