A Flexible Framework for Cubic Regularization Algorithms for Nonconvex Optimization in Function Space

References

• Conn, A. R., Gould, N. I. M., Toint, P. L. (2000). Trust-Region Methods. SIAM.
   Google Scholar
• Griewank, A. (1981). The Modification of Newton’s Method for Unconstrained Optimization by Bounding Cubic Terms. Technical Report NA/12, Department of Applied Mathematics and Theoretical Physics. University of Cambridge.
   Google Scholar
• Weiser, M., Deuflhard, P., Erdmann, B. (2007). Affine conjugate adaptive Newton methods for nonlinear elastomechanics. Opt. Meth. Softw. 22(3):413–431.
   Web of Science ®Google Scholar
• Deuflhard, P., Weiser, M. (1997). Local inexact Newton multilevel FEM for nonlinear elliptic problems. In: Bristeau, M.-O., Etgen, G., Fitzgibbon, W., Lions, J.-L., Periaux, J., Wheeler, M., eds. Computational Science for the 21st Century. Wiley, pp. 129–138.
   Google Scholar
• Deuflhard, P., Weiser, M. (1998). Global inexact Newton multilevel FEM for nonlinear elliptic problems. In: W. Hackbusch and G. Wittum, eds. Multigrid Methods V. Lecture Notes in Computational Science and Engineering. Springer, pp. 71–89.
   Google Scholar
• Cartis, C., Gould, N. I. M., Toint, P. L. (2009). Adaptive cubic regularisation methods for unconstrained optimization. part i: motivation, convergence and numerical results. Math. Program. 1–51.
   Web of Science ®Google Scholar
• Cartis, C., Gould, N. I. M., Toint, P. L. (2010). Adaptive cubic regularisation methods for unconstrained optimization. part ii: worst-case function- and derivative-evaluation complexity. Math. Program. 1–25.
   Web of Science ®Google Scholar
• Toint, P. L. (2013). Nonlinear stepsize control, trust regions and regularizations for unconstrained optimization. Optim. Methods Softw. 28(1):82–95.
   Web of Science ®Google Scholar
• Toint, P. L. (1988). Global convergence of a class of trust-region methods for nonconvex minimization in Hilbert space. IMA J. Num. Anal. 8(2):231–252.
   Web of Science ®Google Scholar
• Kelley, C. T., Sachs, E. W. (1999). A trust region method for parabolic boundary control problems. SIAM J. Optim. 9(4):1064–1081.
   Web of Science ®Google Scholar
• Ulbrich, M., Ulbrich, S., Heinkenschloss, M. (1999). Global convergence of trust-region interior-point algorithms for infinite-dimensional nonconvex minimization subject to pointwise bounds. SIAM J. Control Optim. 37(3):731–764.
   Web of Science ®Google Scholar
• Ulbrich, M. (2011). MOS-SIAM series on optimization 11. SIAM. Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces.
   Google Scholar
• Zeidler, E. (1990). Nonlinear Functional Analysis and Its Applications, Vol. II/A. Springer, New York.
   Google Scholar
• Schechter, E. (1997). Handbook of Analysis and Its Foundations. Academic Press.
   Google Scholar
• Adams, R. A. (1975). Sobolev Spaces. Academic Press.
   Google Scholar
• Ekeland, I., Témam, R. (1999). Convex Analysis and Variational Problems. Number 28 in Classics in Applied Mathematics. SIAM.
   Google Scholar
• Dacorogna, B. (2008). Direct Methods in the Calculus of Variations. 2nd ed. Springer.
   Google Scholar
• Zeidler, E. (1990). Nonlinear Functional Analysis and Its Applications, Vol. II/B. Springer, New York.
   Google Scholar
• Ladyzhenskaya, O. A., Uraltseva, N. N. (1968). Linear and Quasilinear Elliptic Equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. New York: Academic Press.
   Google Scholar
• Zeidler, E. (1985). Nonlinear Functional Analysis and Its Applications, Vol.III. Springer, New York.
   Google Scholar
• Lions, J. L. (1971). Optimal Control of Systems Governed by Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, Vol. 170. Springer.
   Google Scholar
• Tröltzsch, F. (2010). Optimal Control of Partial Differential Equations, Vol. 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels.
   Google Scholar
• Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S. (2009). Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, Vol. 23. New York: Springer.
   Google Scholar