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In the past twenty years, a strong interplay has developed between convex optimization and algebraic geometry. Algebraic geometry provides necessary tools to analyze the behavior of solutions, the geometry of feasible sets, and to develop new relaxations for hard non-convex problems. On the other hand, numerical solvers for convex optimization have led to new fast algorithms in real algebraic geometry. In Chapter 1, we introduce some of the necessary background in convex optimization and real algebraic geometry and discuss some of the important results and questions in their intersection. One of the biggest of which is: when can a convex closed semialgebraic set be the feasible set of a semidefinite program and how can one construct such a representation?In Chapter 2, we explore the consequences of an ideal having a real radical initial ideal, both for the geometry its real variety and as an application to sums of squares representations of polynomials. We show that if the initial ideal of an ideal is real radical for a vector in the tropical variety, then this vectors belongs to logarithmic set of its real variety. We also give algebraic sufficient conditions for a ray to be in the logarithmic limit set of a more general semialgebraic set. If, in addition, the ray has positive coordinates, then the corresponding quadratic module is stable, which has consequences for problems in polynomial optimization. In particular, if an ideal has a is real radical initial ideal for some positive weight vector, then the preorder generated by the ideal is stable. This provides a method for checking the conditions for stability given by Powers and Scheiderer. In Chapter 3, we examine fundamental objects in convex algebraic geometry, such as definite determinantal representations and sums of squares, in the special case of plane quartics. A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 repre

In the past twenty years, a strong interplay has developed between convex optimization and algebraic geometry. Algebraic geometry provides necessary tools to analyze the behavior of solutions, the geometry of feasible sets, and to develop new relaxations for hard non-convex problems. On the other hand, numerical solvers for convex optimization have led to new fast algorithms in real algebraic geometry. In Chapter 1, we introduce some of the necessary background in convex optimization and real algebraic geometry and discuss some of the important results and questions in their intersection. One of the biggest of which is: when can a convex closed semialgebraic set be the feasible set of a semidefinite program and how can one construct such a representation?In Chapter 2, we explore the consequences of an ideal having a real radical initial ideal, both for the geometry its real variety and as an application to sums of squares representations of polynomials. We show that if the initial ideal of an ideal is real radical for a vector in the tropical variety, then this vectors belongs to logarithmic set of its real variety. We also give algebraic sufficient conditions for a ray to be in the logarithmic limit set of a more general semialgebraic set. If, in addition, the ray has positive coordinates, then the corresponding quadratic module is stable, which has consequences for problems in polynomial optimization. In particular, if an ideal has a is real radical initial ideal for some positive weight vector, then the preorder generated by the ideal is stable. This provides a method for checking the conditions for stability given by Powers and Scheiderer. In Chapter 3, we examine fundamental objects in convex algebraic geometry, such as definite determinantal representations and sums of squares, in the special case of plane quartics. A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 repre