A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
Lectures on modern convex optimization: analysis, algorithms, and engineering applications
Lectures on modern convex optimization: analysis, algorithms, and engineering applications
On Polyhedral Approximations of the Second-Order Cone
Mathematics of Operations Research
Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets
SIAM Journal on Optimization
Subset Algebra Lift Operators for 0-1 Integer Programming
SIAM Journal on Optimization
On the copositive representation of binary and continuous nonconvex quadratic programs
Mathematical Programming: Series A and B
On the Complexity of Nonnegative Matrix Factorization
SIAM Journal on Optimization
Theta Bodies for Polynomial Ideals
SIAM Journal on Optimization
Symmetry matters for the sizes of extended formulations
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
An upper bound for nonnegative rank
Journal of Combinatorial Theory Series A
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In this paper, we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or lift of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices. We show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual. This generalizes a theorem of Yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts of convex sets can also be characterized similarly. When the cones live in a family, our results lead to the definition of the rank of a convex set with respect to this family. We present results about this rank in the context of cones of positive semidefinite matrices. Our methods provide new tools for understanding cone lifts of convex sets.