Abstract

The paper offers second order necessary and sufficient conditions for a sufficiently general class of optimization problems in Banach spaces. The reader can easily see that this is a ``no gap'' pair of conditions, although sufficient conditions (as usual) hold under some additional assumptions, mainly aimed to guarantee that tangent sets to the non-functional constraint set approximate the latter sufficiently well. In the last section we apply the results to a mathematical programming problem that contains, along with functional equality and inequality constraints, also a non-functional set constraint.

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Optimization problem
  • mathematical programming
  • local solution
  • tangent cone
  • critical cone

Mathematics Subject Classification

49K05

References

  1. J. P. Aubin, H. Frankowska: Set-Valued Analysis, Birkhäuser, Basel (1990).Google Scholar
  2. J. F. Bonnans, A. Shapiro: Perturbation Analysis of Optimization Problems, Springer, Berlin (2000).https://doi.org/10.1007/978-1-4612-1394-9Google Scholar
  3. R. Cominetti: Metric regularity, tangent sets and second order optimality conditions, Appl. Math. Optimization 21 (1990) 265–287.https://doi.org/10.1007/BF01445166Google Scholar
  4. H. Gfrerer, J. J. Ye, J. Zhow: Second order optimality conditions for non-convex set-constrained optimization problems, Math. Oper. Research 47 (2022) 2344–2365.https://doi.org/10.1287/moor.2021.1211Google Scholar
  5. H. Frankowska, N. P. Osmolovskii: Second order optimality conditions for minimization on a general set. I: Applications to mathematical programming, J. Math. Analysis Appl. 529/2 (2024), art. no. 127384, 34p.https://doi.org/10.1016/j.jmaa.2023.127384Google Scholar
  6. A. D. Ioffe: Variational Analysis of Regular Mappings, Springer, Berlin (2017).https://doi.org/10.1007/978-3-319-64277-2Google Scholar
  7. A. D. Ioffe: Towards the theory of strong minimum in calculus of variations and optimal control: a view from variational analysis, Calc. Var. Partial Diff. Equations 59/2 (2020), art. no. 83, 25p.https://doi.org/10.1007/s00526-020-01736-2Google Scholar
  8. A. D. Ioffe, V. M. Tihomirov: Theory of Extremal Problems (Russian), Nauka, Moscow (1974); English translation: North Holland, Amsterdam (1979).Google Scholar
  9. J.-P. Penot: Second-order conditions for optimization problems with constraints, SIAM J. Control Optimization 37/1 (1998) 303–318.https://doi.org/10.1137/S0363012996311095Google Scholar
  10. L. Thibault: Unilateral Variational Analysis in Banach Spaces Set. Vol. 1: General Theory, World Scientific, Singapore (2023).https://doi.org/10.1142/12797Google Scholar

Suggested citation

A. D. Ioffe. “On Second Order Sufficient Conditions for a Minimum.” Journal of Convex Analysis 31 (2024), No. 4, 1151–1163. https://doi.org/10.68381/jca31056

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.