[Paper Review] Complete Solutions and Triality Theory to a Nonconvex Optimization Problem with Double-Well Potential in R^n

Daniel Morales-Silva, David Yang Gao|arXiv (Cornell University)|Oct 3, 2011

Advanced Optimization Algorithms Research16 references8 citations

TL;DR

This paper resolves the open problem of double-min duality in canonical duality theory for a nonconvex optimization problem with a double-well potential in $\mathbb{R}^n$. By proving a convergence theorem for the linear perturbation canonical dual method, it establishes a complete set of analytical solutions, identifies both the global minimizer and the largest local maximizer, and provides a foundation for proving triality theory in general global optimization problems.

ABSTRACT

The main purpose of this research note is to show that the triality theory can always be used to identify both global minimizer and the biggest local maximizer in global optimization. An open problem left on the double-min duality is solved for a nonconvex optimization problem with double-well potential in $ eal^n$, which leads to a complete set of analytical solutions. Also a convergency theorem is proved for linear perturbation canonical dual method, which can be used for solving global optimization problems with multiple solutions. The methods and results presented in this note pave the way towards the proof of the triality theory in general cases.

Motivation & Objective

• To resolve the open problem of double-min duality in triality theory for nonconvex optimization with a double-well potential.
• To provide a complete set of analytical solutions for the global minimization problem with a fourth-order polynomial potential in $\mathbb{R}^n$.
• To prove a convergence theorem for the linear perturbation canonical dual method to identify global minimizers and local extrema.
• To establish a framework for proving triality theory in general global optimization problems.

Proposed method

• Formulates the primal problem as minimizing $\Pi(x) = W(x) - \langle x, f \rangle$ with $W(x) = \frac{\alpha}{2}\left(\frac{1}{2}|x|^2 - \lambda\right)^2$.
• Applies canonical duality theory to transform the nonconvex primal problem into a one-dimensional canonical dual problem.
• Uses the complementary-dual principle to reconstruct all critical points of the primal problem from dual solutions.
• Employs a linear perturbation method to handle cases with multiple global minimizers and proves convergence to global and local extrema.
• Applies the triality theory to identify both the global minimizer and the largest local maximizer via dual solutions.
• Analyzes asymptotic behavior of dual sequences to confirm convergence to global minimizers and local maximizers.

Experimental results

Research questions

• RQ1 Can the triality theory be used to identify both the global minimizer and the largest local maximizer in nonconvex optimization?
• RQ2 Does the double-min duality statement in triality theory hold without additional constraints for the double-well potential problem?
• RQ3 Can the linear perturbation canonical dual method converge to all critical points, including global minimizers and local maximizers, in nonconvex problems with multiple solutions?
• RQ4 What is the asymptotic behavior of dual sequences in the canonical dual method when the primal problem has multiple global minimizers?
• RQ5 How does the canonical dual method perform in identifying local extrema along different search directions in the primal space?

Key findings

• The open problem of double-min duality in triality theory is resolved for the double-well potential problem in $\mathbb{R}^n$.
• The linear perturbation canonical dual method converges to the global minimizer and the largest local maximizer, with convergence proven via asymptotic analysis of dual sequences.
• For $f = 0$, the global minimizers lie on the sphere $|x|^2 = 6$ and the origin is the local maximizer, confirmed by dual function analysis.
• When $|f|^2 = 8\alpha^2\lambda^3/27$, the dual function has a double root at $\varsigma_2 = \varsigma_3 = -2\lambda/3$, indicating a degenerate case with two identical local maximizers.
• When $|f|^2 > 8\alpha^2\lambda^3/27$, only one real dual solution exists, corresponding to the unique global minimizer, and the method converges to it via perturbation.
• The asymptotic analysis shows $x_{1,k}$ and $x_{2,k}$ converge to global minimizers $\sqrt{2\lambda}\frac{f_o}{|f_o|}$, while $x_{3,k}$ converges to the local maximizer at the origin.