Second-order conditions and constraint qualifications in stability and sensitivity analysis of solutions to optimization problems in Hilbert spaces (Q1186091)

The author considers the following parametric optimization problem \((P_ h):\) minimize \(F(z,h)\) s.t. \(-\varphi(z,h)\in K\), where \(Z\), \(Y\) are Hilbert spaces, \(H\) is a Banach space, \(G\subset H\) is open, \(K\subset Y\) is a closed convex cone, \(F\) and \(\varphi\) are twice Frechet differentiable on \(Z\times G\) and \(P_ h\) has a local optimal solution. Under stronger forms of linear independence of gradients of active constraints condition and strong second-order sufficient optimality condition, the author obtains the Lipschitz continuity of the map \(h\to (z_ h,\lambda_ h)\), where \(z_ h\) is the optimal solution of \((P_ h)\) and \(\lambda_ h\) is the (unique) corresponding Lagrange multiplier. Under additional hypotheses one obtains directional differentiability of solutions. An application to optimal control problems is given.