Quadratic Growth and Stability in Convex Programming Problems with Multiple Solutions Quadratic Growth and Stability in Convex Programming Problems with Multiple Solutions

Given a convex program with C 2 functions and a convex set S of solutions to the problem, we give a second order condition which guarantees that the problem does not have solutions outside of S. This condition is interpreted as a characterization for the quadratic growth of the cost function. The crucial role in the proofs is played by a theorem describing a certain uniform regularity property of critical cones in smooth convex programs. We apply these results to the discussion of stability of solutions of a convex program under possibly nonconvex perturbations. croissance quadratique et stabilit e en optimisation convexe avec solutions multiples R esum e : Dans le cadre de probl emes d'optimisation convexe a donn ees C 2 , nous formulons une condition du deuxi eme ordre qui garantit que le probl eme n'a pas de solution en dehors d'un ensemble S de solutions. Cette condition s'interpr ete comme une caract erisation de la croissance quadratique du co^ ut. La cl e de la d e-monstration est un th eoreme d ecrivant une propri et e de regularit e uniforme pour les cones critiques en optimisation convexe lisse. Nous appliquons ces resultats a la discussion de la stabilit e des solutions d'un probl eme d'optimisation convexe sous une perturbation non convexe.