High-Dimensional Optimization: Set Exploration in the Non-Asymptotic Regime

Noonan, Jack, Zhigljavsky, Anatoly

High-Dimensional Optimization: Set Exploration in the Non-Asymptotic Regime
  • 出版商: Springer
  • 出版日期: 2024-06-11
  • 售價: $2,370
  • 貴賓價: 9.5 折 $2,252
  • 語言: 英文
  • 頁數: 143
  • 裝訂: Quality Paper - also called trade paper
  • ISBN: 3031589084
  • ISBN-13: 9783031589089

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商品描述

This book is interdisciplinary and unites several areas of applied probability, statistics, and computational mathematics including computer experiments, optimal experimental design, and global optimization. The bulk of the book is based on several recent papers by the authors but also contains new results. Considering applications, this brief highlights multistart and other methods of global optimizations requiring efficient exploration of the domain of optimization. This book is accessible to a wide range of readers; the prerequisites for reading the book are rather low, and many numerical examples are provided that pictorially illustrate the main ideas, methods, and conclusions.

The main purpose of this book is the construction of efficient exploration strategies of high-dimensional sets. In high dimensions, the asymptotic arguments could be practically misleading and hence the emphasis on the non-asymptotic regime. An important link with global optimization stems from the observation that approximate covering is one of the key concepts associated with multistart and other key random search algorithms. In addition to global optimization, important applications of the results are computer experiments and machine learning.

It is demonstrated that the asymptotically optimal space-filling designs, such as pure random sampling or low-discrepancy point nets, could be rather inefficient in the non-asymptotic regime and the authors suggest ways of increasing the efficiency of such designs. The range of techniques ranges from experimental design, Monte Carlo, and asymptotic expansions in the central limit theorem to multivariate geometry, theory of lattices, and numerical integration.

This book could be useful to a wide circle of readers, especially those specializing in global optimization, numerical analysis, computer experiments, and computational mathematics. As specific recipes for improving set exploration schemes are formulated, the book can also be used by the practitioners interested in applications only.

商品描述(中文翻譯)

這本書是跨學科的，結合了應用概率、統計和計算數學的幾個領域，包括計算機實驗、最佳實驗設計和全域優化。書中的大部分內容基於作者的幾篇近期論文，但也包含了新的研究結果。考慮到應用，本書重點介紹了多啟動（multistart）和其他需要有效探索優化領域的全域優化方法。這本書對於廣泛的讀者來說是可讀的；閱讀本書的前提條件相對較低，並提供了許多數值範例，以圖示方式說明主要思想、方法和結論。

本書的主要目的是構建高維集合的有效探索策略。在高維空間中，漸近論證可能會在實際應用中產生誤導，因此強調非漸近範疇。全域優化的一個重要聯繫來自於觀察到近似覆蓋是與多啟動和其他關鍵隨機搜索算法相關的關鍵概念之一。除了全域優化，這些結果的重要應用還包括計算機實驗和機器學習。

本書展示了漸近最優的空間填充設計，例如純隨機抽樣或低差異點網，在非漸近範疇中可能相當低效，作者建議了提高這類設計效率的方法。技術範圍涵蓋了實驗設計、蒙地卡羅方法、中心極限定理中的漸近展開到多變量幾何、格理論和數值積分。

這本書對於廣泛的讀者群體可能會有幫助，特別是那些專注於全域優化、數值分析、計算機實驗和計算數學的專業人士。由於具體的改善集合探索方案的建議被提出，本書也可以被僅對應用感興趣的實務工作者使用。

作者簡介

Jack Noonan is a postdoctoral researcher at Cardiff University School of Mathematics, UK. At Cardiff University, he received a PhD on applied probability and statistics in 2021 and received a BSc in Mathematics, Operational Research and Statistics in 2017. His areas of research include high-dimensional optimization and inference, change-point detection, group testing, modelling of epidemics and missing data.

Anatoly Zhigljavsky is a professor of mathematics and statistics at Cardiff University, UK. He holds this post since 1997. He received PhD (and then habilitation) on applied probability and computational mathematics in 1981 (respectively, in 1986) at St. Petersburg State University. He is the author or co-author of 12 monographs on the topics of stochastic global optimization (five), time series analysis (four), optimal experimental design (two) and dynamical systems (one); editor/co-editor of 12 books or special issues of journals on the topics above, the author of more than 200 research papers in refereed journals, organizer of several major conferences on kernel methods in machine learning, time series analysis, experimental design, and global optimization. Professor Zhigljavsky is a recipient of a prestigious Constantine Caratheodory award (2019) by the International Society for Global Optimization for his life-time achievement in the field of stochastic global optimization.

作者簡介(中文翻譯)

Jack Noonan 是英國卡迪夫大學數學學院的博士後研究員。他於2021年在卡迪夫大學獲得應用概率與統計的博士學位，並於2017年獲得數學、運籌學與統計的學士學位。他的研究領域包括高維優化與推斷、變更點檢測、群體測試、流行病建模及缺失數據。

Anatoly Zhigljavsky 是英國卡迪夫大學的數學與統計學教授，自1997年以來一直擔任此職位。他於1981年在聖彼得堡國立大學獲得應用概率與計算數學的博士學位（並於1986年獲得資格認證）。他是12本專著的作者或合著者，主題涵蓋隨機全局優化（五本）、時間序列分析（四本）、最佳實驗設計（兩本）和動態系統（一本）；同時也是12本書籍或期刊特刊的編輯/共同編輯，並在同行評審的期刊上發表了超過200篇研究論文。他還組織了幾個關於機器學習中的核方法、時間序列分析、實驗設計和全局優化的主要會議。Zhigljavsky教授因其在隨機全局優化領域的終身成就而獲得國際全局優化學會頒發的著名康斯坦丁·卡拉泰奧多里獎（2019年）。

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