We study spurious local minima in a nonconvex low-rank formulation of sum-of-squares optimization on a real variety X. We reformulate the problem of finding a spurious local minimum in terms of syzygies of the underlying linear series, and also bring in topological tools to study this problem. When the variety X is of minimal degree, there exist spurious local minima if and only if both the dimension and the codimension of the variety are greater than one, answering a question by Legat, Yuan, and Parrilo. Moreover, for surfaces of minimal degree, we provide sufficient conditions to exclude points from being spurious local minima. In particular, all spurious local minima on the Veronese surface, corresponding to ternary quartics, lie on the boundary and can be written as a binary quartic, up to a linear change of coordinates, complementing work by Scheiderer on decompositions of ternary quartics as a sum of three squares. For general varieties of higher degree, we give examples and characterizations of spurious local minima in the interior, which can potentially lead to efficient low-rank sum-of-squares algorithms.

We consider the solution of nonconvex quadratic optimization problems using an outer approximation of the set-copositive
cone that is iteratively strengthened with conic constraints and cutting planes. Our methodology utilizes an
MILP-based oracle for a generalization of the copositive cone that considers additional linear equality constraints. In numerical testing we evaluate our algorithm on a variety of different nonconvex quadratic problems.

We investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to conic integer programming. We show that the semigroup associated with nonpolyhedral cones can sometimes have a notion of finite generating set. We show this is true for the cone of positive semidefinite matrices (PSD) and the second order cone (SOC). Both cones have a finite generating set of integer points, similar in spirit to Hilbert bases, but require the action of a finitely generated group. We also extend notions of total dual integrality, Gomory-Chv\'{a}tal closure, and Carath\'{e}odory rank to integer points in arbitrary cones.

The problem of inferring Markov random fields (MRFs) with a sparsity or robustness prior can be naturally modeled as a mixed-integer program. This motivates us to study solution methods for a general class of convex submodular optimization problems with indicator variables, which we show to be polynomially solvable in this paper. The key insight is that, possibly after a suitable reformulation, indicator constraints preserve submodularity. Fast computations of the associated Lovasz extensions are also discussed under certain smoothness conditions, and can be implemented using only linear-algebraic operations in the case of quadratic objectives.