Rank-constrained Hyperbolic Programming

TL;DR

This paper extends rank-constrained optimization to hyperbolic programs, exploring new problems like rank-constrained SOCP and QCQP, analyzing their NP-hardness, and demonstrating sparsity results that enable efficient solutions.

Contribution

It introduces rank constraints to general hyperbolic programs, analyzes their computational complexity, and extends sparsity results to new classes like SOCP and QCQP.

Findings

Rank-constrained SOCP includes Max-Cut and nonconvex QP as special cases.

Rank-constrained SOCP and QCQP are NP-hard.

Existence of sparse solutions with bounded cardinality for SOCP and QCQP.

Abstract

We extend rank-constrained optimization to general hyperbolic programs (HP) using the notion of matroid rank. For LP and SDP respectively, this reduces to sparsity-constrained LP and rank-constrained SDP that are already well-studied. But for QCQP and SOCP, we obtain new interesting optimization problems. For example, rank-constrained SOCP includes weighted Max-Cut and nonconvex QP as special cases, and dropping the rank constraints yield the standard SOCP-relaxations of these problems. We will show (i) how to do rank reduction for SOCP and QCQP, (ii) that rank-constrained SOCP and rank-constrained QCQP are NP-hard, and (iii) an improved result for rank-constrained SDP showing that if the number of constraints is $m$ and the rank constraint is less than $2^{1/2-\epsilon} \sqrt{m}$ for some $\epsilon>0$, then the problem is NP-hard. We will also study sparsity-constrained HP and extend results on LP sparsification to SOCP and QCQP. In particular, we show that there always exist (a) a solution to SOCP of cardinality at most twice the number of constraints and (b) a solution to QCQP of cardinality at most the sum of the number of linear constraints and the sum of the rank of the matrices in the quadratic constraints; and both (a) and (b) can be found efficiently.