k -Winner-Take-All Competition Based on Novel Dynamic Neural Networks
提出一种结构简化的动态神经网络，将k胜者全得问题转化为约束二次规划，用平滑S形函数编码不等式约束，实现实时求解，理论证明全局收敛与最优性，实验验证快速收敛和鲁棒性。
The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i>-winner-takes-all (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i>-WTA) problem involves selecting the top <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> agents with the highest inputs from a set of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> candidates. This problem plays a fundamental role in modeling competitive behaviors in social systems and economic environments. In this article, we propose a structurally simplified dynamic neural network to solve the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i>-WTA problem efficiently. The original <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i>-WTA task is first reformulated as a constrained quadratic programming (QP) problem. A smooth sigmoid function is then introduced to encode inequality constraints implicitly, simplifying the representation. Based on this formulation, we develop a continuous-time neural dynamic model capable of solving the problem in real time. The proposed model is theoretically proven to achieve global convergence and optimality with respect to the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i>-WTA solution. Extensive numerical experiments, including tests on real-world data, validate the effectiveness of the proposed approach, demonstrating fast convergence, robustness, and practical applicability.