具体描述
Optimization is one of the most important areas of modern applied mathematics, with applications in fields from engineering and economics to finance, statistics, management science, and medicine. While many books have addressed its various aspects, Nonlinear Optimization is the first comprehensive treatment that will allow graduate students and researchers to understand its modern ideas, principles, and methods within a reasonable time, but without sacrificing mathematical precision. Andrzej Ruszczynski, a leading expert in the optimization of nonlinear stochastic systems, integrates the theory and the methods of nonlinear optimization in a unified, clear, and mathematically rigorous fashion, with detailed and easy-to-follow proofs illustrated by numerous examples and figures.
The book covers convex analysis, the theory of optimality conditions, duality theory, and numerical methods for solving unconstrained and constrained optimization problems. It addresses not only classical material but also modern topics such as optimality conditions and numerical methods for problems involving nondifferentiable functions, semidefinite programming, metric regularity and stability theory of set-constrained systems, and sensitivity analysis of optimization problems.
Based on a decade's worth of notes the author compiled in successfully teaching the subject, this book will help readers to understand the mathematical foundations of the modern theory and methods of nonlinear optimization and to analyze new problems, develop optimality theory for them, and choose or construct numerical solution methods. It is a must for anyone seriously interested in optimization.
Reviews:
"This book offers a very good introduction to differentiable and nondifferentiable nonlinear optimization theory and methods. With no doubt the major strength of this book is the clear and intuitive structure and systematic style of presentation. This book can be recommended as a material for both self study and teaching purposes, but because of its rigorous style it works also as a valuable reference for research purposes."--Mathematical Modeling and Operational Research
"This is one of the best textbooks on nonlinear optimization I know. Focus is on both theory and algorithmic solution of convex as well as of differentiable programming problems."--Stephan Dempe, Zentralblatt MATH Database
"In summary, this book competes with the topmost league of books on optimization. The wide range of topics covered and the thorough theoretical treatment of algorithms make it not only a good prospective textbook, but even more a reference text (which I am happy to have on my shelf.)"--Franz Rendl, Operations Research Letters
"Throughout the book the writing style is very clear, compact and easy to follow, but at the same time mathematically rigorous. The proofs are easy to follow because the author usually carefully explains every move. In addition the meaning of the most central results is usually demonstrated with examples and in many cases explanations are also supported by visualizations...This book offers a very good introduction to differentiable and nondifferentiable nonlinear optimization theory and methods...Recommended as a material for both self study and teaching purposes"--Petri Eskelinen, Mathematical Methods of Operation Research
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这本书的视角非常独特，它似乎更侧重于从信息论和信息几何的角度去审视优化过程的效率，而不是传统上侧重的收敛速度。我发现书中关于信息矩阵的构建和应用部分极具启发性，它提供了一种全新的思路来量化模型参数的不确定性对最优解的影响。这对于处理高维、数据稀疏的现代问题尤其关键。阅读过程中，我感觉到作者在试图构建一个统一的框架，将传统的梯度方法与更现代的贝叶斯优化思想巧妙地融合起来。虽然某些章节需要反复阅读才能领会其深层含义，但一旦理解，你对“搜索空间”的认识将不再局限于简单的几何概念。对于那些希望在优化算法前沿进行探索的科研人员，这本书提供了一种突破现有思维定式的工具箱，它鼓励我们用更广阔的视野去设计下一代更智能的优化器。
评分读完这本关于数值方法的专著，我深感它在理论深度和广度上的平衡把握得恰到五奇。它不仅仅停留在对数学公式的罗列，而是花费了大量篇幅来探讨数值稳定性、计算复杂性以及如何处理实际数据中的不确定性。书中对于大规模问题的处理策略，例如如何利用预处理技术加速迭代过程，提供了非常具有洞察力的见解。我特别欣赏作者对“黑箱”算法的解构过程，它强迫读者跳出简单调用库函数的舒适区，去真正理解每一步计算背后的物理意义和数值限制。对于那些在工业界从事优化模型构建的工程师而言，这本书提供的不仅仅是理论框架，更是一套实用的“排错指南”。比如，当一个迭代过程陷入停滞或发散时，书中对敏感参数的讨论，能立刻指引我们回到问题的根源进行诊断。它不是一本轻松的入门读物，需要读者具备一定的数学背景和计算经验，但一旦攻克，其回报是巨大的——对计算流程的掌控力会提升到一个全新的水平。
评分我必须承认，最初拿起这本书时，对其篇幅感到有些压力，但一旦进入状态，便发现它的组织结构极具说服力。这本书的强大之处在于，它成功地将理论的深度和教学的友好性结合了起来，使得即便是跨学科的研究人员也能顺利入门。它没有跳过任何关键的数学推导，但又总能在关键转折点提供直观的几何解释或物理类比，极大地降低了理解的门槛。相比市面上其他偏重某一特定算法（如强化学习中的策略梯度）的书籍，这本著作的优势在于其包容性，它为所有主流的优化范式提供了统一的数学语言。特别是对约束处理机制的细致剖析，让我彻底明白了松弛变量和拉格朗日乘子在实际求解中的微妙作用。如果你想写出健壮、可解释、性能可靠的优化求解器，这本书绝对是必须精读的参考资料，它传授的不是技巧，而是解决问题的底层思维方式。
评分这本书简直是为那些痴迷于算法优雅性的读者准备的盛宴。它的叙事风格带着一种冷静而精确的美感，将复杂的优化难题层层剥开，展示出其内在的简洁结构。其中关于KKT条件和对偶性的讨论，简直可以用“教科书级别”来形容，作者的讲解清晰到几乎不需要借助外部参考资料。我尤其赞赏书中对全局最优性检验方法的梳理，这在很多实际应用中是容易被忽略但至关重要的一环。对于学习理论数学的本科高年级或研究生来说，这本书提供了一个完美的视角，去欣赏优化问题是如何从一个抽象的数学描述，通过一系列精妙的迭代和逼近，最终转化为可以在计算机上执行的具体步骤。它没有过多纠缠于过于晦涩的纯数学证明，而是将重点放在了“如何有效求解”这一核心目标上，体现了极强的工程导向性，使得理论知识拥有了坚实的落地基础。
评分这是一本关于计算科学的经典著作，深得领域内研究者和实践者的推崇。它以一种严谨而全面的方式，梳理了优化理论的基石，从凸集、凸函数这些基础概念出发，逐步深入到线性规划、二次规划等经典模型。作者在阐述理论的同时，非常注重与实际应用的结合，书中包含了大量的案例分析和算法实现细节，对于希望将优化方法应用于工程、金融、机器学习等领域的读者来说，无疑是一本极佳的参考手册。书中对于算法收敛性的分析深入而透彻，这一点对于需要设计或改进优化算法的研究人员尤为重要。无论是初学者试图建立扎实的理论基础，还是资深专家希望回顾或查阅特定问题的求解策略，都能在这本书中找到价值。特别是它对内点法、牛顿法等核心算法的详尽介绍，使得读者能够清晰地理解这些强大工具背后的数学原理和计算效率考量。全书结构清晰，逻辑性强，阅读体验流畅，是理解现代优化技术不可或缺的工具书。
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