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Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in Rn

- Morales-Silva, Daniel, Gao, David

**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**Full Text:****Reviewed:****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.

**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**Full Text:****Reviewed:****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.

On the triality theory for a quartic polynomial optimization problem

**Authors:**Gao, David , Wu, Changzhi**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Industrial and Management Optimization Vol. 8, no. 1 (2012), p. 229-242**Full Text:**false**Reviewed:****Description:**This paper presents a detailed proof of the triality theorem for a class of fourth-order polynomial optimization problems. The method is based on linear algebra but it solves an open problem on the double-min duality. Results show that the triality theory holds strongly in the tri-duality form for our problem if the primal problem and its canonical dual have the same dimension; otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a symmetrical form. Some numerical examples are presented to illustrate that this theory can be used to identify not only the global minimum, but also the local minimum and local maximum.

- Gao, David, Ruan, Ning, Pardalos, Panos

**Authors:**Gao, David , Ruan, Ning , Pardalos, Panos**Date:**2012**Type:**Text , Book chapter**Relation:**Sensors: Theory, Algorithms, and applications optimization and its applications p. 37-56**Full Text:**false**Reviewed:****Description:**This chapter presents a canonical dual approach for solving a general sum of fourth-order polynomial minimization problem. This problem arises extensively in engineering and science, including database analysis, computational biology, sensor network communications, nonconvex mechanics, and ecology. We first show that this global optimization problem is actually equivalent to a discretized minimal potential variational problem in large deformation mechanics. Therefore, a general analytical solution is proposed by using the canonical duality theory developed by the first author. Both global and local extremality properties of this analytical solution are identified by a triality theory. Application to sensor network localization problem is illustrated. Our results show when the problem is not uniquely localizable, the “optimal solution” obtained by the SDP method is actually a local maximizer of the total potential energy. However, by using a perturbed canonical dual approach, a class of Euclidean distance problems can be converted to a unified concave maximization dual problem with zero duality gap, which can be solved by well-developed convex minimization methods. This chapter should bridge an existing gap between nonconvex mechanics and global optimization.

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