Perturbed optimization in Banach spaces II: A theory based on a strong directional constraint qualification

J. Frédéric Bonnans, Roberto Cominetti

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9 Scopus citations


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.

Original languageEnglish
Pages (from-to)1172-1189
Number of pages18
JournalSIAM Journal on Control and Optimization
Issue number4
StatePublished - Jul 1996


  • Approximate solutions
  • Directional constraint qualification
  • Marginal function
  • Sensitivity analysis
  • Square root expansion


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