## Preprint 2013-13

## Gabriel Carcamo, Fabián Flores-Bazán:

### A geometric characterization of strong duality in nonconvex quadratic programming with linear and nonconvex quadratic constraints

### Abstract:

We first establish a relaxed version of Dines theorem associated to quadratic minimization problems with finitely many linear equality and a single (nonconvex) quadratic inequality constraints. The case of unbounded optimal valued is also discussed. Then, we characterize geometrically the strong duality, and some relationships with the conditions employed in Finsler theorem are established. Furthermore, necessary and sufficient optimality conditions with or without the Slater assumption are derived. Our results can be used to situations where none of the results appearing elsewhere are applicable. In addition, a revisited theorem due to Frank and Wolfe along with that due to Eaves is established for asymptotically linear sets.

This preprint gave rise to the following definitive publication(s):

**Gabriel CARCAMO, Fabián FLORES-BAZáN: ***A geometric characterization of strong duality in nonconvex quadratic programming with linear and nonconvex quadratic constraints*. Mathematical Programming, vol. 145, 1-2, pp. 263-290, (2014).