Submodularity in Conic Quadratic Mixed 0–1 Optimization
We describe strong convex valid inequalities for conic quadratic mixed 0–1 optimization. These inequalities can be utilized for solving numerous practical nonlinear discrete optimization problems from value-at-risk minimization to queueing system design, from robust interdiction to assortment optimization through appropriate conic quadratic mixed 0–1 relaxations. The inequalities exploit the submodularity of the binary restrictions and are based on the polymatroid inequalities over binaries for the diagonal case. We prove that the convex inequalities completely describe the convex hull of a single conic quadratic constraint as well as the rotated cone constraint over binary variables and unbounded continuous variables. We then generalize and strengthen the inequalities by incorporating additional constraints of the optimization problem. Computational experiments on mean-risk optimization with correlations, assortment optimization, and robust conic quadratic optimization indicate that the new inequalities strengthen the convex relaxations substantially and lead to significant performance improvements.
Authors:
;
Award ID(s):
Publication Date:
NSF-PAR ID:
10188591
Journal Name:
Operations Research
Volume:
68
Issue:
2
Page Range or eLocation-ID:
609-630
ISSN:
0030-364X
3. Abstract We study the low-rank phase retrieval problem, where our goal is to recover a $d_1\times d_2$ low-rank matrix from a series of phaseless linear measurements. This is a fourth-order inverse problem, as we are trying to recover factors of a matrix that have been observed, indirectly, through some quadratic measurements. We propose a solution to this problem using the recently introduced technique of anchored regression. This approach uses two different types of convex relaxations: we replace the quadratic equality constraints for the phaseless measurements by a search over a polytope and enforce the rank constraint through nuclear norm regularization. The result is a convex program in the space of $d_1 \times d_2$ matrices. We analyze two specific scenarios. In the first, the target matrix is rank-$1$, and the observations are structured to correspond to a phaseless blind deconvolution. In the second, the target matrix has general rank, and we observe the magnitudes of the inner products against a series of independent Gaussian random matrices. In each of these problems, we show that anchored regression returns an accurate estimate from a near-optimal number of measurements given that we have access to an anchor matrix of sufficient quality. We also showmore »