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首页凸优化:理论与应用
凸优化:理论与应用
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"这是一本关于凸优化的书籍,由斯坦福大学的Stephen Boyd和加州大学洛杉矶分校的Lieven Vandenberghe合著。本书探讨了凸优化这一数学优化问题的特殊类别,该类别包括最小二乘法和线性规划问题。凸优化理论完备,应用广泛,并且在数值求解上非常高效。" 凸优化是数学优化领域的一个子集,它专注于那些具有特定性质(如凸性)的问题。凸函数和凸集的概念是理解凸优化的关键。凸函数在所有方向上的局部极小值都是全局极小值,这使得解决凸优化问题相对简单且有保证找到全局最优解。 本书详细介绍了凸优化的基本概念,包括凸集、凸函数、凸组合以及相关的性质。它涵盖了各种凸优化问题的形式化表述,例如线性规划、二次规划、锥规划和几何编程等。此外,书中还讨论了如何通过梯度下降法、拟牛顿法、内点法等算法有效地求解这些问题。 作者们强调了凸优化在实际应用中的重要性,如在工程、经济、统计学和机器学习等领域。例如,在信号处理中,凸优化可用于最小化误差平方和,从而进行数据拟合;在机器学习中,支持向量机的训练问题可以转化为凸优化问题来解决。 书中的内容不仅包含理论分析,还包括了数值实验和实例,帮助读者理解和掌握凸优化技术。此外,还讨论了软件工具,如CVX,这是一个用于指定和求解凸优化问题的MATLAB接口,使得非专业程序员也能方便地应用凸优化。 "Convex Optimization" 是一本深入浅出的教材,适合对优化感兴趣的研究生、研究人员和工程师阅读。它提供了对凸优化理论的全面介绍,以及实用的求解策略,是这个领域的经典之作。通过这本书,读者可以学习到如何利用凸优化解决实际问题,提升问题解决能力。
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2 1 Introduction
for all x, y ∈ R
n
and al l α , β ∈ R with α + β =1,α ≥ 0, β ≥ 0. Comparing (1.3)
and (1.2), we se e 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 optimizationproblem,
we can consider convex optimization to be a generalization oflinearprogramming.
1.1.1 Applications
The optimization prob l e m (1.1) is an abs t r action of the problem of making the best
possible choice of a vector in R
n
from a set of candidate choices. The variable x
represents the choice made; the constraints f
i
(x) ≤ b
i
represent firm requirements
or specifications that limit the possible choices, and the objective value f
0
(x)rep-
resents the cost of choosing x.(Wecanalsothinkof−f
0
(x) as representing the
value, or utility , of choosing x .) A s olution of the optimization problem (1.1) corre-
sponds to a choice that has minimum cost (or maximum utility),amongallchoices
that meet the firm r equir ements.
In portfolio optimization,forexample,weseekthebestwaytoinvestsome
capital in a set of n assets. The variable x
i
represents the investment in the ith
asset, so the vector x ∈ R
n
describes the overall portfolio allocation across the set of
assets. The constraints might represent a limit on the budget(i.e.,alimitonthe
total amount to be invested), the requirement that investments are nonnegative
(assuming short positions are not allowed), and a minimum acceptable value of
expected r etur n 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 thefirmrequirements.
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 ensurethatthecircuitcan
operate reliably at a specified speed, and a limit on the total area of the circui t. A
common objective in a device sizing problem is the total powerconsumedbythe
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,thetaskistofindamodel,fromafamilyofpotentialmodels,
that best fits some observed data and prior information. Here the variables are the
parameters in the model, and the constraints can repr esent 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 oftheunlikelinessor
implausibility of the parameter values. The optimization problem (1.1) is to find
the model parameter values that are consistent with the priorinformation,andgive
the smallest misfit or pr ediction error with the observed data(or,inastatistical
1.1 Mathematical optimization 3
framework, are most likely).
An amazing variety of practical problems involving d ecisionmaking(orsystem
design, analysis, and operation) can be cast in the form of a mathematical opti-
mization problem, or some variation such as a multicriterionoptimizationproblem.
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 l ist of applications is still steadily expanding.
For most of these applications, mathematical optimization is used as an aid to
ahumandecisionmaker,systemdesigner,orsystemoperator,whosupervisesthe
process, checks the results, and mo difies the problem (or the solution approach)
when neces s ar y. This human decision maker als o carries out any actions suggested
by the optimization problem, e.g.,buyingorsellingassetstoachievetheoptimal
portfolio.
Arelativelyrecentphenomenonopensthepossibilityofmanyotherapplications
for mathematical optimization. With the proliferation of computers embedded in
products, we have seen a rapid growth in embedded optimization.Intheseem-
bedded applications, optimization is used to automaticallymakereal-timechoices,
and even carry out the associated actions, with no (or li ttle)humaninterventionor
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.,aninstance 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.,ourabilitytosolvetheoptimizationprob-
lem (1.1), varies considerably, and depends on factors s uch as the particu l ar forms
of the objective and constraint functions, how many vari ab les and constraints there
are, an d special structur e, such as sparsity.(Aproblemissparse if each constraint
function depends on only a small number of the variables).
Even when the objective and constraint functions are s mooth (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 thes e methods are disc u ssed in §1.4.
There are, however, some important exceptions to the generalrulethatmost
optimization problems are difficult to solve. For a few problemclasseswehave
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, thereareveryeffective
algorithms that can reliab l y 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 comple t e 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 objecti ve which is a sum of squares of terms of the form a
T
i
x − b
i
:
minimize f
0
(x)=∥Ax − b∥
2
2
=
!
k
i=1
(a
T
i
x − b
i
)
2
.
(1.4)
Here A ∈ R
k×n
(with k ≥ n), a
T
i
are the rows of A,andthevectorx ∈ R
n
is the
optimization variable.
Solving least-squares problems
The solution of a least-squares problem (1.4) can be re d uced to solving a set of
linear equations,
(A
T
A)x = A
T
b,
so we have the analytical solution x =(A
T
A)
−1
A
T
b.Forleast-squaresproblems
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
2
k,withaknownconstant. Acurrent
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 t h e future, accor d i ng 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 pr oblems, by exploiting
some special structure in the coefficient matrix A.Suppose,forexample,thatthe
matrix A is sparse,whichmeansthatithasfarfewerthankn nonzero entries. By
exploiting sparsity, we can usually solve the least-squaresproblemmuchfasterthan
order n
2
k. A current desktop computer can solve a spars e least-squaresproblem
1.2 Least-squares and linear programming 5
with tens of thousands of variables, and hundreds of thousands of terms, in around
aminute(althoughthisdependsontheparticularsparsitypattern).
For extr e me ly 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 th at 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,thatcanbereliablyusedbymanypeoplewhodo
not know, and do not need to know, the details.
Using least-squares
The leas t - squares problem is the bas is for re gression analysis, optimal control, and
many parameter estimation and data fitting methods. It has a number of statistical
interpretations, e.g. ,asmaximumlikelihoodestimationofavectorx,givenlinear
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 asso ciated quadratic form is positive semidefinite). While the
basic least-squares problem has a simple fixed form, several standard techniques
are use d to increase its flexibility in applications.
In weighted least-squares,theweightedleast-squarescost
k
"
i=1
w
i
(a
T
i
x − b
i
)
2
,
where w
1
,...,w
k
are positive, is minimized. (This problem is readily cast and
solved as a standard least-squares problem.) Here the weights w
i
are chosen to
reflect differing levels of concern about the sizes of the terms a
T
i
x − b
i
,orsimply
to influence the solution. In a statistical setting, weighted least-squares arises
in estimation of a vector x,givenlinearmeasurementscorruptedbyerrorswith
unequal variances.
Another technique in least-squares is regularization,inwhichextratermsare
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:
k
"
i=1
(a
T
i
x − b
i
)
2
+ ρ
n
"
i=1
x
2
i
,
where ρ > 0. (This problem too can be formulated as a standard least-squares
problem.) The extra terms penalize large values of x,andresultinasensible
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
!
k
i=1
(a
T
i
x−b
i
)
2
small, while keeping
!
n
i=1
x
2
i
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 opt i mi zation problems is linear programming, in which
the objective and all constraint functions are linear:
minimize c
T
x
subject to a
T
i
x ≤ b
i
,i=1,...,m.
(1.5)
Here the vectors c, a
1
,...,a
m
∈ R
n
and scalars b
1
,...,b
m
∈ R are problem pa-
rameters that specify the objective and constraint functions.
Solving linear programs
There is no si mp l e 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 require d 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
2
m (assuming m ≥ n)butwithaconstantthatis
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-squ ares 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)linearprograms
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
|a
T
i
x − b
i
|.
(1.6)
Here x ∈ R
n
is the variable, and a
1
,...,a
k
∈ R
n
, b
1
,...,b
k
∈ R are parameters
that specify the problem instance. Note the resemblance to the least-squares pr ob-
lem (1.4). For both problems, the objective is a measure of thesizeoftheterms
a
T
i
x − b
i
.Inleast-squares,weusethesumofsquaresofthetermsasobjective,
whereas in Chebyshev approximation, we use the maximum of theabsolutevalues.
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