Boyd的ADMM讲义：非凸优化与投资组合风险平摊中的应用

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本文档"admm_4776.pdf"主要探讨了交替方向乘子法（ADMM）在非凸优化中的应用，特别关注的是它在二次平方函数（也称为二次最小二乘问题）和风险平摊投资组合选择中的运用。该方法最初由优化科学界的学者们研究多年，但 Stephen Boyd 和 Yinyu Ye 等人在2011年发表的论文《Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers》（《机器学习趋势》, 2011, 3(1): 1-122）中对ADMM进行了深入挖掘和推广，使之成为工程界关注的焦点。 ADMM是一种强大的算法，其基本思想是通过分解大规模优化问题为多个小规模子问题，然后交替求解这些子问题，直到达到收敛。这种方法的优势在于它能够处理带有约束条件的优化问题，特别是在涉及非线性和非凸函数时，如在风险平摊投资组合优化中，这种特性显得尤为重要。风险平摊是一种投资策略，旨在分散风险，同时保持一定的收益水平，而非简单的最大化期望回报或最小化方差。论文中，作者 Xi Bai 和 Katya Scheinberg 深入探讨了如何针对这类具有特殊结构的非线性、非凸函数进行二次平方函数的优化。他们提出了交替方向和交替线性化的方法，并分析了这些方法的收敛性和复杂性。由于问题的特殊结构，每次迭代的核心步骤转换成了解决可解的凸优化子问题，简化了实际求解过程。作者不仅提供了关于他们提出的优化方法的收敛率结果，还展示了如何利用全局松弛技术来扩展方法的应用范围，这有助于在实际问题中找到更有效的解决方案。这份讲义是理解ADMM在非凸优化中的核心应用以及如何将其应用于金融领域如风险平摊投资组合选择的重要资源，对于从事相关领域的研究人员和工程师来说，具有很高的参考价值。

where f (x) and g(x) are convex, smooth and nonnegative. Then we deﬁne F

(x, ¯x)=f(x)g(¯x) and

(¯x, x)=f(¯x)g(x), which implies that F

and F

satisfy Assumption and we have

(x, ¯x)=f(x)g(¯x)+f (¯x) hrg(¯x),x ¯xi+

2µ

||x  ¯x||

(¯x, x)=f(¯x)g(x )+g(¯x) hrf(¯x),x ¯xi +

2µ

||x  ¯x||

(2.9)

Finally, we present another general setting which inc l ud es the objective fun ct i on of (1.2) and

satisﬁes Assumption 2.2.

Example 2.3. Assume that h =

((f

)

 c

)

and thus

F (x)=

(x)

(x)  c

)

, (2.10)

where each f

(x)=[f

(x),f

(x),...,f

(x)] and g

(x)=[g

(x),g

(x),...,g

(x)] is an aﬃne function

of x, R

! R

It is easy to see, that is functions f

and g

are aﬃne and homogeneous (that i s they do not

contain a constant term), then this form is equivalent to (1.2), with an appropriate choic e of M

and

parameters, ot h er wi se formulation (1.4) applies with q(x) = 0. Clearly, when q(x) is a smooth

convex function, then h =

 c

)

+ q(x) should be considered .

In the risk parity case, in particular, we have

min

x,✓

F (x)=

i=1

x  ✓)

s.t. x 2 X,

which can be written as (2. 10) with f

(x, ✓)=[x, 1] and g

(x, ✓)=[M

x, ✓], i =1,...,n are

(n + 1)-dimensional aﬃne vector functions of x and ✓ and c

= 0, 8i. For any given ¯x: F

(x, ¯x,

✓)=

i=1

¯x 

✓)

and F

(¯x, x)=

i=1

(¯x

x  ✓)

. Both F

and F

are convex quadratic

functions and Assumption 2.2 is satisﬁed.

3 Variable splitting and augmented L a gr a ngi a n based methods

In this section, we discuss several alternating direction methods , all of which are based on the

augmented Lagrangi an framework with variable splitting. Augmented Lagrangi an method (with

variable splitting) and its variants have been increasingly popular in recent literat u r e [1, 7, 12, 28,

29].

In particular observe that (2.1) can be equivalently written as

min

x2X,y

F (x, y)=h(f(x),g(y))

s.t. x = y,

(3.1)

where x, y 2 R

. In other words, we map the dimension of decision variable from n in (2.1) to 2n

in (3.1).

Consider problem in the form of (3.1). Provided a penalty parameter 1/µ (µ>0), we have the

following augmented Lagrangian function:

(x, y; )=F (x, y)  

(x  y)+

2µ

kx  yk

, (3.2)

and, hence, (3.1) can be solved by the augmented Lagrangian method described in Algorithm 1.

Algorithm 1 Augmented Lagrangian method (AL)

1. Choose µ

, 

, and x

= y

;

2. for k =0, 1,..., do

k+1

] := arg min

x2X,y

(x, y; 

);

update the multiplier 

k+1

= 



k+1

 y

k+1

);

possibly choose new penalty parameter µ

k+1

Algorithm 2 Alternating direction methods of multipliers (ADMM)

1. Choose µ, 

, and x

= y

;

2. for k =0, 1,..., do

k+1

:= arg min

x2X

(x, y

; 

);

k+1

:= arg min

k+1

,y; 

);

update the multiplier 

k+1

= 



k+1

 y

k+1

);

Example 3.1. In the second-order least-squares case, we split the variables as follows

min

x,y

F (x)=

i=1

y  c

)

s.t. x = y

x 2 X.

(3.3)

Then the augmented Lagrangian function is deﬁned as

(x, y; )=

i=1

y  c

)

 

(x  y)+

2µ

kx  yk

The convergence of augmented Lagrangian method (see Algorithm 1) has b e en well s t ud i ed (see,

for instance, [5, 30]). Furthermore, the minimization of the augment Lagrangian in Al gori t hm 1,

can be performed by applying block coordinate decent method (BCD) until ﬁrst-order optimality is

guaranteed. Convergence of block coordinate decent method, under t h e assumption of uniqueness

of minimizers over blocks, and for simple convex constraints is studied, for instance, in [13].

3.1 Alternating direction methods of multipliers

Alternating direction methods of multipliers (ADMM), or alternating direction augmented La-

grangian method (ADAL), can be regarded as a variant of augmented Lagrangi an with subproblems

solved inexactly by BCD. ADMM and other alternating direction methods (ADMs) can be tracked

back to the Douglas-Rachford method in the 1950s [9] and ADMs for solving variational problems

associated with PDEs in the 1970s [10, 11]. In ADMM the multiplier is updated after only one

minimization step over each x and y blocks instead of after min im iz i ng the augmented Lagrangian

over x and y jointly. A simple framework of ADMM for s olv i n g (3.1) is given in Algorithm 2.

ADMM has been widely used and well studied, in the large-scale convex optimization setting

in the case of composite stru ct u re of the objective function (see [7] for a review). In the nonconvex

setting, ADMM has been applied to obtain KKT solutions often obtaining competitive results,

empirically. However, its theoreti c al properties are not well understood in the non convex case.

Typically, convergence is shown under the assumption that the successive di↵erences of the iterate s

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