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Nonsmooth Optimization Algorithms

Quasidifferentiable and Codifferentiable Optimization
  • Vladimir F. Dem’yanov
  • Georgios E. Stavroulakis
  • Ludmila N. Polyakova
  • Panagiotis D. Panagiotopoulos
Chapter
  • 134 Downloads
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 10)

Abstract

Quasidifferentiable and codifferentiable optimization algorithms are based on gradientlike, descent, iterative techniques whereas gradient information is replaced by the setvalued quasidifferential or the codifferential. Then the steepest descent finding subproblems are appropriately replaced by quadratic programming subproblems with a polyhedral approximation of the aforementioned set—valued quantities. Since supergradients (resp. hyperdifferentials) pose a combinatorial problem in the descent direction finding subproblem, which can effectively be treated after making the polyhedral approximation by repeated solution of a number of similar subproblems or simply by solving one of them (supergradient—like technique), the basic methods used are the ones of hypodifferential optimization. These techniques will be described in the sequel (for more details we refer to [3], [9], [5]). It should be mentioned here that first order quasidifferential and codifferential optimization schemes treat more effectively, in a correct way vertical branches of laws and boundary conditions in mechanical problems, or equivalently, the nonsmoothness of the respective potentials. If at a neighbourhood of the solution the problem is essentially smooth, i.e. the solution lies far away from a point of nondifferentiability, classical methods of nonlinear computational mechanics (e.g. Newton—type methods and its derivates) can be used for the refinement of the accuracy and for speeding up the rate of convergence. Nevertheless if multiple points of nondifferentiability (cusps) have to be passed along a given loading path the general methods presented here must be used (see also Chapter 8).

Keywords

Convex Function Regularity Condition Descent Direction Exact Penalty Exact Penalty Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Auchmuty G. (1989), Duality algorithms for nonconvex variational principles. Num. Functional Analysis and Optimization, 10, 211–264.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bertsekas D.P. (1982), Constrained optimization and Lagrange multiplier methods, Academic Press, New York.Google Scholar
  3. 3.
    Demyanov V.F. and Vasiliev L.N. (1985), Nondifferentiable Optimization, Optimization Software, New York.CrossRefzbMATHGoogle Scholar
  4. 4.
    Demyanov V.F. and Rubinov A.M. (1990), Foundations of Nonsmooth Analysis. Quasidifferential Calculus, (in Russian), Nauka, Moscow, 431 p.Google Scholar
  5. 5.
    Demyanov V.F and Rubinov A.M. (1995), Introduction to Constructive Nonsmooth Analysis, Peter Lang Verlag, Frankfurt a.M. — Bern — New York, 414 p.Google Scholar
  6. 6.
    Hiriart-Urruty J.-B. (1985), Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. In: Convexity and duality in optimization, Ed. J. Ponstein, 37–50, Lect. Notes in Economics and Mathematical Systems Vol. 256, Springer.Google Scholar
  7. 7.
    Di Pillo G. and Facchinei F. (1992), Regularity conditions and exact penalty functions in Lipschitz programming problems, In: Nonsmooth optimization methods and applications, Ed. F. Giannessi, 107–120, Gordon and Breach, Amsterdam.Google Scholar
  8. 8.
    Polyakova L.N. (1981), Necessary conditions for an extremum of quasidifferentiable functions, Vestnik Leningrad Univ. Math. 13, 241–249.zbMATHGoogle Scholar
  9. 9.
    Polyakova L.N. (1986), On minimizing the sum of a convex function and a concave function, Mathematical Programming Study 29, 69–73.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rockafellar R.T. (1970), Convex Analysis, Princeton University Press, Princeton.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Vladimir F. Dem’yanov
    • 1
  • Georgios E. Stavroulakis
    • 2
  • Ludmila N. Polyakova
    • 1
  • Panagiotis D. Panagiotopoulos
    • 3
    • 4
  1. 1.Department of MathematicsSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Lehr- und Forschungsgebiet für Mechanik; Lehrstuhl C für MathematikRWTHAachenGermany
  3. 3.Department of Civil EngineeringAristotle UniversityThessalonikiGreece
  4. 4.Faculty of Mathematics and PhysicsRWTHAachenGermany

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