[Paper Review] Global Solutions to Large-Scale Spherical Constrained Quadratic Minimization via Canonical Dual Approach

Yi Chen, David Yang Gao|arXiv (Cornell University)|Aug 20, 2013

Matrix Theory and Algorithms20 references6 citations

TL;DR

This paper presents a canonical dual approach to solve large-scale spherical constrained quadratic minimization problems, particularly addressing the 'hard case' where standard methods fail. By reformulating the problem into a one-dimensional canonical dual problem without duality gap, it provides sufficient and necessary conditions for identifying the hard case and introduces a perturbation method to achieve global convergence.

ABSTRACT

This paper presents global optimal solutions to a nonconvex quadratic minimization problem over a sphere constraint. The problem is well-known as a trust region subproblem and has been studied extensively for decades. The main challenge is the so called 'hard case', i.e., the problem has multiple solutions on the boundary of the sphere. By canonical duality theory, this challenging problem is able to reformed as an one-dimensional canonical dual problem without duality gap. Sufficient and necessary conditions are obtained by the triality theory, which can be used to identify whether the problem is hard case or not. A perturbation method and the associated algorithms are proposed to solve this hard case problem. Theoretical results and methods are verified by large-size examples.

Motivation & Objective

• To address the global solution of large-scale spherical constrained quadratic minimization problems, particularly the challenging 'hard case' where multiple solutions exist on the sphere boundary.
• To overcome numerical difficulties in the hard case by developing a perturbation-based method that restores the existence of unique solutions.
• To establish sufficient and necessary conditions for identifying the hard case using triality theory.
• To provide a globally convergent algorithm for large-scale problems via canonical duality and perturbation techniques.

Proposed method

• Reformulates the original nonconvex quadratic minimization problem into a canonical dual problem via canonical dual transformation, eliminating duality gap.
• Applies triality theory to derive sufficient and necessary conditions for identifying the hard case, based on the structure of the dual function and its critical points.
• Introduces a perturbation method by adding small parameters αi to the objective vector f, ensuring ∑αi² ≠ 0 to restore the existence of a unique solution in the perturbed problem.
• Uses the perturbed problem to approximate the solution of the original hard case problem, with convergence guaranteed under specific bounds on the perturbation parameters.
• Employs the dual function Pd(σ) = ∑_{i=k+1}^n [f̂i² / (λi + σ)²] - r² to locate the critical point σ̄, which corresponds to the optimal dual variable.
• Solves the primal solution via x̄ = G_a(σ̄)^{-1}f, where G_a(σ) is the transformed Hessian matrix, ensuring global optimality.

Experimental results

Research questions

• RQ1 Under what conditions does the spherical constrained quadratic minimization problem admit a unique solution, and when does it become a 'hard case'?
• RQ2 Can the canonical dual approach provide a global solution to the hard case without duality gap?
• RQ3 How can perturbation parameters be chosen to ensure convergence of the perturbed solution to the true solution of the hard case?
• RQ4 What theoretical conditions guarantee the existence and uniqueness of the critical point in the canonical dual function?

Key findings

• The canonical dual problem is reformulated as a one-dimensional optimization without duality gap, enabling global solution recovery.
• The hard case is identified when ∑_{i=1}^k f̂i² = 0 and ∑_{i=k+1}^n f̂i² / (λi - λ1)² ≤ r², which causes the dual function to lack a critical point.
• For the hard case, a perturbation method is proposed by introducing non-zero αi to restore the existence of a unique critical point in the dual function.
• The perturbation condition ∑αi² ≤ (λ2 - λ1)²(r² - ∑_{i=k+1}^n f̂i² / (λi - λ1)²)(1/√(2(1 - cos(ε/r)) - 1)^{-2} ensures that the perturbed solution converges to the true solution within ε tolerance.
• Theoretical analysis confirms that the critical point σ̄ of the dual function Pd(σ) is unique and corresponds to a globally optimal primal solution x̄ = G_a(σ̄)^{-1}f.
• Numerical experiments on large-size problems verify the effectiveness and convergence of the proposed method, even in the hard case.