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.
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Part of the results were presented at ICM-2010 held in Hyderabad, India; thanks to the local support from the Organizing Committee and to IMU for the partial travel grant awarded. This research, for the first author, was supported in part by CONICYT-Chile through FONDECYT 110-0667 and FONDAP and BASAL Projects, CMM, Universidad de Chile.
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Flores-Bazán, F., Flores-Bazán, F. & Vera, C. A complete characterization of strong duality in nonconvex optimization with a single constraint. J Glob Optim 53, 185–201 (2012). https://doi.org/10.1007/s10898-011-9673-6
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DOI: https://doi.org/10.1007/s10898-011-9673-6