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David Papp(dpappncsu.edu) Abstract: In a common formulation of semiinfinite programs, the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval. If this function is sufficiently closely approximable by a polynomial or a rational function, then the semiinfinite program can be reformulated as an equivalent semidefinite program. Solving this semidefinite program is challenging if the polynomials involved are of high degree, due to numerical difficulties and bad scaling arising both from the polynomial approximations and from the fact that the semidefinite programming constraints coming from the sumofsquares representation of nonnegative polynomials are badly scaled. We combine rational function approximation techniques and polynomial programming to overcome these numerical difficulties, using sumofsquares interpolants. Specifically, it is shown that the conditioning of the reformulations using sumofsquares interpolants does not deteriorate with increasing degrees, and problems involving sumofsquares interpolants of hundreds of degrees can be handled without difficulty. The proposed reformulations are sufficiently well scaled that they can be solved easily with every commonly used semidefinite programming solver, such as SeDuMi, SDPT3, and CSDP. Motivating applications include convex optimization problems with semiinfinite constraints and semidefinite conic inequalities, such as those arising in the optimal design of experiments. Numerical results align with the theoretical predictions; in the problems considered, available memory was the only factor limiting the degrees of polynomials, to approximately 1000. Keywords: semiinfinite programming, semidefinite programming, sumofsquares, interpolants, design of experiments Category 1: Infinite Dimensional Optimization (Semiinfinite Programming ) Category 2: Linear, Cone and Semidefinite Programming (Semidefinite Programming ) Category 3: Applications  Science and Engineering (Statistics ) Citation: Submitted. Download: [PDF] Entry Submitted: 12/21/2015 Modify/Update this entry  
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