Branch-and-Cut Approaches for Nonconvex Nonlinear and Mixed-Integer Nonlinear Programming

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This grant provides funds for the development of new theory and computational methods to solve a number of nonconvex nonlinear and mixed-integer nonlinear programming problems (NCNLP and MINLP) through the branch-and-cut (B&C) approach, which are of significant theoretical and practical interest. Specifically, new formulation, pre-processing, branching scheme, primal heuristic, and cutting planes will be developed for them. The project will focus on the following directions: (1) piecewise linear optimization to solve NCNLP and MINLP approximately; (2) linear complementarity problem to solve nonconvex quadratic programming; (3) cutting planes for bounded purely integer (as opposed to mixed-integer) nonlinear programming; (4) specially structured problems, including bimatrix games and portfolio optimization. Part of the funds will be applied towards building new undergraduate and graduate curricula and educational materials in the area of discrete nonlinear optimization for students in engineering, management, economics, mathematics, and science. The project contributes to the knowledge of algorithms for solving such optimization models that combine discrete variables and nonlinear functions. As such, it will open new venues of research in the combined fields of discrete and nonlinear optimization. The results of the project will provide significant enhancements to the optimization tools that are used in practice. The final outcome will be efficient B&C strategies that will advance our ability to solve models that stand as some of the most difficult in fields such as finance, bioinformatics, and data mining, and therefore are at the core of our economy and welfare. The results and tools developed through the project, including the new theoretical developments, software, and the new teaching resources, will be publicly available through a website that will be created for the project. The development and dissemination of the educational material related to the project will contribute to the formation of personnel in the field of operations research who are essential both in industry and academia.