Nested monotony for variational inequalities over product of spaces and convergence of iterative algorithms.

*(English)*Zbl 0628.90069The auxiliary problem principle has been proposed by the first author as a framework to describe and analyze iterative algorithms such as gradient as well as decomposition/coordination algorithms for optimization problems and variational inequalities. The key assumption to prove the global and strong convergence of such algorithms, as well as of most of the other algorithms proposed in the literature, is the strong monotony of the operator involved in the variational inequalities. In this paper, we consider variational inequalities defined over a product of spaces and we introduce a new property of strong nested monotony, which is weaker than the ordinary overall strong monotony generally assumed. In some sense, this new concept seems to be a minimal requirement to insure convergence of the algorithms alluded to above. A convergence theorem based on this weaker assumption is given. Applications of this result to the computation of Nash equilibria can be found in another paper of the first author [“Nash equilibria: Gradient and decomposition algorithms”, to appear].

##### MSC:

90C30 | Nonlinear programming |

65K10 | Numerical optimization and variational techniques |

91A40 | Other game-theoretic models |

##### Keywords:

auxiliary problem principle; decomposition/coordination; strong monotony; variational inequalities; product of spaces; strong nested monotony; computation of Nash equilibria
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\textit{G. Cohen} and \textit{F. Chaplais}, J. Optim. Theory Appl. 59, No. 3, 369--390 (1988; Zbl 0628.90069)

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##### References:

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