A Stochastic Algorithm for Finding Points Satisfying Infinitely Many Inequality Constraints

  • Y. Wardi
Conference paper
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 83)


An optimization algorithm for finding a point satisfying infinitely many inequality constraints is presented. The algorithm approximates the maximum of an infinite number of inequalities at a point by performing a set of random experiments, resulting in a finite number of inequalities, over which the maximum is taken. It uses a constraint-dropping scheme, by which it eliminates points from a constraint-set at hand, which are felt to be irrelevant. At each point the algorithm constructs, it evaluates a measure of optimality, which indicates how close the point is to satisfying all of the constraints. It uses this measure to determine the number of random experiments performed. The number of such experiments tends to be small initially, when the point at hand is far from satisfying all of the constraints, and it increases gradually as a solution point is approached.


Infinite Number Inequality Constraint Accumulation Point Solution Point Algorithm Construct 
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  1. 1.
    Gonzaga, C., and Polak, E., “On Constraint Dropping Schemes and Optimality Functions for a Class of Outer Approximation Algorithms, ” SIAM Journal on Control and Optimization Vol. 17, No. 4, pp. 477–493, 1979.CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Heunis, A. J., “Use of a Monte Carlo Method in an Algorithm which Solves a Set of Functional Inequalities,” Journal of Optimization Theory and Applications Vol. 45, No. 1, pp. 89–99, 1985.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Mayne, D. Q., Polak, E., and Trahan, R., “An Outer Approximation Algorithm for Computer-Aided Design Problems, ” Journal of Optimization Theory and Applications Vol. 28, No. 3, pp. 331–351, 1979.CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Mayne, D. Q., and Polak, E., “Feasible Direction Algorithms for Optimization Problems with Equality and Inequality Constraints, ” Mathematical Programming Vol. 11, pp. 67–80, 1976.CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Polak, E., and Wardi, Y., “Nondifferentiable Optimization Algorithm for Designing Control Systems Having Singular Value Inequalities, ” Automatica Vol. 18, pp. 267–283, 1982.CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    D. Q. Mayne, E. Polak, and A. J. Heunis, “Solving Nonlinear Inequalities in a Finite Number of Iterations,” JOTA, Vol. 33, No. 2, 1981.Google Scholar
  7. 7.
    Y. Wardi, “A Stochastic Algorithm for Optimization Problems with Continua of Inequalities,” submitted to JOTA.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1986

Authors and Affiliations

  • Y. Wardi
    • 1
  1. 1.School of Electrical EngineeringGeorgia Institute of TechnologyAtlantaUSA

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