Basics of Minimax Algorithms

  • E. Polak
Part of the Ettore Majorana International Science Series book series (EMISS, volume 43)


Minimax problems are a very important class of nonsmooth optimization problems. They occur in curve fitting, engineering design, optimal control and many other situations (see [26]) for some specific examples). They are also among the best understood nonsmooth optimization problems, particularly when they involve maxima of smooth functions. There is now a considerable literature dealing with minimax problems and we present a selected list of publications in our references section (see [2], [4], [6], [7], [8], [9], [11], [12], [18], [20], [21], [22], [23], [24], [25], [26], [28], [29], [31], [32]). Looking over these papers, the reader will find that several approaches to minimax algorithms are possible, some of which yield first order methods, while others yield superlinearly converging ones. In this chapter we examine a particularly simple approach to the construction of minimax algorithms, which yields first order methods only.


Optimal Control Problem Search Direction Steep Descent Accumulation Point Directional Derivative 
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  1. [1]
    L. Armijo. “Minimization of functions having continuous partial derivatives”. Pacific J. Math., 16 (1966), pp. 1–3.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    T. Baker. “Algorithms for optimal control of systems described by partial and ordinary differential equations”. Ph.D. Thesis, University of California, Berkeley (1988).Google Scholar
  3. [3]
    R.O. Barr. “An efficient computational procedure for a generalized quadratic programming problem”. SIAM J. Control, 7, No. 3 (1969).Google Scholar
  4. [4]
    C. Berge. “Topological spaces”. Macmillan, New York, N.Y. ( 1963 ). WileyInterscience, New York, N.Y. (1983).Google Scholar
  5. [5]
    C. Charalambous and A.R. Conn. “An efficient method to solve the minimax problem directly”. SIAM J. Numerical Analysis, 15 (1978), pp. 162–187.MathSciNetADSCrossRefMATHGoogle Scholar
  6. [6]
    F.H. Clarke. “Optimization and nonsmooth analysis”. Wiley-Interscience, New York (1983).MATHGoogle Scholar
  7. [7]
    J.M. Danskin. “The theory of minimax with applications”. SIAM J. Appl. Math., 14 (1966), pp. 641–655.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    V.A. Daugavet and V.N. Malozemov. “Quadratic rate of convergence of a linearization method for solving discrete minimax problems”. U.S.S.R. Comput. Maths. Math. Phys., 21, No. 4 (1981), pp. 19–28.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    R. Fletcher. “A model algorithm for composite nondifferentiable optimization problems”. Mathematical Programming Studies, 17 (1982), pp. 67–76.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    M. Frank and P. Wolfe. “An algorithm for quadratic programming”. Naval Research Logistics Quarterly, 3 (1956), pp. 95–110.MathSciNetCrossRefGoogle Scholar
  11. [11]
    C. Gonzaga, E. Lopak and R. Trahan. “An improved algorithm for optimization problems with functional inequality constraints”. IEEE Trans. on Automatic Control, AC-25, No. 1 (1979), pp. 49–54.Google Scholar
  12. [12]
    J. Hald and K. Madsen. “Combined LP and quasi-Newton methods for minimax optimization” Mathematical Programming, 20 (1981), pp. 49–62.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    J.E. Higgins and E. Polak. “Minimizing pseudo-convex functions on convex compact sets”. University of California, Berkeley, Electronics Research Laboratory Memo No. UCB/ERL M88/22. To appear in Jou. Optimization Th..Appl. on 1988.Google Scholar
  14. [14]
    B. von Hohenbalken. “Simplicial decomposition in nonlinear programming algorithms”. Mathematical Programming, 13 (1977), pp. 49–68.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    P. Huard. “Programmation mathematic convex”. Rev. Fr. Inform. Rech. Operation, 7 (1968), pp. 43–59.MathSciNetGoogle Scholar
  16. [16]
    P.E. Gill, S.J. Hammarling, W. Murray, M.A. Saunders and M.H. Wright. “User’s guide for LSSOL (version 1.0): a fortran package for constrained linear least-squares and convex quadratic programming”. Technical report SOL 86–1, Department of Operations Research, Stanford University, Stanford (1986).Google Scholar
  17. [17]
    E.G. Gilbert. “An iterative procedure for computing the minimum of a quadratic forn on a convex set”. SIAM J. Control, 4, No. 1 (1966).Google Scholar
  18. [18]
    K.C. Kiwiel. “Methods of descent for nondifferentiable optimization”. Springer-Verlag, Berlin-Heidelberg-New York-Tokyo (1985).MATHGoogle Scholar
  19. [19]
    D.G. Kuenberger.“Introduction to linear and nonlinear programming’, Addison-Wesley, New York (1983).Google Scholar
  20. [20]
    K. Madsen and H. Schjaer-Jacobsen. “Linearly constrained minimax optimization;;. Mathematical Programming, 14 (1978), pp. 208–223.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    D.Q. Mayne and E. Polak. “An exact penalty function algorithm for optimal control problems with control and terminal inequality constraints, Part 1”. Jou. Optimization Th. Appl., 32, No. 2 (1980), pp. 211–246.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    D.Q. Mayne and E. Polak. “An exact penalty function algorithm for optimal problems with control and terminal inequality constraints, Part 2”. Jou. Optimization Th. Appl., 32, No. 2 (1980), pp. 345–363.CrossRefMATHGoogle Scholar
  23. [23]
    W. Murray and M.L. Overton. “A projected Lagrangian algorithm for nonlinear minimax optimization”. SIAM J. Sci. Stat. Cumput., 1, No. 3 (1980).Google Scholar
  24. [24]
    O. Pironneau and E. Polak. “On the rate of convergence of certain methods of centers”. Mathematica Programming, 2, No. 1 (1972), pp. 230–258.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    O. Pironneau and E. Polak. “A dual method for optimal control problems with initial and final boundary constraints”. SIAM J. Control, 11, No. 3 (1973), pp. 534–349.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    E. Polak. “On the mathematical foundations of nondifferentiable optimization in engineering design”. SIAM Review (1987), pp. 21–91.Google Scholar
  27. [27]
    E. Polak. “Computational methods in optimization: A unified approach”. Academic Press (1971).Google Scholar
  28. [28]
    E. Polak and J.W. Wiest. “Variable metric techniques for the solution of affimely parametrized nondifferentiable optimal design problems”. University of California, Berkeley, Electronics Research Laboratory Memo No. UCB/ERL M88 /42 (1988).Google Scholar
  29. [29]
    E. Polak, D.Q, Mayne and J. Higgins. “A superlinearly convergent algorithm for min-max problems”. Proc. 1988 IEEE Conf. on Dec. and Control (1988)Google Scholar
  30. [30]
    E. Polak, R. Trahan and D.Q. Mayne. “Combined rhase I- rhase II methods of feasible directions”. Mathematical Programming No. 1 (1979), pp. 32–61MathSciNetGoogle Scholar
  31. [31]
    B.N. Pshenichnyi and Yu M. Danilin. “Numerical methods in extremal problems”. Nauka, Moscow (1975).Google Scholar
  32. [32]
    R.S. Womersley and R. Fletcher. “An algorithm for composite nonsmooth optimization problems”. Jou. Optimization Th. Appl., 48, No. 3 (1986).Google Scholar
  33. [33]
    P. Wolfe. “Finding the nearest point in a polytope”. Mathematical Programming, 11 (1976), pp. 128–149.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • E. Polak
    • 1
  1. 1.Dept. of Electrical Engineering and Computer ScienceUniv. of CaliforniaBerkeleyUSA

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