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Research interests, research group, publications

Research interests

Characterization

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.

Members of the research group

Dipl. math. Jan Heerda bis 2011, Dipl. math. Philipp Puffer (wiss. Assist.), Robert Grothe (stud. Hilfskraft)

Publications

Books

  1. "Spiele auf Graphen." pdf-file (6.1MB)

    Deutscher Verlag d. Wissenschaften, Berlin

    und Birkhäuser, Basel 1979 (ISNM Series, Vol 44);

    Translation in Russian: MIR, Moskau 1982

  2. "Nonlinear Parametric Optimization." pdf-file (30MB)

    (with B. Bank, J. Guddat, D. Klatte, K. Tammer)

    Akademie Verlag Berlin, 1982

  3. "Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications."

    (with D. Klatte)

    Kluwer, Series: Nonconvex Optimization and its Applications, Dortrecht-Boston-London, 2002

    Corrections and Summary: pdf-file

Recent Papers (B. Kummer and co-authors)

The new topics of the papers 3, 4 and 5: Basic stability properties of (generalized, non-differentiable) equations (in particular calmness and metric regularity) are characterized by linear convergence of appropriate solution methods for the related problem. This avoides the usually applied vehicle of generalized (co-)derivatives (which neither can be determined nor allow precise stability characterizations in arbitrary Banach spaces).

  1. "Convergence of primal-dual solutions for the nonconvex log-barrier method without LICQ."

    C. Grossmann, D. Klatte, B. Kummer

    Kybernetika, 40: 571-584, 2004.

    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.

  2. "Strong Lipschitz stability of stationary solutions for nonlinear programs and variational inequalities."

    D. Klatte, B. Kummer

    SIAM J. Optimization, 16: 96-119, 2005.

    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.

  3. "Stability of inclusions: Characterizations via suitable Lipschitz functions and algorithms."

    D. Klatte, B. Kummer

    Optimization, 55 (Issue 5 & 6): 627-660, Oct. 2006 (D. Pallaschke collection)

    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.

  4. "Calmness of Banach space mappings."

    J. Heerda, B. Kummer

    (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.

  5. "Optimization Methods and Stability of Inclusions in Banach Spaces."

    D. Klatte, B. Kummer

    Math. Programming, B; Vol. 117, 2009, 305-330 (S.M. Robinson collection) cf. 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.

  6. "How fast is fast fictitious play?"

    B. Kummer

    (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.

  7. "Newton methods for stationary points: an elementary view of regularity conditions and solution schemes."

    D. Klatte, B. Kummer

    Optimization, 56 (No. 4): 441-462, Aug. 2007 (F. Nozicka collection)

    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.

  8. "Inclusions in general spaces: Hoelder stability, Solution schemes and Ekeland's principle"

    B. Kummer

    J. Math. Anal. Appl. 358 (2009) 327-344, cf. 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.

  9. "On second-order Taylor expansion of critical values"

    S. Buetikofer, D. Klatte, B. Kummer

    Kybernetika, Vol. 46, no.3, 2010; 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.

  10. "From convergence principles to stability and optimality conditions"

    D. Klatte, A. Kruger and B. Kummer

    Journal of Convex Analysis 19 (2012), No. 4

    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.

Articles concerning the current research of the group:

Running Programs

cf. implement. SOFTWARE

Concerns:

Projects supported by DFG

Journals

Graduations

  1. Dissertation A, Dr. rer. nat. 1975, Humboldt - Universität

    "Diskrete Positionsspiele und eine Verallgemeinerung des von Neumann-Morgensternschen Lösungsbegriffes für klassische Kooperativspiele."

  2. Dissertation B, Dr. sc. nat. 1977, Humboldt - Universität

    "Globale Stabilitätsuntersuchungen für parametrische Optimierungsaufgaben."

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