FRG: Collaborative Research: Semidefinite optimization and convex algebraic geometry

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The goal in this proposal is to develop the mathematical foundations andassociated computational methods for the study of convex sets in realalgebraic geometry. This work requires a combination of ideas andmathematical tools from optimization, analysis, algebra and combinatorics.The proposed program will lead not only to theoretical insights, but also tonew algorithms and software that will enable novel applications inmathematics, engineering, and beyond. The work is organized in five mainthrusts: semidefinite programming and sums of squares, convex semi-algebraicsets, sparsity and graphical structure, numerical polynomial optimizationand applications, and deformations and variation of parameters. The PIs willfocus on the development of a comprehensive theory and practical newalgorithms for convex sets defined by polynomial inequalities. Specificproblems and techniques include the formulation of semidefinite descriptionsof convex hulls of real algebraic varieties, determinantal representationsof hyperbolic polynomials, sparse polynomials and their symmetries, tropicalgeometry and homotopy techniques, and geometric programming.Many areas in mathematics, as well as applications in engineering, financeand the sciences, require a thorough understanding of convex sets. This is aclass of geometric shapes, with several different but complementaryinterpretations. The goal in this project is to achieve a betterunderstanding of how these geometric properties emerge from their algebraicdescriptions in terms of polynomial equations, and the correspondingcomputational implications. One of the main motivations is the possibilityof applying these results in the context of optimization. The proposedresearch will contribute to existing knowledge, both in algebraic-geometrictechniques as well as in mathematical optimization. It will create synergiesbetween different branches of applied mathematics, and their engineering andscientific applications (e.g., in computational biology and statisticalmodeling). Successful completion of this project should contribute to theavailability of efficient and reliable computational tools for solvingpolynomial systems, which have clear technological and economic interest.Other key features of this proposal include its integration with curriculumdevelopment, undergraduate research projects, training of graduate studentsand postdocs, and the development of new software tools for computationaloptimization.