Convex Analysis: (PMS-28) (Princeton Landmarks in Mathematics and Physics) by Ralph Tyrell Rockafellar (PDF)

Description

Available for the first time in paperback, R. Tyrrell Rockafellar’s classic study presents readers with a coherent branch of nonlinear mathematical analysis that is especially suited to the study of optimization problems. Rockafellar’s theory differs from classical analysis in that differentiability assumptions are replaced by convexity assumptions. The topics treated in this volume include: systems of inequalities, the minimum or maximum of a convex function over a convex set, Lagrange multipliers, minimax theorems and duality, as well as basic results about the structure of convex sets and the continuity and differentiability of convex functions and saddle- functions. This book has firmly established a new and vital area not only for pure mathematics but also for applications to economics and engineering. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book. There is also a guide for the reader who may be using the book as an introduction, indicating which parts are essential and which may be skipped on a first reading.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐In a nutshell, an exceptional book, ideal for use as a reference (completeness and organization stands out), but also excellent for self-studying too! In fact, the last part came as a pleasant surprise to me.PhD Candidates conducting theoretical research could definitely learn a great deal about writing elegant and good math from this book.You just need to understand thata) you would rather have previous exposition to abstract mathematics (otherwise I doubt it is fit for you),b) The first few sections quickly introduce you to convex analysis, but the book is huge and it is extremely ambitious to try to read it from cover to cover.c) The book is about convex ANALYSIS, NOT CONVEX GEOMETRY. It (intentionally) does not focus on the geometric interpretation of convexity and for a good reason. Many convex settings involve multiple dimensions (e.g. thousands for convex optimization problems). A geometric account is more intuitive but does not safely and readily extend to multiple dimensions, where intuition is lost or becomes error prone. That is where analysis shines, as it abstracts the geometric intuition into algebraic relations and properties. So don’t expect to find fancy figures and illustrations (it has none).d) The book contains theoretical results pertaining to convex optimization, and is certainly written, in large, with that in mind. But remember, it is about the theory, NOT ABOUT THE ALGORITHMS etc. You need it to gain profound knowledge on the theoretical aspects of convexity. If you need to focus on convex optimization see e.g. the book from Stephen Boyd on Convex Optimization (also available for free on his website).PRESENTATION STYLEHe explains and motivates the introduction of every new definition, and although he is very meticulous in the proofs, most of them are really simplified and well presented. He actually breaks most results into small theorems, and he progressively builds upon previous results to prove the more complex theorems. This helps keep proofs really short and intuitively appealing.TARGET AUDIENCERemarkable clarity and depth in the exposition. However, as the author does mention, this book requires mathematical maturity to understand. The reader would rather have even a superficial exposition to abstract mathematics.COVERAGERegarding the coverage of the subject matter, I doubt there is anything missing, literally! If what you need is not there verbatim, then chances are you can readily deduce it from one of the existing theorems.ORGANIZATIONThe author has done a phenomenal job at organizing the content of the books. It really stands out! It is separated in 7 parts and 39 Sections. He also quite successfully managed to make every section as independent as possible from the rest. That, in combination with the excellent coverage, make this book ideal as a reference.SOME MINOR DRAWBACKSPrinting quality: The book was written in the 60s, and this is a scanned reprint. The scanning quality itself is rather poor, and some symbols are rarely faded (especially small subscripts). Nothing you won’t get easily used to, but I thought I should mention.Also you may notice (very very rarely) minor mistakes (e.g. duplication of letters etc) that one would expect to be corrected from such an old, classic book.

⭐This is a good book for the first year in PhD studies. I recommend amply this book, it’s very clear in the explanation, if you have any doubts about topology, Rockafellar explained in this book very simple the theory and all you need about Topology.

⭐cover almost all aspect; it’s easy to understand because things are discussed in R^n (rather than hilbert space, which is also a con of this book)

⭐This is the most important and influential book ever written on convex analysis and optimization. Based on the works of Fenchel and other mathematicians from the 50s and early 60s (such as the Princeton school), Rockafellar takes the subject to a new level, with a deep and comprehensive synthesis, focused primarily on a definitive development of duality theory, and of the convex analysis that supports it. This is the place to start if you are looking for a result on the theory and convex sets and functions, or duality theory; the book is comprehensive. This is not an easy book to read, and it would benefit from illustrations and exercises (it has none). However, its value and profound influence on the field are hard to overestimate.

⭐A serious piece, but actually very readable. Its packed with precious jewels if you already have an exposure to convexity.

⭐I bought the old version. I’m sure the new version is a little easier to read, maybe with some pictures included. The old version is very dense, but this can be good if you already know a bit about the field.

⭐This book is a classic. It is probably the best reference book although it is tough to read from the beginning untill the end. The style is heavy and you need strong mathematical background to understand it.Anyway, if you need a result on convex functions or convex analysis it is very likely that you will find it in ths book.

⭐convex programming is a beautiful topic which admits amazing geometric interpretation.books like this manage to destroy one’s appreciation of the topic by not providing even one (gasp!) figure. damn Bourbaki style.

⭐@INTRO@É un testo splendido che ho utilizzato come riferimento matematico per la tesi di laurea triennale in Ingegneria Elettrica ( si,é stata una follia!). Personalmente lo trovo completo, dettagliato e ben organizzato ma…troppo complesso per affrontarlo così presto(rispetto al mio percorso di studi) e senza una curata preparazione matematica formale.Il libro è stato scritto negli anni ’60 e questa è la versione scansionata dell’originale; infatti, la qualità risulta in fin dei conti piuttosto scarsa con alcuni simboli sbiaditi.Si possono incontrare alcuni piccoli errori come, per esempio, la duplicazione di lettere. Questi errori,trattandosi di un testo piuttosto datato, ci si aspetta debbano essere corretti.@INDICE@Vedi foto@CONCLUSIONE@Mi sento di consigliarlo agli studenti di dottorato ma non a studenti universitari in generale. D’altra parte se siete svegli e appassionati perché non affrontarlo prima!Spero di esservi stato d’aiuto. Buono studio!Ringrazio Amazon per la Sua professionalità e per la spedizione impeccabile.

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⭐un clásico que cualquiera que trabaje en matemática posiblemente debería tener en formato papel…calidad del texto de segunda mano excelente…

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⭐言うまでもなく本書は凸解析の世界的大家による押しも押されもせぬ古典的名著であり，1970 年の刊行後半世紀近い年月を閲する現在でも凸解析といえばまずこの本である．本書が捧げられているところの著者の師 Fenchel の講義ノート (1951, mimeo) 以来体系立った教科書のなかったこの分野に最初の綿密な導入を企てた功績は大であり，同時期 (1970 年代) の類書には本書のあと Roberts and Varberg (1973) や Ekeland and Temam (1974, 英訳 1976) が出たがいまではあまり参照されることがないこと，また最近では Hiriart-Urruty and Lemarechal (2001) や Boyd and Vandenberghe (2004) のような目新しい話題と応用に詳しい後発の書にいくらか役割を譲り渡していることを考えても，なおも読みつがれていることには相応の理由がある．いまもって本書にしか取り扱いのない話題も少なくない．したがってその価値はおのずから示されており，いまさら一般の読者がとりたてて弁護するのもおこがましいことである．もちろん学習者は必要に応じて前掲したような新しい教科書もそろえあわせて学ばれるのがよかろう．そうしたなかあえて本レビューを草するのは，古いレビュワーによる日本の Amazon で唯一の評 (13 年以上前！) が「内容は数学的に厳密で、やや読みにくい方」とおっしゃることに多少の違和感を覚えたからである．(欧米の教科書に多く共通する特徴ではあるが) 本書の記述は比較的に冗長であり行間が少なく，定理間には低次元 (多くは 2 次元集合や 2 変数関数 = エピグラフは 3 次元) での具体例も豊富である．「演習問題」や「明らか」として読者に委ねられる箇所もないではないが，主観的にはそれらは邦語の数学書に比べて要求の程度が低く無理難題はない．もとより数学的にごまかしがあるというわけではないが，「厳密で読みにくい」という表現から受ける印象とは異なっているように思う．したがって，評判に尻込みせずひもといてみれば案外にとりつきやすい本であって，「古典」というのも「誰も読まない本」という意味ではない．余談だが，定理や系の言明中に「必要十分条件は……だが，実際には」とか「じつはこのとき」とかいったつけたしの主張が多く，独立の系にすればよいものをと個人的には感じる．また上記の具体例は本文中に証明と行間を空けず組まれており，最近のきれいなレイアウトの本のように囲まれていたり例番号がついていたりもしない．こうした点では「読みにくい」という感覚は共有できる．誤植は少なからず見受けるが，その大部分はささいな筆の勢い (添字が多い箇所で添字の必要ない文字にまでついてしまっている (定理 2.3 の証明中の λ_1, …, λ_r や補題 7.3 の証明中の f_1, …, f_r) とか，“i ≧ i_0, j ≧ i_0” と書きたいものを “i ≧ i_0, j ≧ j_0” となっている (定理 10.8 の証明中) とか) と組版上のミス (場合分けの条件文の配置がおかしい (定理 4.5 の証明の直後や第 9 節冒頭の式) とか，コンマが宙に浮いている (定理 10.8) とか) であり，致命的なものは見つけていない (もっとも重大な例でも，たとえば定理 8.3 の言明中の最後の x と y の取り違いのごときがせいぜいである)．細かな記法の不統一 (集合族が {C_i, i ∈ I} だったり {C_i | i ∈ I} だったり，cl (dom f) や conv (cl C) で括弧があったりなかったり (定理 10.6 の証明中)) も校正の不徹底を感じさせるが，変数や関数名の文字は統一されているのでたいした問題ではないだろう．なお評者は以前にペーパーバック版を購入したが諸事情で目下手もとにないため大学図書館で借りたハードバックの第 1 刷 (1970 年) をもっぱら用いており，誤植の指摘はその版にもとづいているがそのいくつかはのちの刷で直っているかもしれない (ただ「なか見！検索」で見る 1996 年のリプリント版はまるっきり初版のままのように見えるが)．

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