References
- Fiege S, Walther A, Griewank A. An algorithm for nonsmooth optimization by successive piecewise linearization. Math Program Ser A. 2018. DOI:https://doi.org/10.1007/s10107-018-1273-5.
- Griewank A. On stable piecewise linearization and generalized algorithmic differentiation. Optim Methods Softw. 2013;28(6):1139–1178. doi: https://doi.org/10.1080/10556788.2013.796683
- Griewank A, Walther A. Relaxing kink qualifications and proving convergence rates in piecewise smooth optimization. SIAM J Optim. 2019;29(1):262–289. doi: https://doi.org/10.1137/17M1157623
- Scholtes S. Introduction to piecewise differentiable functions. Berlin: Springer; 2012.
- Dodds PG, Dodds TK, de Pagter B, et al. Lipschitz continuity of the absolute value and Riesz projections in symmetric operator spaces. J Funct Anal. 1997;148(1):28–69. doi: https://doi.org/10.1006/jfan.1996.3055
- Griewank A. The modification of Newton's method for unconstrained optimization by bounding cubic terms. University of Cambridge; 1981. (Technical report; NA/12).
- Griewank A, Fischer J, Bosse T. Cubic overestimation and secant updating for unconstrained optimization of c2,1 functions. Optim Methods Softw. 2014;29(5):1075–1089. doi: https://doi.org/10.1080/10556788.2013.863308
- Knossalla M. Concepts on generalized ϵ-subdifferentials for minimizing locally lipschitz continuous functions. J Nonlinear Var Anal. 2017;1:265–279.
- Muu LD, Quy NV. Dc-gap function and proximal methods for solving nash-cournot oligopolistic equilibrium models involving concave cost. J Appl Numer Optim. 2019;1:13–24.
- Eckstein J, Bertsekas D. On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math Program. 1992;55(3(A)):293–318. doi: https://doi.org/10.1007/BF01581204
- Guler O. On the convergence of the proximal point algorithm for convex minimization. SIAM J Control Optim. 1991;29(2):403–419. doi: https://doi.org/10.1137/0329022
- Rockafellar RT. Monotone operators and the proximal point algorithm. SIAM J Control Optim. 1976;14:877–898. doi: https://doi.org/10.1137/0314056
- Tung NL, Luu DV. Optimality conditions for nonsmooth multiobjective optimization problems with general inequality constraints. Nonlinear Funct Anal. 2018;2018: Article ID 2.
- Van Ackooij W, Bello Cruz JY, De Oliveira W. A strongly convergent proximal bundle method for convex minimization in hilbert spaces. Optimization. 2016;65:145–167. doi: https://doi.org/10.1080/02331934.2015.1004549
- Correa R, Lemaréchal C. Convergence of some algorithms for convex minimization. Math Program. 1993;62:261–275. doi: https://doi.org/10.1007/BF01585170
- Hertlein L, Ulbrich M. An inexact bundle algorithm for nonconvex nondifferentiable functions in hilbert space. Technische Universität München; 2018. (Technical report; SPP1962-084).
- Clason C, Nhu VH, Rösch A. Optimal control of a non-smooth quasilinear elliptic equation. Preprint SPP1962-101, 12 2018.
- Ulbrich M. Nonsmooth Newton-like methods for variational inequalities and constrained optimization problems in function spaces [Habilitation]. Technische Universität München; 2002.
- Casas E, Herzog R, Wachsmuth G. Optimality conditions and error analysis of semilinear elliptic control problems with L1 cost functional. SIAM J Optim. 2012;22(3):795–820. doi: https://doi.org/10.1137/110834366
- Stadler G. Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Comput Optim Appl. 2007 Nov;44(2):159. doi: https://doi.org/10.1007/s10589-007-9150-9
- Christof C, Clason C, Meyer C, et al. Optimal control of a non-smooth semilinear elliptic equation. 2017. (Technical report; SPP1962-020, DFG SPP 1962).
- Clarke F. Optimization and nonsmooth analysis. Philadelphia (PA): SIAM; 1990.
- Clarke F. A new approach to lagrange multipliers. Math Oper Res. 1976;1:167–174. doi: https://doi.org/10.1287/moor.1.2.165
- Jahn J. Introduction to the theory of nonlinear optimization. Berlin: Springer; 2007.
- Absil P-A, Mahony R, Andrews B. Convergence of the iterates of descent methods for analytic cost functions. SIAM J Optim. 2005;16(2):531–547. doi: https://doi.org/10.1137/040605266
- Attouch H, Bolte J. On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math Program Ser B. 2009;116(1–2):5–16. doi: https://doi.org/10.1007/s10107-007-0133-5
- Clarke F. Generalized gradients of lipschitz functionals. Adv Math (NY). 1981;40:52–67. doi: https://doi.org/10.1016/0001-8708(81)90032-3
- Clarke F. Generalized gradients and applications. Trans Am Math Soc. 1975;205:247–262. doi: https://doi.org/10.1090/S0002-9947-1975-0367131-6
- Zeidler E. Applied functional analysis: applications to mathematical physics. Berlin: Springer; 1995.