[Paper Review] The Constant Trace Property in Noncommutative Optimization

Ngoc Hoang Anh, Abhishek Bhardwaj+1 more|arXiv (Cornell University)|Feb 3, 2021

Advanced Optimization Algorithms Research27 references4 citations

TL;DR

This paper establishes that semidefinite relaxations of noncommutative polynomial optimization problems (NCPOP) with noncommutative ball or polydisk constraints possess the constant trace property (CTP), enabling efficient solution via first-order methods like CGAL. The key contribution is a constructive proof of CTP in the nc setting, leading to significant computational gains over interior-point solvers like Mosek, especially for large-scale problems with improved convergence and reduced memory usage.

ABSTRACT

In this article, we show that each semidefinite relaxation of a ball-constrained noncommutative polynomial optimization problem can be cast as a semidefinite program with a constant trace matrix variable. We then demonstrate how this constant trace property can be exploited via first order numerical methods to solve efficiently the semidefinite relaxations of the noncommutative problem.

Motivation & Objective

• To establish the constant trace property (CTP) in semidefinite relaxations of noncommutative polynomial optimization problems (NCPOP) with nc ball or polydisk constraints.
• To extend first-order numerical methods, such as the conditional gradient-based augmented Lagrangian (CGAL), to efficiently solve large-scale NCPOP relaxations by exploiting the CTP.
• To demonstrate computational advantages of CTP-based solvers over interior-point methods like Mosek, particularly in memory efficiency and scalability for high-degree or high-variable problems.
• To develop and release a Julia package for solving NCPOPs with CTP, supporting both dense and correlative sparse NCPOP instances.
• To validate the framework on benchmark problems, including eigenvalue and trace minimization, showing competitive or superior convergence and accuracy.

Proposed method

• Prove that semidefinite relaxations of NCPOP with nc ball or polydisk constraints yield matrices with constant trace, leveraging the structure of nc Hankel matrices and moment matrices.
• Construct a hierarchy of semidefinite programs (SDPs) for NCPOP using sums of hermitian squares and trace/eigenvalue optimization, ensuring the CTP holds under ball or polydisk constraints.
• Utilize first-order methods—specifically the conditional gradient-based augmented Lagrangian (CGAL)—to solve CTP-structured SDPs, avoiding expensive Hessian computations of interior-point methods.
• Implement a change of basis transformation to maintain the constant trace property when applying first-order solvers, enabling efficient and scalable optimization.
• Integrate the CTP framework into a Julia package that supports both dense NCPOPs and NCPOPs with correlative sparsity, enabling practical deployment.
• Validate the method on randomly generated QCQPs and eigenvalue minimization problems, comparing CGAL and Mosek in terms of solution quality, runtime, and memory usage.

Experimental results

Research questions

• RQ1 Does the constant trace property (CTP) hold for semidefinite relaxations of noncommutative polynomial optimization problems (NCPOP) with nc ball or polydisk constraints?
• RQ2 Can first-order methods like CGAL be effectively applied to solve CTP-structured SDPs arising from NCPOP relaxations, and how do they compare to interior-point solvers like Mosek?
• RQ3 What are the computational and memory advantages of using CTP-aware solvers for large-scale NCPOP instances, especially when Mosek fails due to memory limits?
• RQ4 How does the CTP framework extend to NCPOPs with correlative sparsity, and what performance gains are achievable?
• RQ5 Can the CTP-based approach provide reliable lower bounds for challenging problems in quantum information, such as Bell inequality violations or ground state energy estimation?

Key findings

• The constant trace property (CTP) holds for semidefinite relaxations of NCPOP when nc ball or polydisk constraints are present, enabling efficient first-order optimization.
• For $k=2$ relaxations, CGAL outperforms Mosek in runtime and memory usage, with Mosek failing due to memory overflow for $u \geq 15$ in dense QCQPs.
• In eigenvalue minimization problems, CGAL achieves relative optimality gaps of less than 1% compared to Mosek, with improved lower bounds for $k=2$.
• For randomly generated QCQPs with correlative sparsity and nc ball constraints on cliques, CGAL achieves faster convergence and better scalability than Mosek, especially at higher $k$ levels.
• The Julia package successfully solves large-scale NCPOPs with CTP, demonstrating practical utility and computational efficiency beyond theoretical frameworks.
• Preliminary results suggest that CTP-based solvers can improve existing bounds for quantum information problems, such as Bell inequality violations, where Mosek fails due to memory constraints.