A projection activation inertial BAM neural network for solving fixed points of extremal mappings with coupled constraints

Yunlai Zhong; Li Wang; Juhe Sun; Xu Wu; Yunlai Zhong; Li Wang; Juhe Sun; Xu Wu

doi:10.3934/math.20261357

AIMS Mathematics

2026, Volume 11, Issue 10: 34849-34883. doi: 10.3934/math.20261357

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Research article

A projection activation inertial BAM neural network for solving fixed points of extremal mappings with coupled constraints

1.
College of Artificial Intelligence, Shenyang Aerospace University, No. 37 Daoyi South Avenue, Shenyang 110136, Liaoning, China
2.
School of Science, Shenyang Aerospace University, No. 37 Daoyi South Avenue, Shenyang 110136, Liaoning, China

Received: 15 August 2026 Revised: 17 September 2026 Accepted: 09 October 2026 Published: 26 October 2026
MSC : 34K20, 90C30, 68T07

This paper develops a projection activation inertial bidirectional associative memory (BAM) neural network allowing delayed feedback for computing fixed points of extremal mappings with coupled constraints. In this problem, the unknown point is both a parameter and an optimizer of an inner constrained optimization problem. By combining the Karush–Kuhn–Tucker (KKT) conditions with the projection form of normal cone inclusions, the fixed point problem is reformulated as a projection residual equation. Different from BAM neural networks with standard activation functions, the proposed model uses the projection mapping itself as the activation mapping, so that the constraint set, multiplier nonnegativity, and residual structure are embedded into the neural dynamics. Under a matching condition on the connection matrices, the equilibria of the proposed network are proved to coincide with the zeros of the projection residual, and therefore correspond to the fixed points of the original extremal mapping. A conditional exponential stability criterion on a prescribed operating domain is further established by using matrix measure estimates and Halanay's inequality. Numerical examples and a multi-actuator output allocation application confirm the theoretical stability condition and demonstrate the convergence accuracy, boundary projection ability, scalability, and delay-dependent performance of the proposed method.
- fixed points of extremal mappings,
- coupled constraints,
- projection residual,
- projection activation,
- inertial BAM neural network,
- exponential stability
Citation: Yunlai Zhong, Li Wang, Juhe Sun, Xu Wu. A projection activation inertial BAM neural network for solving fixed points of extremal mappings with coupled constraints[J]. AIMS Mathematics, 2026, 11(10): 34849-34883. doi: 10.3934/math.20261357

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Abstract

This paper develops a projection activation inertial bidirectional associative memory (BAM) neural network allowing delayed feedback for computing fixed points of extremal mappings with coupled constraints. In this problem, the unknown point is both a parameter and an optimizer of an inner constrained optimization problem. By combining the Karush–Kuhn–Tucker (KKT) conditions with the projection form of normal cone inclusions, the fixed point problem is reformulated as a projection residual equation. Different from BAM neural networks with standard activation functions, the proposed model uses the projection mapping itself as the activation mapping, so that the constraint set, multiplier nonnegativity, and residual structure are embedded into the neural dynamics. Under a matching condition on the connection matrices, the equilibria of the proposed network are proved to coincide with the zeros of the projection residual, and therefore correspond to the fixed points of the original extremal mapping. A conditional exponential stability criterion on a prescribed operating domain is further established by using matrix measure estimates and Halanay's inequality. Numerical examples and a multi-actuator output allocation application confirm the theoretical stability condition and demonstrate the convergence accuracy, boundary projection ability, scalability, and delay-dependent performance of the proposed method.

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