CVX用户指南：优化问题解决方案

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"CVX manual from Boyd_optimization" CVX 是一个强大的数学建模语言，用于在MATLAB环境中解决线性规划、二次锥规划和半定规划等优化问题。CVX手册由Michael C. Grant和Stephen P. Boyd撰写，是CVXResearch, Inc.发布的一个详细指南，帮助用户理解和应用CVX解决实际的优化问题。手册首先介绍了CVX的基本概念。CVX是一种基于“纪律化凸编程”（Disciplined Convex Programming, DCP）的方法，它允许用户以自然的数学表达式来构建优化问题，而无需手动转换成标准形式。CVX不是一种独立的求解器，而是对现有求解器（如MOSEK、Gurobi等）的接口，它自动将用户的模型转换为这些求解器可以处理的形式。在安装部分，手册详述了CVX支持的操作系统平台，并指导如何安装CVX专业版许可证，同时也列出了随CVX一起提供的求解器。CVX支持多种求解器，这些求解器能够处理不同类型的优化问题。快速入门章节通过几个简单的例子展示了CVX的使用，例如最小子平方法、带有约束的最小二乘问题、其他范数和函数、额外的约束以及绘制最优权衡曲线。每个示例都解释了如何使用cvx_begin和cvx_end定义优化问题，以及如何声明变量、设置目标函数和约束。 CVX的基础部分深入讨论了变量、目标函数、约束、函数、集合成员资格、对偶变量、赋值和表达式持有者等核心概念。这部分内容对于理解CVX的工作原理至关重要。 DCP规则集是CVX的核心，它定义了构建有效凸模型的规则。规则集包括关于曲率的分类、顶级规则、约束规则、表达式规则、函数规则以及它们的组合。这些规则确保了用户构建的模型是有效的，且可以被CVX正确处理。手册还涵盖了半定规划模式，这是处理涉及矩阵变量的优化问题的重要工具。此外，还介绍了几何编程模式，这是一种特定类型的非线性优化，特别适合处理比例和几何关系。总而言之，CVX手册是学习和应用CVX解决优化问题的宝贵资源，无论你是初学者还是经验丰富的研究人员，都能从中获益。它详细阐述了CVX的功能和用法，使得复杂的优化问题可以更直观、更方便地进行建模和求解。

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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