CVX MATLAB工具箱：凸优化入门与进阶指南

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"CVX是用于MATLAB的凸优化工具箱，它提供了一种方便的方式来表达和求解凸优化问题。这个用户指南详细介绍了CVX的基本用法、规则集以及如何添加新的函数到原子库。" 在凸优化领域，CVX工具箱是一个强大的工具，它使得用户能够用自然的数学表达式来定义优化问题，而无需手动转换为标准形式。以下是关于CVX的一些关键知识点： 1. **凸优化**：凸优化是优化理论的一个分支，它研究的是找到在凸集内的局部最优解，这些解也是全局最优解。CVX允许用户处理这类问题，而无需深入理解复杂的优化算法。 2. **CVX工具箱**：CVX是基于MATLAB的，它将用户的优化模型转换为标准的数学形式，然后调用求解器（如SDPT3或MOSEK）进行求解。这简化了用户的工作流程，使得他们可以专注于问题的建模，而不是求解细节。 3. **Disciplined Convex Programming (DCP)**：这是CVX的核心概念，它是一组规则，用于自动验证用户定义的优化问题是凸的。DCP确保了模型的正确性，并且能自动推导出问题的对偶形式。 4. **cvxbegin和cvxend**：这两个命令用于包围优化问题的定义。在cvxbegin和cvxend之间的代码定义了优化问题的变量、目标函数和约束。 5. **变量数据类型**：CVX支持不同类型的变量，如实数、整数、非负数等。这些变量在定义约束时非常重要。 6. **目标函数**：用户可以通过简单的MATLAB表达式定义目标函数，例如最小化平方误差或最大化某些性能指标。 7. **约束**：用户可以设置各种类型的约束，包括线性、二次和非线性约束。CVX会自动处理这些约束的凸性。 8. **函数和集合**：CVX支持多种预定义的函数（如norm、exp等）和集合（如非负向量、对角矩阵等），这些在构建优化问题时非常有用。 9. **对偶变量和表达式持有者**：CVX自动处理对偶变量，这些变量与原始问题的约束有关。表达式持有者则允许用户存储和重用表达式，提高效率。 10. **DCP规则集**：该规则集定义了如何构造和组合凸函数，以保持问题的凸性。它包括关于函数的曲率、顶级规则、约束和表达式规则。 11. **添加新函数到原子库**：CVX允许用户通过遵循DCP规则集来扩展其内置函数库，这样可以处理更广泛的优化问题。通过这个用户指南，用户不仅可以学习如何使用CVX解决基本的凸优化问题，还可以了解如何扩展CVX以适应更复杂的应用场景。对于希望在MATLAB环境中进行凸优化研究或应用的工程师和学者来说，CVX是一个非常有价值的工具。

With cvx, the same problem is speciﬁed as follows:

108 cvx_begin

109 variable x(n);

110 minimize( norm(A*x-b,Inf) );

111 cvx_end

The code based on linprog, and the cvx spe ciﬁcation above will both solve the

Chebyshev norm minimization 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 c an 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:

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

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

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

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

143 x_l1 = xzz(1:n,:);

The cvx version is, not surprisingly,

149 cvx_begin

150 variable x(n);

151 minimize( norm(A*x-b,1) );

152 cvx_end

cvx automatically transforms both of these problems into LPs, not unlike those gen-

erated manually for linprog.

The advantage that automatic transformation provides is magniﬁed if we consider

functions (and their resulting transformations) that are less we ll-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 w ill omit it he re. 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:

179 k = 5;

180 cvx_begin

181 variable x(n);

182 minimize( norm_largest(A*x-b,k) );

183 cvx_end

Now let us add an equality constraint and a nonlinear inequality constraint to the

original least-squares problem:

232 cvx_begin

233 variable x(n);

234 minimize( norm(A*x-b) );

235 subject to

236 C*x == d;

237 norm(x,Inf) <= 1;

238 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 in line 236 above 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 c onstruction 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.

Inequality constraints of the form f(x) ≤ g(x) or g(x) ≥ f(x) are accepted only if f

can be veriﬁed as convex and g veriﬁed as concave. So a constraint such as

norm(x,Inf) >= 1;

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 b e satis ﬁed closely, but often not exactly.

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