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The problem of optimizing over the cone of nonnegative polynomials is afundamental problem in computational mathematics, with applications topolynomial optimization, control, machine learning, game theory, andcombinatorics, among others. A number of breakthrough papers in the early 2000sshowed that this problem, long thought to be out of reach, could be tackled byusing sum of squares programming. This technique however has proved to beexpensive for large-scale problems, as it involves solving large semidefiniteprograms (SDPs).  In the first part of this thesis, we present two methods for approximatelysolving large-scale sum of squares programs that dispense altogether withsemidefinite programming and only involve solving a sequence of linear orsecond order cone programs generated in an adaptive fashion. We then focus onthe problem of finding tight lower bounds on polynomial optimization problems(POPs), a fundamental task in this area that is most commonly handled throughthe use of SDP-based sum of squares hierarchies (e.g., due to Lasserre andParrilo). In contrast to previous approaches, we provide the first theoreticalframework for constructing converging hierarchies of lower bounds on POPs whosecomputation simply requires the ability to multiply certain fixed polynomialstogether and to check nonnegativity of the coefficients of their product.  In the second part of this thesis, we focus on the theory and applications ofthe problem of optimizing over convex polynomials, a subcase of the problem ofoptimizing over nonnegative polynomials. (See manuscript for the rest of theabstract.)

Optimization over Nonnegative and Convex Polynomials With and Without Semidefinite Programming