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The cones of nonnegative polynomials and sums of squares arise as central objects in convex algebraic geometry and have their origin in the seminal work of Hilbert ([Hil88]). Depending on the number of variables n and the degree d of the polynomials, Hilbert famously characterizes all cases of equality between the cone of nonnegative polynomials and the cone of sums of squares. This equality precisely holds for bivariate forms, quadratic forms and ternary quartics ([Hil88]). Since then, a lot of work has been done in understanding the difference between these two cones, which has major consequences for many practical applications such as for polynomial optimization problems. Roughly speaking, minimizing polynomial functions (constrained as well as unconstrained) can be done efficiently whenever certain nonnegative polynomials can be written as sums of squares (see Section 2.3 for the precise relationship). The underlying reason is the fundamental difference that checking nonnegativity of polynomials is an NP-hard problem whenever the degree is greater or equal than four ([BCSS98]), whereas checking whether a polynomial can be written as a sum of squares is a semidefinite feasibility problem (see Section 2.2). Although the complexity status of the semidefinite feasibility problem is still an open problem, it is polynomial for fixed number of variables. Hence, understanding the difference between nonnegative polynomials and sums of squares is highly desirable both from a theoretical and a practical viewpoint.
This work is concerned with two topics at the intersection of convex algebraic geometry and optimization.
We develop a new method for the optimization of polynomials over polytopes. From the point of view of convex algebraic geometry the most common method for the approximation of polynomial optimization problems is to solve semidefinite programming relaxations coming from the application of Positivstellensätze. In optimization, non-linear programming problems are often solved using branch and bound methods. We propose a fused method that uses Positivstellensatz-relaxations as lower bounding methods in a branch and bound scheme. By deriving a new error bound for Handelman's Positivstellensatz, we show convergence of the resulting branch and bound method. Through the application of Positivstellensätze, semidefinite programming has gained importance in polynomial optimization in recent years. While it arises to be a powerful tool, the underlying geometry of the feasibility regions (spectrahedra) is not yet well understood. In this work, we study polyhedral and spectrahedral containment problems, in particular we classify their complexity and introduce sufficient criteria to certify the containment of one spectrahedron in another one.