Abstract
This paper pursues a twofold goal. Firstly, we aim at deriving novel second-order characterizations of important robust stability properties of perturbed Karush–Kuhn–Tucker systems for a broad class of constrained optimization problems generated by parabolically regular sets. Secondly, the obtained characterizations are applied to establish well-posedness and superlinear convergence of the basic sequential quadratic programming method to solve parabolically regular constrained optimization problems.
Similar content being viewed by others
References
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)
Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
Mohammadi, A., Mordukhovich, B.S., Sarabi, M.E.: Parabolic regularity in geometric variational analysis. (2019). arxiv.org/abs/1909.00241
Mohammadi, A., Mordukhovich, B.S., Sarabi, M.E.: Variational analysis of composite models with applications to continuous optimization. to appear in Math. Oper. Res. (2019). arXiv:1905.08837v2
Mordukhovich, B.S., Sarabi, M.E.: Criticality of Lagrange multipliers in variational systems. SIAM J. Optim. 29, 1524–1557 (2019)
Bonnans, J.F.: Local analysis of Newton-type methods for variational inequalities and nonlinear programming. Appl. Math. Optim. 29, 161–186 (1994)
Ding, C., Sun, D., Zhang, L.: Characterization of the robust isolated calmness for a class of conic programming problems. SIAM J. Optim. 27, 67–90 (2017)
Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Cham, Switzerland (2018)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006)
Mordukhovich, B.S.: Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Am 340, 1–35 (1993)
Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings: A View from Variational Analysis, 2nd edn. Springer, New York (2014)
Cibulka, R., Dontchev, A.L., Kruger, A.: Strong metric subregularity of mappings in variational analysis and optimization. J. Math. Anal. Appl. 457, 1247–1282 (2018)
Cui, Y., Sun, D., Toh, K.-C.: On the R-superlinear convergence of the KKT residues generated by the augmented Lagrangian method for convex composite conic programming. Math. Program. 178, 381–415 (2019)
Dontchev, A.L., Rockafellar, R.T.: Characterizations of Lipschitzian stability in nonlinear programming. In: Fiacco, V. (ed.) Mathematical Programming With Data Perturbations (A, pp. 65–82. Marcel Dekker, New York (1997)
Gfrerer, H.: First-order and second-order characterizations of metric subregularity and calmness of constraint set mappings. SIAM J. Optim. 21, 1439–1474 (2011)
Hang, N.T.V., Mordukhovich, B.S., Sarabi, M.E.: Second-order variational analysis in second-order cone programming. Math. Program. 180, 75–116 (2020)
Henrion, R., Outrata, J.V.: Calmness of constraint systems with applications. Math. Program. 104, 437–464 (2005)
Izmailov, A.F., Solodov, M.V.: Newton-Type Methods for Optimization and Variational Problems. Springer, New York (2014)
Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods, and Applications. Kluwer, Boston, MA (2002)
Robinson, S.M.: Some continuity properties of polyhedral multifunctions. Math. Program. Stud. 14, 206–214 (1981)
Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22, 977–997 (1997)
Zheng, X.Y., Ng, K.F.: Metric subregularity and calmness for nonconvex generalized equations in Banach spaces. SIAM J. Optim. 20, 2119–2136 (2010)
Mordukhovich, B.S., Sarabi, M.E.: Critical multipliers in variational systems via second-order generalized differentiation. Math. Program. 169, 605–648 (2018)
Izmailov, A.F.: On the analytical and numerical stability of critical Lagrange multipliers. Comput. Math. Math. Phys. 45, 930–946 (2005)
Mordukhovich, B.S., Outrata, J.V., Ramírez, H.: Graphical derivatives and stability analysis for parameterized equilibria with conic constraints. Set-Valued Var. Anal. 23, 687–704 (2015)
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementary Problems. published in two volumes, Springer, New York (2003)
Bonnans, J.F., Gilbert, J.C., Lemaréchal, C., Sagastizabal, C.: Numerical Optimization: Theoretical and Practical Aspects, 2nd edn. Springer, Berlin (2006)
Josephy, N.H.: Newton’s method for generalized equations and the PIES energy model. Ph.D. Dissertation, Department of Industrial Engineering, University of Wisconsin-Madison (1979)
Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 3–62 (1980)
Robinson, S.M.: Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear programming algorithms. Math. Program. 7, 1–16 (1974)
Kato, H., Fukushima, M.: An SQP-type algorithm for nonlinear second-order cone programs. Optim. Lett. 1, 129–144 (2007)
Wang, Y., Zhang, L.: Properties of equation reformulation of the Karush-Kuhn-Tucker condition for nonlinear second-order cone optimization problems. Math. Meth. Oper. Res. 70, 195–218 (2009)
Izmailov, A.F., Solodov, M.V.: Newton-type methods: A broader view. J. Optim. Theory Appl. 164, 577–620 (2015)
Fusek, P.: Isolated zeros of Lipschitzian metrically regular \({\mathbb{R}}^n\)-functions. Optimization 49, 425–446 (2001)
Acknowledgements
The authors are grateful to two anonymous referees and the handling editor for carefully reading the paper and for their insightful comments that allowed us to improve the original presentation. Research of the first author was partly supported by the US National Science Foundation under grant DMS-1808978 and by the US Air Force Office of Scientific Research under grant #15RT0462. Research of the second author was partly supported by the US National Science Foundation under grants DMS-1512846 and DMS-1808978, by the US Air Force Office of Scientific Research under grant #15RT0462, and by the Australian Research Council Discovery Project DP-190100555.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Marcin Studniarski.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mohammadi, A., Mordukhovich, B.S. & Sarabi, M.E. Superlinear Convergence of the Sequential Quadratic Method in Constrained Optimization. J Optim Theory Appl 186, 731–758 (2020). https://doi.org/10.1007/s10957-020-01720-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-020-01720-y
Keywords
- Variational analysis
- Constrained optimization
- KKT systems
- Metric subregularity and calmness
- Critical and noncritical multipliers
- SQP methods
- Superlinear convergence