CVX用户指南：MATLAB中的凸优化工具

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"CVX.pdf 是一份关于CVX的用户指南，这是一款在MATLAB环境中使用的凸优化工具。这份文档由Michael C. Grant和Stephen P. Boyd撰写，由CVX Research, Inc.发布，日期为2018年12月15日。CVX不仅介绍了如何安装和使用，还详细阐述了其基本概念、约束类型、解决器以及几何编程模式等高级特性。" CVX是一款专用于凸优化问题的MATLAB软件包，它允许用户以一种自然的数学表达方式来构建和求解凸优化模型。文档首先介绍了CVX是什么：它是一个基于MATLAB的建模系统，使得用户能够方便地将凸优化问题以标准的数学形式表示出来，并自动将其转化为求解器可以处理的形式。接着，文档详细解释了什么是“纪律化凸编程”（Disciplined Convex Programming, DCP），这是一种规则化的建模方法，确保了用户构建的模型是凸的，从而保证了求解问题的全局最优解。CVX并不是一个通用的数学求解器，而是一个用于定义、验证和解决凸优化问题的框架。在安装部分，CVX支持多种平台，并提供了专业版许可证的安装步骤。此外，文档还列出了随CVX一同提供的求解器，这些求解器能够处理各种类型的凸优化问题。快速入门章节中，CVX展示了如何解决一些基础的优化问题，如最小二乘问题、带约束的最小二乘问题，以及其他范数和函数的优化。此外，还包括其他类型的约束和优化权衡曲线的构建。在基础知识部分，CVX的建模结构通过`cvx_begin`和`cvx_end`进行包围，用户在此范围内定义变量、目标函数和约束。函数的使用、集成员资格、对偶变量以及赋值和表达式持有者等概念也得到了详细解释。 DCP规则集是CVX的核心部分，它规定了如何构造合法的凸表达式。这部分详细阐述了曲率的分类、顶级规则、约束和表达式规则，以及函数规则，包括组合规则、单调性在非线性组合中的应用，以及标量二次形式的处理。最后，文档还讨论了半定规划模式和几何编程模式，这些高级特性扩展了CVX的适用范围，使其能够解决更复杂的优化问题，比如那些涉及到矩阵变量和不等式约束的问题。 CVX是一款强大的工具，通过提供一个直观的接口，使得研究人员和工程师能够在MATLAB环境中轻松处理各种凸优化任务，从最简单的最小化问题到复杂的矩阵优化问题。这份用户指南详细且全面，是理解、学习和使用CVX的宝贵资源。

The CVX Users’ Guide, Release 2.1

m = 16; n = 8;

A = randn(m,n);

b = randn(m,1);

Then the least-squares solution x = (A

−1

b is easily computed using the backslash operator:

x_ls = A \ b;

Using CVX, the same problem can be solved as follows:

cvx_begin

variable x(n)

minimize( norm(A

x-b) )

cvx_end

(The indentation is used for purely stylistic reasons and is optional.) Let us examine this speciﬁcation line

by line:

• cvx_begin creates a placeholder for the new CVX speciﬁcation, and prepares Matlab to accept

variable declarations, constraints, an objective function, and so forth.

• variable x(n) declares x to be an optimization variable of dimension n. CVX requires that all

problem variables be declared before they are used in the objective function or constraints.

• minimize( norm(A

x-b) ) speciﬁes the objective function to be minimized.

• cvx_end signals the end of the CVX speciﬁcation, and causes the problem to be solved.

Clearly there is no reason to use CVX to solve a simple least-squares problem. But this example serves

as sort of a “Hello world!” program in CVX; i.e., the simplest code segment that actually does something

useful.

When Matlab reaches the cvx_end command, the least-squares problem is solved, and the Matlab variable

x is overwritten with the solution of the least-squares problem, i.e., (A

−1

b. Now x is an ordinary

length-n numerical vector, identical to what would be obtained in the traditional approach, at least to within

the accuracy of the solver. In addition, several additional Matlab variables are created; for instance,

• cvx_optval contains the value of the objective function;

• cvx_status contains a string describing the status of the calculation (see Interpreting the results).

All of these quantities—x, cvx_optval, and cvx_status, etc.—may now be freely used in other

Matlab statements, just like any other numeric or string values.

There is not much room for error in specifying a simple least-squares problem, but if you make one, you

will get an error or warning message. For example, if you replace the objective function with

maximize( norm(A

x-b) );

which asks for the norm to be maximized, you will get an error message stating that a convex function

cannot be maximized (at least in disciplined convex programming):

If you type who or whos at the command prompt, you may see other, unfamiliar variables as well. Any variable that begins

with the preﬁx cvx_ is reserved for internal use by CVX itself, and should not be changed.

10 Chapter 3. A quick start

The CVX Users’ Guide, Release 2.1

??? Error using ==> maximize

Disciplined convex programming error:

Objective function in a maximization must be concave.

3.2 Bound-constrained least squares

Suppose we wish to add some simple upper and lower bounds to the least-squares problem above: i.e.,

minimize kAx − bk

subject to l  x  u

where l and u are given data vectors with the same dimension as x. The vector inequality u  v means

componentwise, i.e., u

≤ v

for all i. We can no longer use the simple backslash notation to solve this

problem, but it can be transformed into a quadratic program (QP) which can be solved without difﬁculty

with a standard QP solver.

Let us provide some numeric values for l and u:

bnds = randn(n,2);

l = min( bnds, [], 2 );

u = max( bnds, [], 2 );

If you have the Matlab Optimization Toolbox, you can use quadprog to solve the problem as follows:

x_qp = quadprog( 2

A, -2

b, [], [], [], [], l, u );

This actually minimizes the square of the norm, which is the same as minimizing the norm itself. In contrast,

the CVX speciﬁcation is given by

cvx_begin

variable x(n)

minimize( norm(A

x-b) )

subject to

l <= x <= u

cvx_end

Two new lines of CVX code have been added to the CVX speciﬁcation:

• The subject to statement does nothing—CVX provides this statement simply to make speciﬁca-

tions more readable. As with indentation, it is optional.

• The line l <= x <= u represents the 2n inequality constraints.

As before, when the cvx_end command is reached, the problem is solved, and the numerical solution is

assigned to the variable x. Incidentally, CVX will not transform this problem into a QP by squaring the

objective; instead, it will transform it into an SOCP. The result is the same, and the transformation is done

automatically.

In this example, as in our ﬁrst, the CVX speciﬁcation is longer than the Matlab alternative. On the other

hand, it is easier to read the CVX version and relate it to the original problem. In contrast, the quadprog

There are also a number of solvers speciﬁcally designed to solve bound-constrained least-squares problems, such as BCLS by

Michael Friedlander.

3.2. Bound-constrained least squares 11

The CVX Users’ Guide, Release 2.1

version requires us to know in advance the transformation to QP form, including the calculations such as

A’

A and -2

A’

b. For all but the simplest cases, a CVX speciﬁcation is simpler, more readable, and

more compact than equivalent Matlab code to solve the same problem.

3.3 Other norms and functions

Now let us consider some alternatives to the least-squares problem. Norm minimization problems involving

the `

∞

or `

norms can be reformulated as LPs, and solved using a linear programming solver such as

linprog in the Matlab Optimization Toolbox; see, e.g., Section 6.1 of Convex Optimization. However,

because these norms are part of CVX’s base library of functions, CVX can handle these problems directly.

For example, to ﬁnd the value of x that minimizes the Chebyshev norm kAx − bk

∞

, we can employ the

linprog command from the Matlab Optimization Toolbox:

f = [ zeros(n,1); 1 ];

Ane = [ +A, -ones(m,1) ; ...

-A, -ones(m,1) ];

bne = [ +b; -b ];

xt = linprog(f,Ane,bne);

x_cheb = xt(1:n,:);

With CVX, the same problem is speciﬁed as follows:

cvx_begin

variable x(n)

minimize( norm(A

x-b,Inf) )

cvx_end

The code based on linprog, and the CVX speciﬁcation above will both solve the Chebyshev norm mini-

mization problem, i.e., each will produce an x that minimizes kAx − bk

∞

. Chebyshev norm minimization

problems can have multiple optimal points, however, so the particular x‘s produced by the two methods can

be different. The two points, however, must have the same value of kAx − bk

∞

Similarly, to minimize the `

norm k · k

, we can use linprog as follows:

f = [ zeros(n,1); ones(m,1); ones(m,1) ];

Aeq = [ A, -eye(m), +eye(m) ];

lb = [ -Inf(n,1); zeros(m,1); zeros(m,1) ];

xzz = linprog(f,[],[],Aeq,b,lb,[]);

x_l1 = xzz(1:n,:) - xzz(n+1:end,:);

The CVX version is, not surprisingly,

cvx_begin

variable x(n)

minimize( norm(A

x-b,1) )

cvx_end

CVX automatically transforms both of these problems into LPs, not unlike those generated manually for

linprog.

12 Chapter 3. A quick start

The CVX Users’ Guide, Release 2.1

The advantage that automatic transformation provides is magniﬁed if we consider functions (and their re-

sulting transformations) that are less well-known than the `

∞

and `

norms. For example, consider the

norm

kAx − bk

lgst,k

= |Ax − b|

[1]

+ ··· + |Ax − b|

[k]

where |Ax − b|

[i]

denotes the ith largest element of the absolute values of the entries of Ax − b. This is

indeed a norm, albeit a fairly esoteric one. (When k = 1, it reduces to the `

∞

norm; when k = m, the

dimension of Ax − b, it reduces to the `

norm.) The problem of minimizing kAx − bk

lgst,k

over x can

be cast as an LP, but the transformation is by no means obvious so we will omit it here. But this norm is

provided in the base CVX library, and has the name norm_largest, so to specify and solve the problem

using CVX is easy:

k = 5;

cvx_begin

variable x(n);

minimize( norm_largest(A

x-b,k) );

cvx_end

Unlike the `

, `

, or `

∞

norms, this norm is not part of the standard Matlab distribution. Once you have

installed CVX, though, the norm is available as an ordinary Matlab function outside a CVX speciﬁcation.

For example, once the code above is processed, x is a numerical vector, so we can type

cvx_optval

norm_largest(A

x-b,k)

The ﬁrst line displays the optimal value as determined by CVX; the second recomputes the same value from

the optimal vector x as determined by CVX.

The list of supported nonlinear functions in CVX goes well beyond norm and norm_largest. For

example, consider the Huber penalty minimization problem

minimize

i=1

φ((Ax − b)

)

with variable x ∈ R

, where φ is the Huber penalty function

φ(z) =

(

|z|

|z| ≤ 1

2|z| − 1 |z| ≥ 1

The Huber penalty function is convex, and has been provided in the CVX function library. So solving the

Huber penalty minimization problem in CVX is simple:

cvx_begin

variable x(n);

minimize( sum(huber(A

x-b)) );

cvx_end

CVX automatically transforms this problem into an SOCP, which the core solver then solves. (The CVX

user, however, does not need to know how the transformation is carried out.)

3.3. Other norms and functions 13

The CVX Users’ Guide, Release 2.1

3.4 Other constraints

We hope that, by now, it is not surprising that adding the simple bounds l  x  u to the problems above is

as simple as inserting the line l <= x <= u before the cvx_end statement in each CVX speciﬁcation.

In fact, CVX supports more complex constraints as well. For example, let us deﬁne new matrices C and d

in Matlab as follows,

p = 4;

C = randn(p,n);

d = randn(p,1);

Now let us add an equality constraint and a nonlinear inequality constraint to the original least-squares

problem:

cvx_begin

variable x(n);

minimize( norm(A

x-b) );

subject to

x == d;

norm(x,Inf) <= 1;

cvx_end

Both of the added constraints conform to the DCP rules, and so are accepted by CVX. After the cvx_end

command, CVX converts this problem to an SOCP, and solves it.

Expressions using comparison operators (==, >=, etc.) behave quite differently when they involve CVX

optimization variables, or expressions constructed from CVX optimization variables, than when they involve

simple numeric values. For example, because x is a declared variable, the expression C

x==d causes a

constraint to be included in the CVX speciﬁcation, and returns no value at all. On the other hand, outside of

a CVX speciﬁcation, if x has an appropriate numeric value—for example immediately after the cvx_end

command—that same expression would return a vector of 1s and 0s, corresponding to the truth or falsity of

each equality.

Likewise, within a CVX speciﬁcation, the statement norm(x,Inf)<=1 adds a nonlinear

constraint to the speciﬁcation; outside of it, it returns a 1 or a 0 depending on the numeric value of x

(speciﬁcally, whether its `

∞

-norm is less than or equal to, or more than, 1).

Because CVX is designed to support convex optimization, it must be able to verify that problems are convex.

To that end, CVX adopts certain rules that govern how constraint and objective expressions are constructed.

For example, CVX requires that the left- and right-hand sides of an equality constraint be afﬁne. So a

constraint such as

norm(x,Inf) == 1;

results in the following error:

??? Error using ==> cvx.eq

Disciplined convex programming error:

Both sides of an equality constraint must be affine.

In fact, immediately after the cvx_end command above, you would likely ﬁnd that most if not all of the values returned

would be 0. This is because, as is the case with many numerical algorithms, solutions are determined only to within some nonzero

numeric tolerance. So the equality constraints will be satisﬁed closely, but often not exactly.

14 Chapter 3. A quick start

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