从拉格朗日乘数法到KKT条件的高等数学约束优化思想教学改革研究

Research on Teaching Reform of Constrained Optimization Thinking in Advanced Mathematics: From the Lagrange Multiplier Method to the KKT Conditions

摘要: 传统高等数学中拉格朗日乘数法教学常止步于等式约束求解，造成约束优化思想的认知断裂。本文以构建拉格朗日乘数法到KKT条件的衔接逻辑为主线，提出贯通式教学设计：通过逐层拓展的案例揭示从等式约束到不等式约束下梯度共线与互补松弛的几何本质，并借助AI工具承担复杂计算与可视化，将认知重心从计算操作转向优化思想的内化。该教学设计旨在帮助学生建立起条件极值→最优性条件→约束优化的认知脉络，为机器学习等前沿课程提供思维接口，探索AI时代高等数学深度教学的一条可复用路径。

Abstract: Traditional teaching of the Lagrange multiplier method in advanced mathematics often ends with solving equality-constrained optimization problems, which results in a fragmented understanding of constrained optimization. Taking the logical transition from the Lagrange multiplier method to the Karush-Kuhn-Tucker (KKT) conditions as the main thread, this paper proposes an integrated teaching design. Through a sequence of progressively extended cases, the design reveals how gradient parallelism and complementary slackness extend their geometric interpretation from equality constraints to inequality constraints. AI-based tools are employed to handle complex computations and visualization, shifting the cognitive focus of students from mechanical calculations to the internalization of optimization concepts. The proposed design helps students establish a coherent cognitive framework linking conditional extrema, optimality conditions and constrained optimization. It further provides a cognitive bridge to advanced courses such as machine learning and explores a replicable model for deep teaching in advanced mathematics in the AI era.