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Convergence w.p.1 for Constrained Systems

  • Harold J. Kushner
  • Dean S. Clark
Part of the Applied Mathematical Sciences book series (AMS, volume 26)

Abstract

In this chapter, we treat four basic SA algorithms for the constrained optimization problem. Many of the techniques of proof are similar to those used in Chapter II and, in order to avoid duplications, the treatment is occasionally a little sketchy. The iterates {Xn} generated by the penalty-multiplier methods of Sections 5.1 and 5.4, and Lagrangian method of Section 5.2 are not necessarily feasible, although “their limits are. The iterates generated by the projection method of Section 5.3 are constrained to lie in the feasible set. The projection ideas of Section 2.3 are quite versatile and special simple forms appear in the analysis of the other algorithms, under various noise conditions. The noise condition A5.1.6 which is used in the main result of Section 5.1 is given in the form that is required in the proof. However, the condition holds under more readily verifiable conditions, as illustrated in Section 5.1.2. The Lagrangian method is particularly useful when the values of the constraint functions cannot be calculated, but can be observed with additive observation noise.

Keywords

Inequality Constraint Noise Condition Lagrangian Method Convergent Subsequence Uniform Continuity 
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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Copyright information

© Springer-Verlag New York, Inc. 1978

Authors and Affiliations

  • Harold J. Kushner
    • 1
  • Dean S. Clark
    • 1
  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

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