斯坦福与加州大学洛杉矶分校合作：详解凸优化理论与实战应用

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《凸优化》是由斯坦福大学电气工程系的Stephen Boyd教授和加州大学洛杉矶分校电气工程系的Lieven Vandenberghe教授合著的一本经典著作。这本书专注于解析在众多领域（如工程、计算机科学、数学、统计学、金融和经济学等）中频繁出现的凸优化问题，并提供了一套详尽的理论框架与实用方法。作者从基本的凸集和凸函数概念出发，引导读者逐步了解各类凸优化问题的结构和特性。本书的核心内容包括以下几个方面： 1. 凸集与凸函数基础：介绍凸集的定义、性质以及凸函数的定义，这些是理解整个凸优化理论的基础。读者将学习如何识别和判断一个函数是否为凸函数，这对于确定问题是否可解至关重要。 2. 凸优化问题分类：涵盖无约束和有约束的最小化问题，如线性规划、二次规划、凸二次规划等。通过实例展示，读者可以掌握这些问题的不同求解策略。 3. 双重性和近似技术：讨论了凸优化中的对偶理论，这是解决复杂问题的重要工具，它允许将原问题转化为更易于处理的形式。此外，书中还涉及一些近似算法，如内点法，这些方法对于大规模优化问题尤其有效。 4. 统计估计技术：凸优化在统计学中有广泛应用，特别是在参数估计和模型选择方面，通过凸优化方法可以找到最有效的估计器或模型。 5. 几何问题探讨：书中结合几何直观，深入剖析凸优化问题的几何性质，帮助读者理解问题的结构及其解的特性。 6. 工作示例与习题：书中包含大量实例分析和练习题，旨在增强读者的实践能力，使他们能在实际问题中熟练运用所学知识。《凸优化》不仅适合正在攻读相关专业研究生的学生，也适合研究者和工程师，因为它提供了理论深度与实践经验的平衡，有助于他们在各自领域内提高解决问题的能力。同时，由于版权原因，本书的所有复制权受到严格的保护，未经许可不得擅自复制。该书于2004年由剑桥大学出版社出版，采用LaTeX排版，体现了作者对学术严谨性的重视。如果你对凸优化感兴趣，这本书无疑是深入学习和应用这一领域的宝贵资源。

Chapter 1

Introduction

In this introduction we give an overview of mathematical optimization, focusing on

the special role of convex optimization. The concepts introduced informally here

will be covered in later chapters, with more care and technical detail.

1.1 Mathematical optimization

A mathematical optimization problem, or just optimization problem, has the form

minimize f

(x)

subject to f

(x) ≤ b

, i = 1, . . . , m.

(1.1)

Here the vector x = (x

, . . . , x

) is the optimization variable of the problem, the

function f

: R

→ R is the objective function, the functions f

: R

→ R,

i = 1, . . . , m, are the (inequality) constraint functions, and the constants b

, . . . , b

are the limits, or bounds, for the constraints. A vector x

is called optimal, or a

solution of the problem (1.1), if it has the smallest objective value among all vectors

that satisfy the constraints: for any z with f

(z) ≤ b

, . . . , f

(z) ≤ b

, we have

(z) ≥ f

We generally consider families or classes of optimization problems, characterized

by particular forms of the objective and constraint functions. As an important

example, the optimization problem (1.1) is called a linear program if the objective

and constraint functions f

, . . . , f

are linear, i.e., satisfy

(αx + βy) = αf

(x) + βf

(y) (1.2)

for all x, y ∈ R

and all α, β ∈ R. If the optimization problem is not linear, it is

called a nonlinear program.

This book is about a class of optimization problems called convex optimiza-

tion problems. A convex optimization problem is one in which the objective and

constraint functions are convex, which means they satisfy the inequality

(αx + βy) ≤ αf

(x) + βf

(y) (1.3)

2 1 Introduction

for all x, y ∈ R

and all α, β ∈ R with α + β = 1, α ≥ 0, β ≥ 0. Comparing (1.3)

and (1.2), we see that convexity is more general than linearity: inequality replaces

the more restrictive equality, and the inequality must hold only for certain values

of α and β. Since any linear program is therefore a convex optimization problem,

we can consider convex optimization to be a generalization of linear programming.

1.1.1 Applications

The optimization problem (1.1) is an abstraction of the problem of making the best

possible choice of a vector in R

from a set of candidate choices. The variable x

represents the choice made; the constraints f

(x) ≤ b

represent ﬁrm requirements

or speciﬁcations that limit the possible choices, and the objective value f

(x) rep-

resents the cost of choosing x. (We can also think of −f

(x) as representing the

value, or utility, of choosing x.) A solution of the optimization problem (1.1) corre-

sponds to a choice that has minimum cost (or maximum utility), among all choices

that meet the ﬁrm requirements.

In portfolio optimization, for example, we seek the best way to invest some

capital in a set of n assets. The variable x

represents the investment in the ith

asset, so the vector x ∈ R

describes the overall portfolio allocation across the set of

assets. The constraints might represent a limit on the budget (i.e., a limit on the

total amount to be invested), the requirement that investments are nonnegative

(assuming short positions are not allowed), and a minimum acceptable value of

expected return for the whole portfolio. The objective or cost function might be

a measure of the overall risk or variance of the portfolio return. In this case,

the optimization problem (1.1) corresponds to choosing a portfolio allocation that

minimizes risk, among all possible allocations that meet the ﬁrm requirements.

Another example is device sizing in electronic design, which is the task of choos-

ing the width and length of each device in an electronic circuit. Here the variables

represent the widths and lengths of the devices. The constraints represent a va-

riety of engineering requirements, such as limits on the device sizes imposed by

the manufacturing process, timing requirements that ensure that the circuit can

operate reliably at a speciﬁed speed, and a limit on the total area of the circuit. A

common objective in a device sizing problem is the total power consumed by the

circuit. The optimization problem (1.1) is to ﬁnd the device sizes that satisfy the

design requirements (on manufacturability, timing, and area) and are most power

eﬃcient.

In data ﬁtting, the task is to ﬁnd a model, from a family of potential models,

that best ﬁts some observed data and prior information. Here the variables are the

parameters in the model, and the constraints can represent prior information or

required limits on the parameters (such as nonnegativity). The objective function

might be a measure of misﬁt or prediction error between the observed data and

the values predicted by the model, or a statistical measure of the unlikeliness or

implausibility of the parameter values. The optimization problem (1.1) is to ﬁnd

the model parameter values that are consistent with the prior information, and give

the smallest misﬁt or prediction error with the observed data (or, in a statistical

1.1 Mathematical optimization 3

framework, are most likely).

An amazing variety of practical problems involving decision making (or system

design, analysis, and operation) can be cast in the form of a mathematical opti-

mization problem, or some variation such as a multicriterion optimization problem.

Indeed, mathematical optimization has become an important tool in many areas.

It is widely used in engineering, in electronic design automation, automatic con-

trol systems, and optimal design problems arising in civil, chemical, mechanical,

and aerospace engineering. Optimization is used for problems arising in network

design and operation, ﬁnance, supply chain management, scheduling, and many

other areas. The list of applications is still steadily expanding.

For most of these applications, mathematical optimization is used as an aid to

a human decision maker, system designer, or system operator, who supervises the

process, checks the results, and modiﬁes the problem (or the solution approach)

when necessary. This human decision maker also carries out any actions suggested

by the optimization problem, e.g., buying or selling assets to achieve the optimal

portfolio.

A relatively recent phenomenon opens the possibility of many other applications

for mathematical optimization. With the proliferation of computers embedded in

products, we have seen a rapid growth in embedded optimization. In these em-

bedded applications, optimization is used to automatically make real-time choices,

and even carry out the associated actions, with no (or little) human intervention or

oversight. In some application areas, this blending of traditional automatic control

systems and embedded optimization is well under way; in others, it is just start-

ing. Embedded real-time optimization raises some new challenges: in particular,

it requires solution methods that are extremely reliable, and solve problems in a

predictable amount of time (and memory).

1.1.2 Solving optimization problems

A solution method for a class of optimization problems is an algorithm that com-

putes a solution of the problem (to some given accuracy), given a particular problem

from the class, i.e., an instance of the problem. Since the late 1940s, a large eﬀort

has gone into developing algorithms for solving various classes of optimization prob-

lems, analyzing their properties, and developing good software implementations.

The eﬀectiveness of these algorithms, i.e., our ability to solve the optimization prob-

lem (1.1), varies considerably, and depends on factors such as the particular forms

of the objective and constraint functions, how many variables and constraints there

are, and special structure, such as sparsity. (A problem is sparse if each constraint

function depends on only a small number of the variables).

Even when the objective and constraint functions are smooth (for example,

polynomials) the general optimization problem (1.1) is surprisingly diﬃcult to solve.

Approaches to the general problem therefore involve some kind of compromise, such

as very long computation time, or the possibility of not ﬁnding the solution. Some

of these methods are discussed in §1.4.

There are, however, some important exceptions to the general rule that most

optimization problems are diﬃcult to solve. For a few problem classes we have

4 1 Introduction

eﬀective algorithms that can reliably solve even large problems, with hundreds or

thousands of variables and constraints. Two important and well known examples,

described in §1.2 below (and in detail in chapter 4), are least-squares problems and

linear programs. It is less well known that convex optimization is another exception

to the rule: Like least-squares or linear programming, there are very eﬀective

algorithms that can reliably and eﬃciently solve even large convex problems.

1.2 Least-squares and linear programming

In this section we describe two very widely known and used special subclasses of

convex optimization: least-squares and linear programming. (A complete technical

treatment of these problems will be given in chapter 4.)

1.2.1 Least-squares problems

A least-squares problem is an optimization problem with no constraints (i.e., m =

0) and an objective which is a sum of squares of terms of the form a

x − b

minimize f

(x) = kAx − bk

i=1

x − b

)

(1.4)

Here A ∈ R

k×n

(with k ≥ n), a

are the rows of A, and the vector x ∈ R

is the

optimization variable.

Solving least-squares problems

The solution of a least-squares problem (1.4) can be reduced to solving a set of

linear equations,

A)x = A

so we have the analytical solution x = (A

−1

b. For least-squares problems

we have good algorithms (and software implementations) for solving the problem to

high accuracy, with very high reliability. The least-squares problem can be solved

in a time approximately proportional to n

k, with a known constant. A current

desktop computer can solve a least-squares problem with hundreds of variables, and

thousands of terms, in a few seconds; more powerful computers, of course, can solve

larger problems, or the same size problems, faster. (Moreover, these solution times

will decrease exponentially in the future, according to Moore’s law.) Algorithms

and software for solving least-squares problems are reliable enough for embedded

optimization.

In many cases we can solve even larger least-squares problems, by exploiting

some special structure in the coeﬃcient matrix A. Suppose, for example, that the

matrix A is sparse, which means that it has far fewer than kn nonzero entries. By

exploiting sparsity, we can usually solve the least-squares problem much faster than

order n

k. A current desktop computer can solve a sparse least-squares problem

1.2 Least-squares and linear programming 5

with tens of thousands of variables, and hundreds of thousands of terms, in around

a minute (although this depends on the particular sparsity pattern).

For extremely large problems (say, with millions of variables), or for problems

with exacting real-time computing requirements, solving a least-squares problem

can be a challenge. But in the vast majority of cases, we can say that existing

methods are very eﬀective, and extremely reliable. Indeed, we can say that solving

least-squares problems (that are not on the boundary of what is currently achiev-

able) is a (mature) technology, that can be reliably used by many people who do

not know, and do not need to know, the details.

Using least-squares

The least-squares problem is the basis for regression analysis, optimal control, and

many parameter estimation and data ﬁtting methods. It has a number of statistical

interpretations, e.g., as maximum likelihood estimation of a vector x, given linear

measurements corrupted by Gaussian measurement errors.

Recognizing an optimization problem as a least-squares problem is straightfor-

ward; we only need to verify that the objective is a quadratic function (and then

test whether the associated quadratic form is positive semideﬁnite). While the

basic least-squares problem has a simple ﬁxed form, several standard techniques

are used to increase its ﬂexibility in applications.

In weighted least-squares, the weighted least-squares cost

i=1

x − b

)

where w

, . . . , w

are positive, is minimized. (This problem is readily cast and

solved as a standard least-squares problem.) Here the weights w

are chosen to

reﬂect diﬀering levels of concern about the sizes of the terms a

x − b

, or simply

to inﬂuence the solution. In a statistical setting, weighted least-squares arises

in estimation of a vector x, given linear measurements corrupted by errors with

unequal variances.

Another technique in least-squares is regularization, in which extra terms are

added to the cost function. In the simplest case, a positive multiple of the sum of

squares of the variables is added to the cost function:

i=1

x − b

)

+ ρ

i=1

where ρ > 0. (This problem too can be formulated as a standard least-squares

problem.) The extra terms penalize large values of x, and result in a sensible

solution in cases when minimizing the ﬁrst sum only does not. The parameter ρ is

chosen by the user to give the right trade-oﬀ between making the original objective

function

i=1

x−b

)

small, while keeping

i=1

not too big. Regularization

comes up in statistical estimation when the vector x to be estimated is given a prior

distribution.

Weighted least-squares and regularization are covered in chapter 6; their sta-

tistical interpretations are given in chapter 7.

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