# American Institute of Mathematical Sciences

January  2005, 1(1): 53-63. doi: 10.3934/jimo.2005.1.53

## Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints

 1 Centre for Industrial and Applied Mathematics, Mawson Lakes Campus, University of South Australia, Mawson Lakes Boulevard, Mawson Lakes, 5095, Australia

Received  June 2004 Revised  December 2004 Published  January 2005

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.
Citation: David Yang Gao. Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints. Journal of Industrial and Management Optimization, 2005, 1 (1) : 53-63. doi: 10.3934/jimo.2005.1.53
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