凸优化：理论与应用探讨

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《凸优化》是一本由Stephen Boyd和Lieven Vandenberghe合著的专业书籍，专为理解与解决凸优化问题提供深入的理论与方法。凸优化是数学优化领域的一个重要分支，它涵盖了诸如最小二乘法和线性规划等特殊问题，这些问题不仅理论完备，而且在实际应用中极为广泛，如信号处理、机器学习、经济学和控制论等领域均有其身影。本书的核心观点是，相比于最简单的最小二平方和线性规划问题，更大的凸优化问题同样具备强大的理论基础和高效的数值求解能力。凸优化的特点在于目标函数在整个定义域内具有凸性，这意味着它没有局部凹陷，从而使得优化过程更为直观和易于分析。凸优化问题通常具有全局最优解，且可以通过诸如梯度下降法、单纯形法或更高级的算法来找到这些解，这些方法往往具有良好的收敛性和稳定性。书中详细探讨了凸集、凸函数的性质、凸优化的基本概念，包括可行域、凸优化问题的表述、最优解的存在性和唯一性，以及经典的求解策略，例如Karush-Kuhn-Tucker条件（KKT条件）的应用。此外，本书还涵盖了现代凸优化的重要分支，如二次规划、分段函数优化、凸二次规划器、凸优化的核方法和半定编程等，这些都是实际工程问题中不可或缺的工具。值得注意的是，凸优化不仅是理论研究的基石，也是工业界广泛应用的实践技巧。它在诸如通信系统设计、机器学习中的支持向量机、图像处理中的能量最小化问题、以及经济模型中的定价问题等领域发挥着关键作用。通过《凸优化》，读者不仅可以掌握这一领域的核心理论，还能了解到如何将理论知识转化为实际的计算技术，以解决复杂工程问题。总体而言，《凸优化》是一本综合性和实用性强的教材，对于希望深入理解优化理论，以及寻求解决实际问题中凸优化挑战的工程师、研究人员和学生来说，是不可或缺的参考资源。随着科技的进步和数据科学的发展，凸优化的重要性在未来只会进一步提升。

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 o ptimal, 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 allo cation 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, s uch 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 us ed 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 s olve 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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