Abstract
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.
Highlights
Yμ be a convex subsetThe inequality constraints in the problem (Pq) can be relaxed by the indicator of Yμ and the primal problem (Pq) takes the unconstrained canonical form (Pμ) : min
This paper presents a duality theory for solving concave minimization problem and nonconvex quadratic programming problem subjected to nonlinear inequality constraints
Duality theory plays an important role in both analysis and mathematical programming
Summary
The inequality constraints in the problem (Pq) can be relaxed by the indicator of Yμ and the primal problem (Pq) takes the unconstrained canonical form (Pμ) : min
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