Abstract
This paper addresses the globalization of the semi-smooth Newton method for non-smooth equations F(x) = 0 in \({\mathbb{R}}^m\) with applications to complementarity and discretized ℓ1-regularization problems. Assuming semi-smoothness it is shown that super-linearly convergent Newton methods can be globalized, if appropriate descent directions are used for the merit function |F(x)|2. Special attention is paid to directions obtained from the primal-dual active set strategy.
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References
Clarke F.H. (1983). Optimization and Nonsmooth Analysis. Wiley, New York
Kunisch K. and de los Reyes J.C. (2004). A comparison of algorithms for control constrained optimal control of the Burgers equation. Calcolo 41: 203–225
Facchinei F. and Pang J.-S. (2003). Finite Dimensional Variational Inequalities and Complementarity Problems, vol. II. Springer, Berlin
Han S.-H., Pang J.-S. and Rangaraj N. (1992). Globally convergent Newton methods for nonsmooth equations. Math. Oper. Res. 17: 586–607
Haraux A. (1977). How to differentiate the projection on a closed convex set in Hilbert space. Some applications to variational inequalities. J. Math. Soc. Jpn. 29: 615–631
Hintermüller M., Ito K. and Kunisch K. (2002). The primal–dual active set strategy as a semi-smooth Newton method. SIAM J. Optim. 13: 865–888
Ito K. and Kunisch K. (2004). The primal–dual active set method for nonlinear optimal control problems with bilateral constraints. SIAM J. Control Optim. 43: 357–376
Kanzow C. (2000). Global optimization techniques for mixed complementarity problems. J. Global Optim. 16: 1–21
Kanzow C. (2004). Inexact semismooth Newton methods for large-scale complementarity problems. Optim. Methods Softw. 19: 309–325
Kunisch K. and Rendl F. (2003). An infeasible active set method for convex problems with simple bounds. SIAM J. Optim. 14: 35–52
Mifflin R. (1977). Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15: 959–972
Pang J.S. (1990). Newton’s method for B-differentiable equations. Math. Oper. Res. 15: 311–341
Qi L. and Sun J. (1993). A nonsmooth version of Newton’s method. Math. Program. 58: 353–367
Qi L. (1993). Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18: 227–244
Robinson S.M. (1987). Local structure of feasible sets in nonlinear programming, part III: Stability and sensitivity. Math. Program. Study 30: 45–66
Shapiro A. (1990). On concepts of directional differentiability. J. Optim. Theory Appl. 13: 477–487
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K. Ito’s research was partially supported by the Army Research Office under DAAD19-02-1-039.
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Ito, K., Kunisch, K. On a semi-smooth Newton method and its globalization. Math. Program. 118, 347–370 (2009). https://doi.org/10.1007/s10107-007-0196-3
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DOI: https://doi.org/10.1007/s10107-007-0196-3