A Graph-based Decomposition Method for Convex Quadratic Optimization with Indicators

In this paper, we consider convex quadratic optimization problems with indicator variables when the matrix Q defining the quadratic term in the objective is sparse. We use a graphical representation of the support of Q, and show that if this graph is a path, then we can solve the associated problem in polynomial time. This … Read more

Blessing of Massive Scale: Spatial Graphical Model Estimation with a Total Cardinality Constraint

We consider the problem of estimating high dimensional spatial graphical models with a total cardinality constraint (i.e., the l0-constraint). Though this problem is highly nonconvex, we show that its primal-dual gap diminishes linearly with the dimensionality and provide a convex geometry justification of this ‘blessing of massive scale’ phenomenon. Motivated by this result, we propose … Read more

Higher Order Maximum Persistency and Comparison Theorems

We address combinatorial problems that can be formulated as minimization of a partially separable function of discrete variables (energy minimization in graphical models, weighted constraint satisfaction, pseudo-Boolean optimization, 0-1 polynomial programming). For polyhedral relaxations of such problems it is generally not true that variables integer in the relaxed solution will retain the same values in … Read more