內容簡介

本書涵蓋非綫性規劃的主要內容，包括無約束優化、凸優化、拉格朗日乘子理論和算法、對偶理論及方法等，包含瞭大量的實際應用案例. 本書從無約束優化問題入手，通過直觀分析和嚴格證明給齣瞭無約束優化問題的*優性條件，並討論瞭梯度法、牛頓法、共軛方嚮法等基本實用算法. 進而本書將無約束優化問題的*優性條件和算法推廣到具有凸集約束的優化問題中，進一步討論瞭處理約束問題的可行方嚮法、條件梯度法、梯度投影法、雙度量投影法、近似算法、流形次優化方法、坐標塊下降法等. 拉格朗日乘子理論和算法是非綫性規劃的核心內容之一，也是本書的重點.

精彩書摘

Optimization over a Convex Set

Contents

3.1.ConstrainedOptimizationProblems.........p.2363.1.1.NecessaryandSu.cientConditionsforOptimality.p.2363.1.2.ExistenceofOptimalSolutions.........p.246

3.2.FeasibleDirections-ConditionalGradientMethod..p.2573.2.1.DescentDirectionsandStepsizeRules......p.2573.2.2.TheConditionalGradientMethod........p.262

3.3.GradientProjectionMethods............p.2723.3.1.FeasibleDirectionsandStepsizeRulesBasedon.....Projection..................p.2723.3.2.ConvergenceAnalysis..............p.283

3.4.Two-MetricProjectionMethods..........p.292

3.5.ManifoldSuboptimizationMethods.........p.298

3.6.ProximalAlgorithms...............p.3073.6.1.RateofConvergence..............p.3123.6.2.VariantsoftheProximalAlgorithm.......p.318

3.7.BlockCoordinateDescentMethods.........p.3233.7.1.VariantsofCoordinateDescent.........p.327

3.8.NetworkOptimizationAlgorithms..........p.331

3.9.NotesandSources................p.338

Inthischapterweconsidertheconstrainedoptimizationproblem

minimizef(x)subjecttox∈X,

where,intheabsenceofanexplicitstatementtothecontrary,weassumethroughoutthat:

(a)

Xisanonemptyandconvexsubsetofn .Whendealingwithalgo-rithms,weassumeinadditionthatXisclosed.

(b)

The function f : n →iscontinuouslydi.erentiableoveranopensetthatcontainsX.

Thisproblemgeneralizestheunconstrainedoptimizationproblemoftheprecedingchapters,whereX=n .Wewillseethatthemainalgorithmicideasforsolvingtheunconstrainedandtheconstrainedproblemsarequitesimilar.

UsuallythesetXhasstructurespeci.edbyequationsandinequal-ities.Ifwetakeintoaccountthisstructure,somenewalgorithmicideas,basedonLagrangemultipliersanddualitytheory,comeintoplay.Theseideaswillnotbediscussedinthepresentchapter,buttheywillbethefocusofsubsequentchapters.

Similartotheunconstrainedcase,themethodsofthischapterarebasedoniterativedescentalongsuitablyobtaineddirections.However,thesedirectionsmusthavetheadditionalpropertythattheymaintainfea-sibilityoftheiterates.Suchdirectionsarecalledfeasible,andaswewillseelater,theyareusuallyobtainedbysolvingcertainoptimizationsubprob-lems.Wewillconsidervariouswaystoconstructfeasibledescentdirectionsfollowingthediscussionofoptimalityconditionsinthenextsection.

3.1 CONSTRAINEDOPTIMIZATIONPROBLEMS

Inthissectionweconsiderthemainanalyticaltechniquesforourproblem,andweprovidesomeexamplesoftheirapplication.

3.1.1 NecessaryandSu.cientConditionsforOptimality

We.rstexpandtheunconstrainedoptimalityconditionsofSection1.1fortheproblemofminimizingthecontinuouslydi.erentiablefunctionfovertheconvexsetX.Recallingthede.nitionsofSection1.1,avectorx∈Xisreferredtoasafeasiblevector,andavectorx. ∈XisalocalminimumoffoverXifitisnoworsethanitsfeasibleneighbors;thatis,ifthereexistsan>0suchthat

f(x.)≤f(x),.x∈Xwithx.x.<.

A vector x . ∈XisaglobalminimumoffoverXifitisnoworsethanallotherfeasiblevectors,i.e.,

f (x .)≤f(x),.x∈X.

前言/序言

Preface to the Third Edition

The third edition of the book is a thoroughly rewritten version of the 1999

second edition. New material was included, some of the old material was

discarded, and a large portion of the remainder was reorganized or revised.

The total number of pages has increased by about 10 percent.

Aside from incremental improvements, the changes aim to bring the

book up-to-date with recent research progress, and in harmony with the major

developments in convex optimization theory and algorithms that have

occurred in the meantime. These developments were documented in three

of my books: the 2015 book “Convex Optimization Algorithms,” the 2009

book “Convex Optimization Theory,” and the 2003 book “Convex Analysis

and Optimization” (coauthored with Angelia Nedi′c and Asuman Ozdaglar).

A major difference is that these books have dealt primarily with convex, possibly

nondifferentiable, optimization problems and rely on convex analysis,

while the present book focuses primarily on algorithms for possibly nonconvex

differentiable problems, and relies on calculus and variational analysis.

Having written several interrelated optimization books, I have come to

see nonlinear programming and its associated duality theory as the lynchpin

that holds together deterministic optimization. I have consequently set as an

objective for the present book to integrate the contents of my books, together

with internet-accessible material, so that they complement each other and

form a unified whole. I have thus provided bridges to my other works with

extensive references to generalizations, discussions, and elaborations of the

analysis given here, and I have used throughout fairly consistent notation and

mathematical level.

Another connecting link of my books is that they all share the same style:

they rely on rigorous analysis, but they also aim at an intuitive exposition that

makes use of geometric visualization. This stems from my belief that success

in the practice of optimization strongly depends on the intuitive (as well as

the analytical) understanding of the underlying theory and algorithms.

Some of the more prominent new features of the present edition are:

(a) An expanded coverage of incremental methods and their connections to

stochastic gradient methods, based in part on my 2000 joint work with

Angelia Nedi′c; see Section 2.4 and Section 7.3.2.

(b) A discussion of asynchronous distributed algorithms based in large part

on my 1989 “Parallel and Distributed Computation” book (coauthored

xvii

xviii Preface to the Third Edition

with John Tsitsiklis); see Section 2.5.

(c) A discussion of the proximal algorithm and its variations in Section 3.6,

and the relation with the method of multipliers in Section 7.3.

(d) A substantial coverage of the alternating direction method of multipliers

(ADMM) in Section 7.4, with a discussion of its many applications and

variations, as well as references to my 1989 “Parallel and Distributed

Computation” and 2015 “Convex Optimization Algorithms” books.

(e) A fairly detailed treatment of conic programming problems in Section

6.4.1.

(f) A discussion of the question of existence of solutions in constrained optimization,

based on my 2007 joint work with Paul Tseng [BeT07], which

contains further analysis; see Section 3.1.2.

(g) Additional material on network flow problems in Section 3.8 and 6.4.3,

and their extensions to monotropic programming in Section 6.4.2, with

references to my 1998 “Network Optimization” book.

(h) An expansion of the material of Chapter 4 on Lagrangemultiplier theory,

using a strengthened version of the Fritz John conditions, and the notion

of pseudonormality, based on my 2002 joint work with Asuman Ozdaglar.

(i) An expansion of the material of Chapter 5 on Lagrange multiplier algorithms,

with references to my 1982 “Constrained Optimization and

Lagrange Multiplier Methods” book.

The book contains a few new exercises. As in the second edition, many

of the theoretical exercises have been solved in detail and their solutions have

been posted in the book’s internet site

http://www.athenasc.com/nonlinbook.html

These exercises have been marked with the symbolsWWW. Many other exercises

contain detailed hints and/or references to internet-accessible sources.

The book’s internet site also contains links to additional resources, such as

many additional solved exercises from my convex optimization books, computer

codes, my lecture slides from MIT Nonlinear Programming classes, and

full course contents from the MIT OpenCourseWare (OCW) site.

I would like to express my thanks to the many colleagues who contributed

suggestions for improvement of the third edition. In particular, let

me note with appreciation my principal collaborators on nonlinear programming

topics since the 1999 second edition: Angelia Nedi′c, Asuman Ozdaglar,

Paul Tseng, Mengdi Wang, and Huizhen (Janey) Yu.

Dimitri P. Bertsekas

June, 2016