Improved Approximation Algorithms for Orthogonally Constrained Problems Using Semidefinite Optimization

Abstract

Building on the blueprint from Goemans and Williamson (1995) for the Max-Cut problem, we construct a polynomial-time approximation algorithm for orthogonally constrained quadratic optimization problems. First, we derive a semidefinite relaxation and propose a randomized rounding algorithm to generate feasible solutions from the relaxation. Second, we derive constant-factor approximation guarantees for our algorithm. When optimizing for $m$ orthonormal vectors in dimension $n$, we leverage strong duality and semidefinite complementary slackness to show that our algorithm achieves a $1/3$-approximation ratio. For any $m$ of the form $2^q$