A complete characterization of strong duality in nonconvex optimization with a single constraint

Authors:
Fabián Flores-Bazán;Fernando Flores-Bazán;Cristián Vera
Affiliations:
CI2MA and Departamento deIngeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Concepción, Chile;Departamento de Matemática, Facultad de Ciencias, Universidad del Bío Bío, Concepción, Chile;Departamento de Matemática y Física Aplicadas, Facultad de Ingeniería, Universidad Católica de laSantísima Concepción, Concepción, Chile
Venue:
Journal of Global Optimization
Year:
2012

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Abstract

We first establish sufficient conditions ensuring strong duality for cone constrained nonconvex optimization problems under a generalized Slater-type condition. Such conditions allow us to cover situations where recent results cannot be applied. Afterwards, we provide a new complete characterization of strong duality for a problem with a single constraint: showing, in particular, that strong duality still holds without the standard Slater condition. This yields Lagrange multipliers characterizations of global optimality in case of (not necessarily convex) quadratic homogeneous functions after applying a generalized joint-range convexity result. Furthermore, a result which reduces a constrained minimization problem into one with a single constraint under generalized convexity assumptions, is also presented.