The paper illustrates connections between classical results of Analysis and Optimization. The focus is on new elementary proofs of Implicit Function Theorem, Lusternik's Theorem, and optimality conditions for equality constrained optimization problems. The proofs are based on Fermat's Theorem and the Weierstrass Theorem and do not use the contraction mapping principle or other advanced results of Real Analysis, so they can be used in any introductory course on Optimization or Real Analysis without the requirement of the advanced background in analysis. The paper also presents a simple proof of Implicit Function Theorem in normed linear spaces.

中文翻译：

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分析优化中一些经典结果的新视角

本文说明了分析和优化的经典结果之间的联系。重点是隐函数定理、Lusternik 定理和等式约束优化问题的最优条件的新基本证明。证明基于费马定理和魏尔斯特拉斯定理，不使用收缩映射原理或实分析的其他高级结果，因此它们可以用于任何优化或实分析的入门课程，无需高级分析背景. 该论文还提出了在范数线性空间中的隐函数定理的简单证明。