CVX用户指南：凸优化入门与应用

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"CVX是用于凸优化的软件包，主要功能是帮助用户解决一系列的数学优化问题。它采用了一种称为‘纪律化凸编程’（Disciplined Convex Programming, DCP）的方法，简化了编写和求解凸优化模型的过程。DCP是一种规则集，确保用户构建的模型是凸的，从而保证了找到的解是全局最优的。CVX不仅包含了基本的优化求解器，还支持多种平台，并提供了专业版的许可证管理。" 在CVX的安装过程中，支持多种操作系统平台，包括常见的Windows、Linux和Mac OS。安装专业版许可证需要按照指定步骤操作，并且CVX自带了一些求解器，如SDPT3、MOSEK等，可用于不同类型的优化问题。快速入门章节展示了如何使用CVX解决最优化问题。例如，CVX可以轻松处理最小二乘问题和带有约束条件的最小二乘问题。此外，它支持各种范数和函数，以及不同类型的约束，如线性、非线性和等式约束。CVX还能绘制出优化问题的最优权衡曲线，帮助理解不同决策变量之间的关系。深入到基础部分，`cvx_begin`和`cvx_end`是CVX用来界定优化问题范围的关键命令。变量在CVX中是声明的，而目标函数和约束条件是问题的核心组成部分。CVX支持各种函数，包括线性和非线性的，以及集合成员关系，如不等式和等式约束。同时，CVX引入了对偶变量的概念，这些变量在解决某些优化问题时非常有用。表达式持有者允许用户创建和存储中间表达式，提高代码的效率。 DCP规则集是CVX的核心，它定义了如何构建有效的凸优化模型。这一规则集包括了关于函数曲率的分类、顶级规则（如加法和乘法规则）、约束规则、表达式规则以及函数规则。DCP还处理了非线性组合的单调性和二次形式的处理，确保了在这些操作中保持凸性。 CVX还支持半定规划模式（Semidefinite Programming mode），适用于处理涉及矩阵变量的优化问题。此外，几何编程模式（Geometric Programming mode）则提供了解决特定类型优化问题的框架，这些问题通常涉及到比例和幂次运算。 CVX是一个强大的工具，它将复杂的凸优化问题转化为易于理解和编程的形式，使得研究人员和工程师能够更便捷地解决实际问题。通过遵循DCP规则，用户可以自信地构建和求解各种凸优化模型，而无需深入研究优化理论的细节。

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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