Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in Rn
- Authors: Morales-Silva, Daniel , Gao, David
- Date: 2013
- Type: Text , Journal article
- Relation: Numerical Algebra, Control and Optimization Vol. 3, no. 2 (2013), p. 271-282
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- Description: 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 ℝn, 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 so- lutions. The methods and results presented in this note pave the way towards the proof of the triality theory in general cases.
Canonical duality theory and triality for solving general global optimization problems in complex systems
- Authors: Morales-Silva, Daniel , Gao, David
- Date: 2015
- Type: Text , Journal article
- Relation: Mathematics and Mechanics of Complex Systems Vol. 3, no. 2 (2015), p. 139-161
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- Description: General nonconvex optimization problems are studied by using the canonical duality-triality theory. The triality theory is proved for sums of exponentials and quartic polynomials, which solved an open problem left in 2003. This theory can be used to find the global minimum and local extrema, which bridges a gap between global optimization and nonconvex mechanics. Detailed applications are illustrated by several examples. © 2015 Mathematical Sciences Publishers.