凸优化：理论与数值方法

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"《凸优化》是一本由斯坦福大学的Stephen Boyd和加州大学洛杉矶分校的Lieven Vandenberghe合著的专业书籍，专注于介绍凸优化理论及其在工程、计算机科学、数学、统计学、金融和经济学等多个领域的应用。这本书不仅涵盖了凸集与凸函数的基础，还详细讲解了各种类型的凸优化问题、对偶理论、近似技术以及统计估计方法。书中通过几何问题的探讨，深入阐述了无约束和约束最小化问题，并重点介绍了内点法。书中的实例和作业练习使读者能够识别并解决实际的凸优化问题，是学生、研究人员和实践者的宝贵资源。" 《凸优化》是凸优化领域的一本权威著作，作者们以其深厚的学术背景，系统地介绍了该主题的各个方面。首先，他们定义和讨论了凸集和凸函数的基本概念，这些是凸优化理论的核心。通过这些基础，他们逐步引入了不同类型的凸优化问题，包括线性规划、二次规划等特殊形式，这些问题在实际应用中极为常见。接着，书中详细讲解了对偶理论，这是凸优化的一个重要工具，它允许我们从不同的视角理解问题，并且经常能提供更高效的求解策略。对偶问题与原问题之间的关系，以及强对偶性和宽对偶性的概念，都在这里得到了清晰的阐述。此外，书中还介绍了近似技术，这对于处理大规模或复杂问题尤其有用，可以找到问题的近似解，而不会过度消耗计算资源。统计估计技术部分则探讨了如何利用凸优化来解决统计学中的问题，如参数估计和假设检验，这些在数据分析和机器学习中具有广泛应用。通过凸优化的方法，我们可以得到统计上有效的解决方案。书中还专门讨论了几何问题，如凸包、凸组合等，这些概念在处理几何优化问题时非常关键。随后，作者们详细分析了无约束和约束最小化问题的解决策略，包括拉格朗日乘子法和Karush-Kuhn-Tucker (KKT) 条件。内点法作为现代优化算法的一种，它的高效性使得处理大规模约束问题成为可能，书中对此进行了深入的探讨和实例演示。《凸优化》不仅提供了理论基础，还强调了实践应用，是学习和研究凸优化的绝佳教材。书中的大量例题和习题可以帮助读者巩固理论知识，提升解决实际问题的能力。无论是初学者还是资深专家，都能从这本书中受益匪浅，提升在优化领域的专业素养。

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.

6 1 Introduction

1.2.2 Linear programming

Another important class of optimization problems is linear programming, in which

the objective and all constraint functions are linear:

minimize c

subject to a

x ≤ b

, i = 1, . . . , m.

(1.5)

Here the vectors c, a

, . . . , a

∈ R

and scalars b

, . . . , b

∈ R are problem pa-

rameters that specify the objective and constraint functions.

Solving linear programs

There is no simple analytical formula for the solution of a linear program (as there

is for a least-squares problem), but there are a variety of very eﬀective methods for

solving them, including Dantzig’s simplex method, and the more recent interior-

point methods described later in this book. While we cannot give the exact number

of arithmetic operations required to solve a linear program (as we can for least-

squares), we can establish rigorous bounds on the number of operations required

to solve a linear program, to a given accuracy, using an interior-point method. The

complexity in practice is order n

m (assuming m ≥ n) but with a constant that is

less well characterized than for least-squares. These algorithms are quite reliable,

although perhaps not quite as reliable as methods for least-squares. We can easily

solve problems with hundreds of variables and thousands of constraints on a small

desktop computer, in a matter of seconds. If the problem is sparse, or has some

other exploitable structure, we can often solve problems with tens or hundreds of

thousands of variables and constraints.

As with least-squares problems, it is still a challenge to solve extremely large

linear programs, or to solve linear programs with exacting real-time computing re-

quirements. But, like least-squares, we can say that solving (most) linear programs

is a mature technology. Linear programming solvers can be (and are) embedded in

many tools and applications.

Using linear programming

Some applications lead directly to linear programs in the form (1.5), or one of

several other standard forms. In many other cases the original optimization prob-

lem does not have a standard linear program form, but can be transformed to an

equivalent linear program (and then, of course, solved) using techniques covered in

detail in chapter 4.

As a simple example, consider the Chebyshev approximation problem:

minimize max

i=1,...,k

x − b

(1.6)

Here x ∈ R

is the variable, and a

, . . . , a

∈ R

, b

, . . . , b

∈ R are parameters

that specify the problem instance. Note the resemblance to the least-squares prob-

lem (1.4). For both problems, the objective is a measure of the size of the terms

x − b

. In least-squares, we use the sum of squares of the terms as objective,

whereas in Chebyshev approximation, we use the maximum of the absolute values.

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