Matlab CVX入门与凸优化教程

需积分: 34 31 浏览量更新于2024-07-09 收藏 575KB PDF 举报

本资源是《CVX用户指南》的Release 2.1版本，由Michael C. Grant和Stephen P. Boyd撰写，由CVXResearch, Inc.于2017年12月26日发布。该指南详细介绍了在MATLAB中的凸优化工具箱CVX的使用方法。CVX是一个强大的工具，用于在解决优化问题时利用其内置的规则集（DCP ruleset）和特定模式，如线性、二次、半定规划和几何编程。 1. **简介**： - CVX的介绍：CVX是一个面向MATLAB用户的图形化界面工具，用于构建和求解凸优化问题，尤其适用于数学模型明确且满足凸性条件的场景。 - **凸优化**：这是一种特殊类型的优化，其中目标函数和约束都是凸函数，使得问题具有全局最优解，并可通过简单的算法求得。 2. **安装与支持平台**： - CVX支持的平台包括Windows、Mac OS和Linux，以及MATLAB的不同版本。 - 安装流程包括确认所使用的平台和专业许可证的安装，以及附带的求解器（如SeDuMi、SDPT3等）。 3. **快速入门**： - 内容涵盖基本的优化任务，如最小二乘问题、有界约束下的最小化、不同范数和函数的使用，以及其他常见约束类型。 - 还演示了如何通过`cvx_begin`和`cvx_end`结构创建优化模型，以及变量、目标函数、约束和函数的定义。 4. **基础用法**： - 详细讲解了变量的声明、目标函数的设置、约束的建立，以及如何使用内置的函数库。 - 提及了CVX对于集合成员资格、对偶变量和表达式处理的支持。 - 介绍了表达式的规则，包括如何构建和处理各种数学形式，如矩阵运算、标量二次形式和严格不等式。 5. **DCP规则集**： - CVX使用DCP（Disciplined Convex Programming）规则来确保输入的优化模型保持凸性，这是其核心优势。 - 规则集涵盖了顶级规则、表达式规则、函数表达式、标量二次形式和严格不等式的处理策略。 6. **特殊模式**： - 介绍了两种特殊的优化模式：半定规划（SDP），用于解决涉及对称矩阵变量的问题；和几何编程（GP），用于优化具有特定形式的目标函数和约束。 7. **高级主题**： - 最后部分可能包括对这些概念的深入讨论，以及如何利用这些特性进行复杂问题的建模和求解。通过阅读这份指南，用户可以系统地学习如何在MATLAB中利用CVX解决凸优化问题，无论是初次接触还是进阶用户，都能从中获益匪浅。理解并遵循DCP规则和特定模式的规则，将极大提升在实际工程或科研项目中的效率和优化问题求解的质量。

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

剩余100页未读，继续阅读

jjz1998

粉丝: 0
资源: 2

Matlab CVX入门与凸优化教程

CVX使用手册.pdf

cvx_usrguide.pdf

cvx-intro.zip_CVX_You Can!_weightjdf

cvx怎么屏蔽运算过程

matlab2018a安装cvx

matlab 安装cvx

CVX求解问题max¦(W) tr(SW) (10) s.t. tr(kh^H W)/γ-σ^2≥0,(11) tr(W)≤P,(12) W≥0,(13)

matlab安装cvx

matlab安装cvx工具箱

cvx.Variable(complex=True)

最新资源