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### Open Access Journals

JIMO

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.
NACO

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.

JIMO

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.

JIMO

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

## Year of publication

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