Nonsmooth (and generalized) equations permit various applications to optimization, game theory and to economical models. Our investigations are related to implicit-function statements, Newton-type methods and corresponding concepts of generalized derivatives. Applications are concered with parametric optimization, algorithms, regularizations and stability. In this context, the concrete structure of assigned nonsmooth functions is essential and must be used.
bis 2011: Dipl. math. Jan Heerda, bis 2012: Dipl. math. Philipp Puffer (wiss. Assistenten), Robert Grothe (stud. Hilfskraft)
"Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications." (with D. Klatte)
Kluwer, Series: Nonconvex Optimization and its Applications, Dortrecht-Boston-London, 2002; cf. iclass.iuea.ac.ug/intranet/E-books/INFORMATION%20TECHNOLOGY/MISCELLANEOUS%20BOOKS/Kluwer%20Academic%20-%20Nonsmooth%20Equations%20in%20Optimization%20-%20Regularity,%20Calculus,%20Methods%20and%20Applications%20-%202002.pdf
The corrected version: (1.9MB) and a recently modified, extended version "Variational Analysis: Constructions ..." (2.2MB); require a book-password, send an email.
Last update 26.02.2014
"Nonlinear Parametric Optimization." (30MB) (with B. Bank, J. Guddat, D. Klatte, K. Tammer)
Akademie Verlag Berlin, 1982; Birkhaeuser Basel, 1983
Deutscher Verlag d. Wissenschaften, Berlin 1979; Birkhäuser, Basel 1980 (ISNM Series, Vol 44); Translation in Russian: MIR, Moskau 1982
"Aubin Property and Uniqueness of Solutions in Cone Constrained Optimization"
Mathematical Methods of OR 77(3): June 2013, 291-304; cf. http://www.optimization-online.org/DB_HTML/2012/07/3526.html
Abstract. We discuss conditions for the Aubin property of solutions to perturbed cone constrained programs, by using and refining results given in \cite{KlaKum02}. In particular, we show that constraint nondegeneracy and hence uniqueness of the multiplier is necessary for the Aubin property of the critical point map. Moreover, we give conditions under which the critical point map has the Aubin property if and only if it is locally single-valued and Lipschitz.
"From convergence principles to stability and optimality conditions."
Journal of Convex Analysis 19 (2012), No. 4, 1043-1072; cf. www.optimization-online.org-DB_FILE-2010-12-2841.pdf
Abstract. We show in a rather general setting that Hoelder and Lipschitz stability properties of solutions to variational problems can be characterized by convergence of more or less abstract iteration schemes. Depending on the principle of convergence, new and intrinsic stability conditions can be derived. Our most abstract models are (multi-) functions on complete metric spaces. The relevance of this approach is illustrated by deriving both classical and new results on existence and optimality conditions, stability of feasible and solution sets and convergence behavior of solution procedures.
"Newton' method for continuous functions ?"
"On second-order Taylor expansion of critical values"
Kybernetika, Vol. 46(3): 472--487, 2010; cf. http://www.kybernetika.cz/content/2010/3/472
Abstract. Studying a critical value function v in parametric nonlinear programming, we recall conditions guaranteeing that v is a C1;1 function and derive second order Taylor expansion formulas including second-order terms in the form of certain generalized derivatives of Dv. Several specializations and applications are discussed. These results are understood as supplements to the well{developed theory of First- and second-order directional differentiability of the optimal value function in parametric optimization.
J. Math. Anal. Appl. 358 (2009) 327-344; cf. link to preprint HUB 2008
Abstract. We present two basic lemmas on exact and approximate solutions of inclusions and equations in general spaces. Its applications involve Ekeland's principle, characterize calmness, lower semicontinuity and the Aubin property of solution sets in some Hoelder-type setting and connect these properties with certain iteration schemes of descent type. In this way, the mentioned stability properties can be directly characterized by convergence of more or less abstract solution procedures. New stability conditions will be derived, too. Our basic models are (multi-) functions on a complete metric space with images in a linear normed space.
"preprint of: Optimization Methods and Stability of Inclusions in Banach Spaces."
Math. Programming, B; Vol. 117, 2009, 305-330 (S.M. Robinson collection); cf. link to preprint HUB 2006
Abstract. Our paper deals with the interrelation of optimization methods and Lipschitz stability of multifunctions in arbitrary Banach spaces. Roughly speaking, we show that linear convergence of several first order methods and Lipschitz stability mean the same. Particularly, we characterize calmness and the Aubin property by uniformly (with respect to certain starting points) linear convergence of descent methods and approximate projection methods. So we obtain, e.g., solution methods (for solving equations or variational problems) which require calmness only. The relations of these methods to several known basic algorithms are discussed, and errors in the subroutines as well as deformations of the given mappings are permitted. We also recall how such deformations are related to standard algorithms like barrier, penalty or regularization methods in optimization.
"Newton methods for stationary points: an elementary view of regularity conditions and solution schemes."
Optimization, 56 (No. 4): 441-462, Aug. 2007 (F. Nozicka collection). cf. http://www.tandfonline.com/doi/pdf/10.1080/02331930701421053#.UqyXHPTuIxA
Abstract. In this paper, we give an elementary view of Newton-type methods and related regularity conditions for a special class of nonsmooth equations arising from necessary optimality criteria for standard nonlinear programs. Different types of linearizations and parameterizations of these equations lead to different iteration schemes, where any abstract calculus of generalized derivatives for nonsmooth mappings is avoided. Based on a general local convergence result on (perturbed) Newton methods for solving Lipschitzian equations, we focus on characterizations which are explicitly given in terms of the original functions and assigned quadratic problems for our special setting. We are particularly interested in certain parameterized Newton equations and in regularity conditions which are weaker than strong regularity.
"Calmness of Banach space mappings."
(18 pg.); 2006 cf. preprint HUB
Abstract. We characterize calmness of multifunctions explicitly by calmness of level sets to globally Lipschitz functions, by convergence of specific solution methods for the related inclusions as well as by solvability of crucial linear systems. As a main tool, a so-called relative slack function will be applied. In this way, also equivalence between calmness and metric regularity of specific subsystems will be derived.
"Stability of inclusions: Characterizations via suitable Lipschitz functions and algorithms."
Optimization, 55 (Issue 5 & 6): 627-660, Oct. 2006 (D. Pallaschke collection); cf. http://www.tandfonline.com/doi/abs/10.1080/02331930600819787#.Uq27ZPTuIxA
Abstract. This paper deals with basic notions of local stability of solutions to generalized equations. The Aubin property, calmness, strong Lipschitz stability and other (Lipschitz) stability concepts are characterized by two classical approaches: by monotonicity of assigned Lipschitz functions and directly via the main applications of stability statements - the behavior of solutions methods. In this way, "stability" will be characterized by several new conditions. We also show how known concepts, which are mainly based on characterizations via generalized derivatives and are widely distributed in the literature, can be derived in an essentially self-contained and straightforward manner.
"How fast is fast fictitious play?"
(16 pg.); 2006 cf. preprint HUB
Abstract. We present a modification of the fictitious play algorithm for matrix games by aggregating trivial original steps. The resulting new algorithm consists of comparable simple steps but has a much better rate of convergence. Modifications for solving large scale LP-programs, numerical results and some conjecture concerning the total effort of elementary steps for ensuring an error < eps will be discussed.
"Strong Lipschitz stability of stationary solutions for nonlinear programs and variational inequalities."
SIAM J. Optimization, 16: 96-119, 2005. cf. link to preprint "http://www2.mathematik.hu-berlin.de/publ/pre/2003/p-list-03.html"
Abstract. The stationary solution map $X$ of a canonically perturbed nonlinear program or variational condition is studied. We primarily aim at characterizing $X$ to be locally single-valued and Lipschitz near some stationary point $x^0$ of an initial problem, where our focus is on characterizations which are explicitly given in terms of the original functions and assigned quadratic problems. Since such criterions are closely related to a non-singularity property of the strict graphical derivative of $X$, explicit formulas for this derivative are presented, too. It turns out that even for convex polynomial problems our stability (and the Aubin property) does not only depend on the derivatives, up to some fixed order, of the problem functions at $x^0$. This is in contrast to various other stability concepts. Further, we clarify completely the relations to Kojima's strong stability and present simplifications for linearly and certain linearly-quadratically constrained problems, convex programs and for the map of global minimizers as well.
"Convergence of primal-dual solutions for the nonconvex log-barrier method without LICQ."
Kybernetika, 40: 571-584, 2004. cf. preprint MATH-NM-22-2003, Nov 2003, Technische Universitaet Dresden
Abstract. This paper characterizes completely the behavior of the logarithmic barrier method under a standard second order condition, strict (multivalued) complementarity and MFCQ at a local minimizer. We present direct proofs, based on certain key estimates and few well--known facts on linear and parametric programming, in order to verify existence and Lipschitzian convergence of local primal-dual solutions without applying additionally technical tools arising from Newton-techniques.
"Lösungsmethoden für Variationsungleichungen."
Diss. Jan. 2003 HU-Berlin
"Second-order characterizations of Lipschitz stability in nonlinear programming."
Journal of Mathematical Sciences 116 (2003), 3231-3252
"Examples and Counterexamples in Lipschitz Analysis."
SIAM J. Control and Cybernetics Vol. 31 No 3 (2002), 471-492
"Constrained minima and Lipschitzian penalties in metric spaces."
SIAM J. Optimization Vol. 13 No 2 (2002), 619-633
"Generalized Newton and NCP-methods: convergence, regularity and actions."
Discussiones Mathematicae - Differential Inclusions Vol. 20 No 2 (2000), 209-244
"Inverse functions of pseudo regular mappings and regularity conditions."
Math. Progr. Ser. B 88 (2000), 313-339
"Isolated zeros of Lipschitzian metrically regular Rn functions."
Optimization Vol. 49 (2001), 425-446
"Eigenschaften pseudo-regulärer Funktionen und einige Anwendungen auf Optimierungsaufgaben."
Diss. Febr. 1999 HU-Berlin
"Contingent derivatives of implicit (multi-)functions and stationary points."
Annals of Operations Research Vol. 101 (2001), 313--331
"Generalized Kojima functions and Lipschitz stability of critical points."
Computational Optimization and Appl. 13 (1999), 61-85
"Strong stability in nonlinear programming revisited."
J. Australian Mathem. Soc. Ser. B 40 (1999), 336-352
"Metric Regularity: Characterizations, Nonsmooth Variations and Successive Approximation."
Optimization Vol. 46 (1999), 247-281
"Parametrizations of Kojimas system and relations to penalty and barrier functions."
Math.Progr. B 76 No 3 (1997), 579-592
"Lipschitzian and Pseudo-Lipschitzian Inverse Functions and Applications to Nonlinear Optimization."
Lecture notes in pure and applied mathematics Vol 195 (1997) (Math. Programming with Data Perturbations, ed. A.V.Fiacco), 201-222.
"On Solvability and Regularity of a Parametrized Version of Optimality Conditions."
ZOR Mathem. Methods of OR 41 (1995), 215-230
"Approximation of multifunctions and superlinear convergence."
In: R. Durier and C. Michelot (Eds.), Recent Developments in Optimization, Ser. Lecture Notes in Economics and Mathematical Systems Vol. 429; Springer, Berlin 1995, 243-251
"Newton's Method based on generalized derivatives for nonsmooth functions: Convergence Analysis."
In: W. Oettli, D. Pallaschke (Eds.), Advances in Optimization, Lecture Notes in Economics and Mathematical Systems 382; Springer, Berlin 1992, 171-194
"On Stability and Newton-type Methods for Lipschitzian equations with Applications to Optimization Problems."
In: P. Kall (Ed.), System Modelling and Optimization, Proceedings of the 15th IPIF Conference Zürich 1991, Lecture Notes in Control and Information Science 180; Springer, Berlin 1992, 3 - 16
"An Implicit Function Theorem for C0.1- Equations and Parametric C1.1 -Optimization."
Journal of Mathematical Analysis & Appl. 158 No 1 (1991), 35-46
"Lipschitzian Inverse Functions, Directional Derivatives and Application in C1.1-Optimization."
JOTA Vol. 70 No 3 (1991), 559-580
"The inverse of a Lipschitz function in Rn: Complete characterization by directional derivatives."
Working Paper WP-89-084 (1989) IIASA Laxenburg Austria
"Conditions for optimality and strong stability in nonlinear programs without assuming twice differentiability of data."
Working Paper WP-89-089 (1989) IIASA Laxenburg Austria
"Newton's method for non-differentiable functions."
In J. Guddat et al. eds, Advances in Math. Optimization, Akademie Verlag Berlin, Ser. Math. Res. 45, Berlin 1988, 114-125
"Stability Properties of Infima and Optimal Solutions of Parametric Optimization Problems."
In: V.Demianov & D. Pallaschke (Eds.), Nondifferentiable Optimization - Motivation and Applications; Springer, Berlin 1985, 215-229
cf. implement. SOFTWARE
Concerns:
1991-1993 with D. Pallaschke (Uni Karlsruhe)
"Quasidifferenzierbare Optimierung" (Pa 219/ 5-1)
1993-1995 with A. Kaplan (Guestrow)
"Proximal-Point- Regularisierung; Theorie und konkrete Anwendungen" (Ku 802/ 2-1)
1997-1998 with K. Tammer (Leipzig) and D. Klatte (Uni Zürich)
"Regularitätsbegrffe nichtglatter (verallgemeinerter) Gleichungen und ihre Anwendungen" (Ku 802/ 4-1)
Co-editor of the Journal "Convex Analysis" and of the Journal "Optimization"
Dissertation A, Dr. rer. nat. 1975, Humboldt - Universität
"Diskrete Positionsspiele und eine Verallgemeinerung des von Neumann-Morgensternschen Lösungsbegriffes für klassische Kooperativspiele."
Dissertation B, Dr. sc. nat. 1977, Humboldt - Universität
"Globale Stabilitätsuntersuchungen für parametrische Optimierungsaufgaben."
Last update October 2008
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