Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints
David Yang Gao
Journal of Industrial & Management Optimization 2005, 1(1): 53-63 doi: 10.3934/jimo.2005.1.53
This paper presents a duality theory for solving concave minimization problem and nonconvex quadratic programming problem subjected to nonlinear inequality constraints. By use of the canonical dual transformation developed recently, two canonical dual problems are formulated, respectively. These two dual problems are perfectly dual to the primal problems with zero duality gap. It is proved that the sufficient conditions for global minimizers and local extrema (both minima and maxima) are controlled by the triality theory discovered recently [5]. This triality theory can be used to develop certain useful primal-dual methods for solving difficult nonconvex minimization problems. Results shown that the difficult quadratic minimization problem with quadratic constraint can be converted into a one-dimensional dual problem, which can be solved completely to obtain all KKT points and global minimizer.
keywords: global optimization quadratic programming NP-hard problems concave minimization duality optimality condition.
Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in $\mathbb{R}^n $
Daniel Morales-Silva David Yang Gao
Numerical Algebra, Control & Optimization 2013, 3(2): 271-282 doi: 10.3934/naco.2013.3.271
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 $\mathbb{R}^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.
keywords: triality Canonical duality theory nonlinear algebraic equations global optimization double-well potential perturbation.
Solutions and optimality criteria to box constrained nonconvex minimization problems
David Yang Gao
Journal of Industrial & Management Optimization 2007, 3(2): 293-304 doi: 10.3934/jimo.2007.3.293
This paper presents a canonical duality theory for solving nonconvex polynomial programming problems subjected to box constraints. It is proved that under certain conditions, the constrained nonconvex problems can be converted to the so-called canonical (perfect) dual problems, which can be solved by deterministic methods. Both global and local extrema of the primal problems can be identified by a triality theory proposed by the author. Applications to nonconvex integer programming and Boolean least squares problems are discussed. Examples are illustrated. A conjecture on NP-hard problems is proposed.
keywords: NP-hard problems. Global optimization; Duality; Nonconvex minimization; Box constraints; Integer programming; Boolean least squares problem
On the triality theory for a quartic polynomial optimization problem
David Yang Gao Changzhi Wu
Journal of Industrial & Management Optimization 2012, 8(1): 229-242 doi: 10.3934/jimo.2012.8.229
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.
keywords: global optimization counter-examples. triality polynomial optimization Canonical duality

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