信号处理与通信中的凸优化技术：经典入门教程

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"凸优化在信号处理与通信中的应用 Convex Optimization in Signal Processing and Communications" 本书由DANIEL P. PALOMAR和YONI N. ELDA编辑，旨在介绍凸优化理论在信号处理和通信系统中的应用，是通信系统凸优化的一个经典入门教程。凸优化是一种数学方法，用于解决具有特定性质（如凸性）的优化问题，它在解决复杂问题时能够保证全局最优解，而非局部最优。在信号处理领域，凸优化广泛应用在滤波器设计、参数估计、信号检测和信号重构等多个方面。例如，最小均方误差（Minimum Mean Square Error, MMSE）估计问题可以通过凸优化来求解，以获得最佳的信号估计。此外，通过凸优化可以设计出性能优良的线性滤波器，如卡尔曼滤波器，用于去除噪声并提取有用信号。在通信系统中，凸优化被用来优化传输效率、功率分配、信道编码和解码策略。例如，在无线通信中，功率分配是一个关键问题，凸优化可以确保在满足服务质量要求的同时，实现能量的有效利用。在多用户通信环境中，资源调度和公平性问题也可以通过凸优化技术得到解决。书中可能涵盖了如拉格朗日乘子法、对偶问题、半定规划（Semidefinite Programming, SDP）、二次锥规划（Quadratic Constrained Programming, QCP）等凸优化的基本工具和算法。这些工具不仅有助于理解和解决实际工程问题，也为理论研究提供了强大的分析框架。此外，书中的例子和练习题可能涵盖各种实际通信系统，如CDMA、OFDM、MIMO（多输入多输出）系统等，让读者能够将理论知识应用于实际场景。同时，书中可能还讨论了最近的进展，如大规模MIMO系统和机器学习在通信中的应用，这些都与凸优化紧密相关。《凸优化在信号处理与通信中的应用》是一本深入浅出的教材，适合通信工程和信号处理领域的学生和研究人员学习，它提供了理论基础和实用技巧，帮助读者掌握如何利用凸优化技术解决实际问题，提升系统性能。

4 Automatic code generation for real-time convex optimization

probabilities of unknown variables are formed soon after new information (in the form

of some known variables) arrives.

Finally, we note that the ideas behind real-time embedded optimization could also

be useful in more conventional situations with no real-time deadlines. The ability to

extremely rapidly solve problem instances from a speciﬁc problem family gives us the

ability to solve large numbers of similar problem instances quickly. Some example uses

of this are listed below.

•

Trade-off analysis.An engineer formulating a design problem as an optimization prob-

lem solves a large number of instances of the problem, while varying the constraints,

to obtain a sampling of the optimal trade-off surface. This provides useful design

guidelines.

•

Global optimization. A combinatorial optimization problem is solved using branch-

and-bound or a similar global optimization method. Such methods require the solution

of a large number of problem instances from a (typically convex, often LP) problem

family. Being able to solve each instance very quickly makes it possible to solve the

overall problem much faster.

•

Monte Carlo performance analysis. With Monte Carlo simulation, we can ﬁnd the

distribution of minimum cost of an optimization problem that depends on some random

parameters. These parameters (e.g., prices of some resources or demands for products)

are random with some given distribution, but will be known before the optimization

is carried out. To ﬁnd the distribution of optimized costs, we use Monte Carlo: we

generate a large number of samples of the price vector (say), and for each one we

carry out optimization to ﬁnd the minimal cost. Here, too, we end up solving a large

number of instances of a given problem family.

1.1.3 Convex optimization

Convex optimization has many advantages over general nonlinear optimization, such as

the existence of efﬁcient algorithms that can reliably ﬁnd a globally optimal solution. A

less appreciated advantage is that algorithms for speciﬁc convex optimization problem

families can be highly robust and reliable; unlike many general purpose optimization

algorithms, they do not have parameters that must be manually tuned for particular

problem instances. Convex optimization problems are, therefore, ideally suited to real-

time embedded applications, because they can be reliably solved.

A large number of problems arising in application areas like signal processing,

control, ﬁnance, statistics and machine learning, and network operation can be cast

(exactly, or with reasonable approximations) as convex problems. In many other prob-

lems, convex optimization can provide a good heuristic for approximate solution of the

problem [13, 14].

In any case, much of what we say in this chapter carries over to local optimization

methods for nonconvex problems, although without the global optimality guarantee, and

with some loss in reliability. Even simple methods of extending the methods of convex

optimization can work very well in pratice. For example, we can use a basic interior-point

1.1 Introduction 5

method as if the problem were convex, replacing nonconvex portions with appropriate

convex approximations at each iteration.

1.1.4 Outline

In Section 1.2, we describe problem families and the speciﬁcation languages used to for-

mally model them, and two general approaches to solving problem instances described

this way: via a parser-solver, and via code generation. We list some speciﬁc example

applications of real-time convex optimization in Section 1.3. In Section 1.4 we describe,

in general terms, some requirements on solvers used in real-time optimization applica-

tions, along with some of the attributes of real-time optimization problems that we can

exploit. We give a more detailed description of how a code generator can be constructed

in Section 1.5, brieﬂy describe a preliminary implementation of a code generator in

Section 1.6, and report some numerical results in Section 1.7. We give a summary and

conclusions in Section 1.8.

1.1.5 Previous and related work

Here we list some representative references that focus on various aspects of real-time

embedded optimization or closely related areas.

Control

Plenty of work focuses on traditional real-time control [15–17], or basic model predictive

control [18–23]. Several recent papers describe methods for solving various associated

QPs quickly. One approach is explicit MPC, pioneered by Bemporad and Morari [24],

who exploit the fact that the solution of the QP is a piecewise linear function of the

problem data, which can be determined analytically ahead of time. Solving instances of

the QPthen reduces to evaluating a piecewise linear function. Interior-point methods [25],

including fast custom interior-point methods [6], can also be used to provide rapid

solutions. For fast solution of the QPs arising in evaluation of control-Lyapunov policies

(a special case of MPC), see [26]. Several authors consider fast solution of nonlinear

control problems using an MPC framework [27–29]. Others discuss various real-time

applications [30, 31], especially those in robotics [32–34].

Signal processing, communications, and networking

Work on convex optimization in signal processing includes 

-norm minimization for

sparse signal recovery, recovery of noisy signals, or statistical estimation [35, 36], or

linear programming for error correction [37]. Goldfarb and Yin discuss interior-point

algorithms for solving total variation image restoration problems [38]. Some combi-

natorial optimization problems in signal processing that are approximately, and very

quickly, solved using convex relaxations and local search are static fault detection [14],

dynamic fault detection [39], query model estimation [40] and sensor selection [13]. In

communications, convex optimization is used in DSL [41], radar [42], and CDMA [43],

to list just a few examples.

6 Automatic code generation for real-time convex optimization

Since the publication of the paper by Kelly et al. [44], which poses the optimal

network ﬂow control as a convex optimization problem, many authors have looked

at optimization-based network ﬂow methods [45–48], or optimization of power and

bandwidth [49, 50].

Code generation

The idea of automatic generation of source code is quite old. Parser-generators such as

Yacc [51], or more recent tools like GNU Bison [52], are commonly used to simplify

the writing of compilers. For engineering problems, in particular, there are a range of

code generators: one widely used commercial tool is Simulink [53], while the open-

source Ptolemy project [54] provides a modeling environment for embedded systems.

Domain-speciﬁc code generators are found in many different ﬁelds [55–58].

Generating source code for optimization solvers is nothing new either; in 1988 Oohori

and Ohuchi [59] explored code generation for LPs, and generated explicit Cholesky fac-

torization code ahead of time. Various researchers have focused on code generation for

convex optimization. McGovern, in his PhD thesis [60], gives a computational complex-

ity analysis of real-time convex optimization. Hazan considers algorithms for on-line

convex optimization [61], and Das and Fuller [62] hold a patent on an active-set method

for real-time QP.

1.2 Solvers and speciﬁcation languages

It will be important for us to carefully distinguish between an instance of an optimization

problem, and a parameterized family of optimization problems, since one of the key

features of real-time embedded optimization applications is that each of the speciﬁc

problems to be solved comes from a single family.

1.2.1 Problem families and instances

We consider continuously parameterized families of optimization problems, of the form

minimize F

(x, a)

subject to F

(x, a) ≤ 0, i = 1, ..., m

(x, a) = 0, i = 1, ..., p,

(1.1)

where x ∈ R

is the (vector) optimization variable, and a ∈ A ⊂ R



is a parameter

or data vector that speciﬁes the problem instance. To specify the problem family (1.1),

we need descriptions of the functions F

, ..., F

, H

, ..., H

, and the parameter set A.

When we ﬁx the value of the parameters, by ﬁxing the value of a, we obtain a problem

instance.

As a simple example, consider the QP

minimize (1/2)x

Px + q

subject to Gx ≤ h, Ax = b,

(1.2)

1.2 Solvers and speciﬁcation languages 7

with variable x ∈ R

, where the inequality between vectors means componentwise. Let

us assume that in all instances we care about, the equality constraints are the same, that

is, A and b are ﬁxed. The matrices and vectors P, q, G, and h can vary, although P must

be symmetric positive semideﬁnite. For this problem family we have

a = (P, q, G, h) ∈ A = S

× R

m×n

× R

where S

denotes the set of symmetric n × n positive semideﬁnite matrices. We can

identify a with an element of R



, with total dimension

 = n(n + 1)/2



 

+ n



+ mn



+ m



In this example, we have

(x, a) = (1/2)x

Px + q

(x, a) = g

x − h

, i = 1, ..., m,

(x, a) =˜a

x − b

, i = 1, ..., p,

where g

is the ith row of G, and ˜a

is the ith row of A. Note that the equality constraint

functions H

do not depend on the parameter vector a; the matrix A and vector b are

constants in the problem family (1.2).

Here we assume that the data matrices have no structure, such as sparsity. But in many

cases, problem families do have structure. For example, suppose that we are interested

in the problem family in which P is tridiagonal, and the matrix G has some speciﬁc

sparsity pattern, with N (possibly) nonzero entries. Then A changes, as does the total

parameter dimension, which becomes

 = 2n − 1





+ n



+ N



+ m



In a more general treatment, we could also consider the dimensions and sparsity

patterns as (discrete) parameters that one speciﬁes when ﬁxing a particular problem

instance. Certainly when we refer to QP generally, we refer to families of QPs with

any dimensions, and not just a family of QPs with some speciﬁc set of dimensions

and sparsity patterns. In this chapter, however, we restrict our attention to continuously

parameterized problem families, as described above; in particular, the dimensions n, m,

and p are ﬁxed, as are the sparsity patterns in the data.

The idea of a parameterized problem family is a central concept in optimization

(although in most cases, a family is considered to have variable dimensions). For exam-

ple, the idea of a solution algorithm for a problem family is sensible, but the idea of

a solution algorithm for a problem instance is not. (The best solution algorithm for a

problem instance is, of course, to output a pre-computed solution.)

8 Automatic code generation for real-time convex optimization

Nesterov and Nemirovsky refer to families of convex optimization problems, with

constant structure and parameterized by ﬁnite dimensional parameter vectors as well

structured problem (families) [63].

1.2.2 Solvers

Asolver or solution method for a problem family is an algorithm that, given the parameter

value a ∈ A, ﬁnds an optimal point x



(a) for the problem instance, or determines that

the problem instance is infeasible or unbounded.

Traditional solvers [1,64,65] can handle problem families with a range of dimensions

(e.g., QPs with the form (1.2), any values for m, n, and p, and any sparsity patterns in the

data matrices). With traditional solvers, the dimensions, sparsity patterns and all other

problem data a are speciﬁed only at solve time, that is, when the solver is invoked. This

is extremely useful, since a single solver can handle a very wide range of problems,

and exploit (for efﬁciency) a wide variety of sparsity patterns. The disadvantage is that

analysis and utilization of problem structure can only be carried out as each problem

instance is solved, which is then included in the per-instance solve time. This also limits

the reasonable scope of efﬁciency gains: there is no point in spending longer looking for

an efﬁcient method than it would take to solve the problem with a simpler method.

This traditional approach is far from ideal for real-time embedded applications, in

which a very large number of problems, from the same continuously parameterized

family, will be solved, hopefully very quickly. For such problems, the dimensions and

sparsity patterns are known ahead of time, so much of the problem and efﬁciency analysis

can be done ahead of time (and in relative leisure).

It is possible to develop a custom solver for a speciﬁc, continuously parameterized

problem family. This is typically done by hand, in which case the development effort

can be substantial. On the other hand, the problem structure and other attributes of

the particular problem family can be exploited, so the resulting solver can be far more

efﬁcient than a generic solver [6,66].

1.2.3 Speciﬁcation languages

A speciﬁcation language allows a user to describe a problem instance or problem family

to a computer, in a convenient, high-level algebraic form. All speciﬁcation languages

have the ability to declare optimization variables; some also have the ability to declare

parameters. Expressions involving variables, parameters, constants, supported operators,

and functions from a library can be formed; these can be used to specify objectives and

constraints. When the speciﬁcation language supports the declaration of parameters, it

can also be used to describe A, the set of valid parameters. (The domains of functions

used in the speciﬁcation may also implicitly impose constraints on the parameters.)

Some speciﬁcation languages impose few restrictions on the expressions that can

be formed, and the objective and constraints that can be speciﬁed. Others impose

strong restrictions to ensure that speciﬁed problems have some useful property such as

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信号处理与通信中的凸优化技术：经典入门教程

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convex optimization for signal processing and communications

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如何通过凸优化技术解决实际中的决策分析问题？请结合《斯坦福大学经典教材：凸优化Convex Optimization》给出具体应用案例。

在实际决策分析中，如何应用凸优化技术来求解资源分配问题？请结合《斯坦福大学经典教材：凸优化Convex Optimization》给出具体应用案例。

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