Perturbed optimization in banach spaces under directional constraint qualification conditions
Roberto Cominetti, J. Frédéric Bonnans · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · 1994
We present an implicit function theorem for systems of inequalities based on directional constraint qualification conditions. The result is used to study the (directional) sensitivity of the optimal value and optimal solutions of perturbed optimization problems in three cases: (1) With existence of multipliers and strong second order conditions: we obtain Lipschitz behavior of optimal solutions, first and second order differentiability of the optimal value and first order expansions for exact and approximate solutions. (2) With existence of multipliers and weak second order conditions: we obtain Hoelder behavior of optimal solutions, first order differentiability of the optimal value and Hoelder expansions for exact and approximate solutions. (3) Without existence of multipliers and weak second order conditions: we get Hoelder expansions for optimal solutions and optimal value function. The results are illustrated on a simple example: the position of minimal potential energy of a hanging chain. Further applications concern differentiability properties of metric projections in Hilbert spaces, using a condition generalizing polyhedricity.