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Stability theorems for 2×2 hypermatrices

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In this paper, stability properties of 2×2 hypermatrices of the form

$$\left[ {\begin{array}{*{20}c} { \in {\rm A} \in {\rm B}} \\ {CD} \\ \end{array} } \right]$$

are investigated, where ε is a small parameter. Stability theorems on these and similar matrices play an important role in the convergence analysis of certain numerical methods of mathematical programming and control theory, as pointed out in Refs. 1–3.

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  1. 1.

    Miele, A.,Recent Advances in Gradient Algorithms for Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 17, pp. 361–430, 1975.

  2. 2.

    Evtushenko, Y.,Generalized Lagrange Multiplier Technique for Nonlinear Programming, Journal of Optimization Theory and Applications, Vol. 21, pp. 121–135, 1977.

  3. 3.

    Gerencsér, L.,On the Use of Stability Theory in the Design of Algorithms for Structured Nonlinear Optimization Problems, Problems of Control and Information Theory, Vol. 7, pp. 471–482, 1978.

  4. 4.

    Gantmacher, F. R.,The Theory of Matrices (in Russian), Nauka, Moscow, USSR, 1966.

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The author is indebted to Mrs. K. Hangos and Miss M. Kovács for the careful reading of the manuscript.

Communicated by R. E. Kalaba

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Gerencsér, L. Stability theorems for 2×2 hypermatrices. J Optim Theory Appl 35, 1–7 (1981). https://doi.org/10.1007/BF00934699

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Key Words

  • Mathematical programming
  • control theory
  • numerical methods
  • convergence theorems
  • stability theorems
  • discrete-time dynamical systems