Improved Approximation Algorithms for Orthogonally Constrained Problems Using Semidefinite Optimization

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 $2^q$ for some integer $q$, we also construct an instance where the performance of our algorithm is exactly $(m+2)/(3m)$, which can be made arbitrarily close to $1/3$ by taking $m \rightarrow + \infty$, hence showing that our analysis is tight.

💡 Research Summary

The paper tackles a broad class of orthogonally constrained quadratic optimization problems, extending the celebrated Goemans‑Williamson (GW) framework originally devised for Max‑Cut. The authors consider the problem of maximizing a quadratic form ⟨A, vec(U) vec(U)ᵀ⟩ over semi‑orthogonal matrices U ∈ ℝⁿˣᵐ (UᵀU = Iₘ), where A ∈ Sⁿᵐ₊ is a positive semidefinite matrix. This formulation subsumes binary quadratic optimization (BQO) as a special case (when m = 1) and is shown to be NP‑hard and approximation‑preserving.

Semidefinite Relaxation.
The authors adopt a Shor‑type SDP relaxation: maximize ⟨A, W⟩ subject to block constraints W(i,i) ≼ Iₙ and trace constraints tr W(j,j′) = δ_{jj′}. The matrix W encodes the outer product of vec(U) with itself; the constraints enforce unit‑norm columns and mutual orthogonality at the level of second moments. While the relaxation is tight for rank‑one optimal solutions (e.g., m = 1), in general the optimal W* has rank up to n + m (Barvinok‑Pataki bound).

Randomized Rounding Procedure.
From an optimal SDP solution W*, the algorithm draws a Gaussian matrix G whose vectorization follows N(0, W*). Even when W* is low‑rank, a constructive sampling method based on a Cholesky‑type decomposition guarantees that vec(G) lies in the span of W*. The matrix G is then orthogonalized via singular value decomposition: G = U Σ Vᵀ, and Q := U Vᵀ is returned. By construction QᵀQ = Iₘ, so Q satisfies the original orthogonality constraints.

Key moment properties of G are established (Lemma 2): E