TY - JOUR
T1 - Perturbed optimization in Banach spaces II
T2 - A theory based on a strong directional constraint qualification
AU - Bonnans, J. Frédéric
AU - Cominetti, Roberto
PY - 1996/7
Y1 - 1996/7
N2 - We study the sensitivity of the optimal value and optimal solutions of perturbed optimization problems in two cases. The first one is when multipliers exist but only the weak (and not the strong) second-order sufficient optimalily condition is satisfied. The second case is when no Lagrange multipliers exist. To deal with these pathological cases, we are led to introduce a directional constraint qualification stronger than in part I of this paper, which reduces to the latter in the important case of equality-inequality constrained problems. We give sharp upper estimates of the cost based on paths varying as the square root of the perturbation parameter and, under a no-gap condition, obtain the first term of the expansion for the cost. When multipliers exist we study the expansion of approximate solutions as well. We show in the appendix that the strong directional constraint qualification is satisfied for a large class of probtems, including regular problems in the sense of Robinson.
AB - We study the sensitivity of the optimal value and optimal solutions of perturbed optimization problems in two cases. The first one is when multipliers exist but only the weak (and not the strong) second-order sufficient optimalily condition is satisfied. The second case is when no Lagrange multipliers exist. To deal with these pathological cases, we are led to introduce a directional constraint qualification stronger than in part I of this paper, which reduces to the latter in the important case of equality-inequality constrained problems. We give sharp upper estimates of the cost based on paths varying as the square root of the perturbation parameter and, under a no-gap condition, obtain the first term of the expansion for the cost. When multipliers exist we study the expansion of approximate solutions as well. We show in the appendix that the strong directional constraint qualification is satisfied for a large class of probtems, including regular problems in the sense of Robinson.
KW - Approximate solutions
KW - Directional constraint qualification
KW - Marginal function
KW - Sensitivity analysis
KW - Square root expansion
UR - http://www.scopus.com/inward/record.url?scp=0030191883&partnerID=8YFLogxK
U2 - 10.1137/S0363012994267285
DO - 10.1137/S0363012994267285
M3 - Article
AN - SCOPUS:0030191883
SN - 0363-0129
VL - 34
SP - 1172
EP - 1189
JO - SIAM Journal on Control and Optimization
JF - SIAM Journal on Control and Optimization
IS - 4
ER -