For constrained equations with nonisolated solutions, we show that if the equation mapping is 2-regular at a given solution with respect to a direction in the null space of the Jacobian, and this direction is interior feasible, then there is an associated domain of starting points from which a family of Newton-type methods is well defined and necessarily converges to this specific solution (despite degeneracy, and despite that there are other solutions nearby). We note that unlike the common settings of convergence analyses, our assumptions subsume that a local Lipschitzian error bound does not hold for the solution in question. Our results apply to constrained and projected variants of the Gauss–Newton, Levenberg–Marquardt, and LP-Newton methods. Applications to smooth and piecewise smooth reformulations of complementarity problems are also discussed.
Constrained equation Complementarity problem Nonisolated solution 2-Regularity Newton-type method Levenberg–Marquardt method LP-Newton method Piecewise Newton method
Mathematics Subject Classification
47J05 90C33 65K15
This is a preview of subscription content, log in to check access.
Research of the first author is supported in part by the Volkswagen Foundation. Research of the second author is supported by the Russian Science Foundation Grant 17-11-01168. The third author is supported in part by CNPq Grant 303724/2015-3 and by FAPERJ Grant 203.052/2016.
Facchinei, F., Fischer, A., Herrich, M.: An LP-Newton method: nonsmooth equations, KKT systems, and nonisolated solutions. Math. Program. 146, 1–36 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
Fischer, A., Herrich, M., Izmailov, A.F., Solodov, M.V.: Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions. Comput. Optim. Appl. 63, 425–459 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
Izmailov, A.F., Solodov, M.V.: On attraction of Newton-type iterates to multipliers violating second-order sufficiency conditions. Math. Program. 117, 271–304 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
Izmailov, A.F., Solodov, M.V.: On attraction of linearly constrained Lagrangian methods and of stabilized and quasi-Newton SQP methods to critical multipliers. Math. Program. 126, 231–257 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
Izmailov, A.F., Solodov, M.V.: Newton-Type Methods for Optimization and Variational Problems. Springer Series in Operations Research and Financial Engineering. Springer International Publishing, Cham (2014)Google Scholar
Oberlin, C., Wright, S.J.: An accelerated Newton method for equations with semismooth Jacobians and nonlinear complementarity problems. Math. Program. 117, 355–386 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
Facchinei, F., Fischer, A., Kanzow, C., Peng, J.-M.: A simply constrained optimization reformulation of KKT systems arising from variational inequalities. Appl. Math. Optim. 40, 19–37 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
Behling, R., Fischer, A., Herrich, M., Iusem, A., Ye, Y.: A Levenberg–Marquardt method with approximate projections. Comput. Optim. Appl. 59, 2–26 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
Behling, R., Fischer, A., Haeser, G., Ramos, A., Schönefeld, K.: On the constrained error bound condition and the projected Levenberg–Marquardt method. Optimization 66, 1397–1411 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)zbMATHGoogle Scholar
Arutyunov, A.V., Izmailov, A.F.: Stability of possibly nonisolated solutions of constrained equations, with applications to complementarity and equilibrium problems. Set-Valued Var. Anal. https://doi.org/10.1007/s11228-017-0459-y