Applications, Algorithms and Theory of Mathematical Programming

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Highly promising mathematical programming techniques applied to medical diagnosis and prognosis as well as to machine learning in general. Five years of experience with a highly accurate system for breast cancer diagnosis, in current use at University of Wisconsin Hospitals, is drawn upon to develop a linear-programming-based prognostic system. Parallel algorithms for the solution of large-scale constrained optimization problems are developed, by distributing ingredients of the problem (constraints, gradients, or/and variables) among parallel processors. Each processor has complete responsibility for varying its own problem ingredients, while allowing the other ingredients to vary in a restricted fashion. The processors share new information computed, then a fast synchronization is performed and the process is repeated. Preliminary algorithm prototypes have been tested successfully with some high parallelization efficiency on publicly available test problems and significant real-world applications. Error bounds are developed for possibly inconsistent systems of inequalities, programs and complementarity problems. The novel idea here is that the system may be unsolvable, and the bounds are meaningful whether the system is solvable or not. In the latter case, it bounds the distance between the point under consideration, and the set of least error solutions of the system. New mathematical programming approaches to some machine learning problems are studied. In particular a new quadratic programming model for a cascade architecture in neural networks is being studied, as is another one for the inherently difficult problem of minimizing the number of misclassified points by a separating plane.