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Volume 10, Issue 4
Minimax Methods for Open-Loop Equilibra in $N$-Person Differential Games Part II: Duality and Penalty Theory

Goong Chen & Quan Zheng

J. Comp. Math., 10 (1992), pp. 305-320.

Published online: 1992-10

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  • Abstract

The equilibrium strategy for $N$-person differential games can be obtained from a min-max problem subject to differential constraints. The differential constraints are treated here by the duality and penalty methods.
We first formulate the duality theory. This involves the introduction of $N+1$ Lagrange multipliers: one for each player and one commonly shared by all players. The primal min-max problem thus results in a dual problem, which is a max-min problem with no differential constraints.
We develop the penalty theory by penalizing $N+1$ differential constraints. We give a convergence proof which generalizes a theorem due to B.T. Polyak.

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@Article{JCM-10-305, author = {Goong Chen and Quan Zheng}, title = {Minimax Methods for Open-Loop Equilibra in $N$-Person Differential Games Part II: Duality and Penalty Theory}, journal = {Journal of Computational Mathematics}, year = {1992}, volume = {10}, number = {4}, pages = {305--320}, abstract = {

The equilibrium strategy for $N$-person differential games can be obtained from a min-max problem subject to differential constraints. The differential constraints are treated here by the duality and penalty methods.
We first formulate the duality theory. This involves the introduction of $N+1$ Lagrange multipliers: one for each player and one commonly shared by all players. The primal min-max problem thus results in a dual problem, which is a max-min problem with no differential constraints.
We develop the penalty theory by penalizing $N+1$ differential constraints. We give a convergence proof which generalizes a theorem due to B.T. Polyak.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9364.html} }
TY - JOUR T1 - Minimax Methods for Open-Loop Equilibra in $N$-Person Differential Games Part II: Duality and Penalty Theory AU - Goong Chen & Quan Zheng JO - Journal of Computational Mathematics VL - 4 SP - 305 EP - 320 PY - 1992 DA - 1992/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9364.html KW - AB -

The equilibrium strategy for $N$-person differential games can be obtained from a min-max problem subject to differential constraints. The differential constraints are treated here by the duality and penalty methods.
We first formulate the duality theory. This involves the introduction of $N+1$ Lagrange multipliers: one for each player and one commonly shared by all players. The primal min-max problem thus results in a dual problem, which is a max-min problem with no differential constraints.
We develop the penalty theory by penalizing $N+1$ differential constraints. We give a convergence proof which generalizes a theorem due to B.T. Polyak.

Goong Chen and Quan Zheng. (1992). Minimax Methods for Open-Loop Equilibra in $N$-Person Differential Games Part II: Duality and Penalty Theory. Journal of Computational Mathematics. 10 (4). 305-320. doi:
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