凸风险函数优化：理论与应用

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本文档《优化凸风险函数》(Optimization of Convex Risk Functions)由Andrzej Ruszczyński和Alexander Shapiro在2006年的《运筹学数学》(Mathematics of Operations Research)上发表，主要探讨了在决策理论中涉及凸风险函数的优化问题。作者利用凸分析和向量空间中的优化理论，发展了新的风险模型表示定理，并在此基础上建立了风险函数相关问题的最优性和对偶性理论。在概率性描述不确定结果的背景下，概率论提供了丰富的概念和技术，如期望效用理论、随机排序以及各种均值-风险模型。本文旨在通过探索风险模型与优化理论之间的联系，对这一研究方向做出贡献。具体而言，作者假设Ω是一个确定的空间，不确定结果通过函数X: Ω→R来表示，这里X的较小值通常代表更好的结果。研究的核心是针对那些随着X的减小而降低的风险函数进行优化。通过引入凸分析的工具，作者能够处理这类非线性、具有约束条件的优化问题，确保找到最小化风险的决策策略。这包括但不限于均值-方差模型，这是一种常用的风险度量方法，它寻求在期望收益与方差之间取得平衡的最优解。论文的主要贡献包括： 1. **风险函数的表示定理**：给出了在凸函数框架下，如何通过可测函数的向量空间来刻画和理解风险模型的新形式，这对于理解和解决实际问题中的风险至关重要。 2. **优化理论的应用**：将优化理论应用于风险函数优化，使得问题的求解更具结构化和效率，有助于找到在满足特定风险容忍度下的最优决策。 3. **最优性和对偶性理论**：构建了与凸风险函数相关的优化问题的最优性条件和对偶性理论，这对于设计算法和评估解决方案的有效性非常有用。 4. **应用领域的拓展**：本文的工作为金融工程、风险管理、统计决策等领域提供了理论支持，尤其是在投资组合选择、保险定价和资产组合管理等实际场景中的决策制定。总结来说，这篇论文深入探讨了如何通过优化技术处理具有凸性质的风险函数，这对于理解和优化不确定环境下的决策过程具有重要的理论和实践意义。通过本文提供的理论框架，决策者可以更好地量化和管理风险，从而做出更明智的选择。

It follows that if ρ is convex and proper, then the representation

ρ(X) = sup

µ∈Y

hµ, Xi−ρ

∗

(µ)

(2.5)

holds if ρ is lower semicontinuous. Conversely, if (2.5) holds for some function ρ

∗

(·),

then ρ is lower semicontinuous and convex. Note also that if ρ is proper, lower

semicontinuous and convex, then its conjugate function ρ

∗

is proper. Let us also

remark that if X is a Banach space and Y := X

∗

is its dual (e.g., X = L

(Ω, F, ¯µ)

and Y = L

(Ω, F, ¯µ)) and ρ is convex, then ρ is lower semicontinuous in the weak

topology iﬀ it is lower semicontinuous in the strong (norm) topology. If the set Ω is

ﬁnite, then the space X is ﬁnite dimensional. In that case ρ is continuous (and hence

lower semicontinuous) if it is real valued.

Theorem 2 If assumptions (A1)–(A3) hold true and the function ρ : X→

R is lower

semicontinuous, then

ρ(X) = sup

µ∈P

hµ, Xi−ρ

∗

(µ)

, ∀X ∈X. (2.6)

Conversely, if ρ can be represented in the form (2.6) for some function ρ

∗

: Y→R,

then ρ is lower semicontinuous and assumptions (A1)–(A3) are satisﬁed.

Proof. Suppose that assumption (A2) holds true. It follows then that ρ

∗

(µ)=+∞

for any measure µ ∈Ywhich is not nonnegative. Indeed, if µ 6∈ Y

, then we have

by condition (C) that hµ,

Xi < 0 for some

X ∈X

. Take an X in the domain of ρ,

i.e., such that ρ(X) is ﬁnite, and consider X

:= X − t

X. Then for t ≥ 0, we have by

assumption (A2) that X ≥ X

, and hence ρ(X) ≥ ρ(X

). Consequently

∗

(µ) ≥ sup

t∈

hµ, X

i−ρ(X

)

≥ sup

t∈

hµ, Xi−thµ,

Xi−ρ(X)

=+∞.

Suppose that assumption (A3) holds. Then, for an X ∈ dom(ρ), we have

∗

(µ) ≥ sup

a∈

hµ, X + ai−ρ(X + a)

= sup

a∈

aµ(Ω) − a + hµ, Xi−ρ(X)

It follows that ρ

∗

(µ)=+∞ for any µ ∈Ysuch that µ(Ω) 6= 1. This shows that, under

the speciﬁed assumptions, it suﬃces to take the supremum in (2.4) with respect to

the set P⊂Yof probability measures, and hence (2.6) follows by the Fenchel-Moreau

theorem.

Conversely, suppose that representation (2.6) holds. Then ρ is given by the supre-

mum of a family of continuous aﬃne functions, and hence is convex and lower semi-

continuous. Now if Y ≥ X, then hµ, Y i−hµ, Xi = hµ, Y − Xi≥0 for any µ ∈P.

Consequently assumption (A2) follows from (2.6). Finally, we have that for any

µ ∈P, hµ, X+ai = hµ, Xi+a, and hence it follows by (2.6) that ρ(X+a)=ρ(X)+a.

The above theorem generalizes the results of F¨ollmer and Schied [6, Theorems 5

and 6].

It is said that the function ρ is positively homogeneous if ρ(tX)=tρ(X) for any

t>0 and X ∈X.Ifρ is convex and positively homogeneous, then its conjugate

∗

is the indicator function of a closed convex set A⊂Ywhich is formed by such

µ ∈Y that hµ, Xi≤ρ(X) for all X ∈X. Moreover, under assumptions (A2)–(A3),

it suﬃces to consider µ ∈P, i.e.,

A = {µ ∈P: hµ, Xi≤ρ(X), ∀ X ∈X}. (2.7)

Therefore Theorem 2 implies the following result.

Corollary 1 Suppose that assumptions (A1)–(A3) hold and the function ρ is lower

semicontinuous and positively homogeneous. Then

ρ(X) = sup

µ∈A

hµ, Xi, ∀X ∈X, (2.8)

where the set A is deﬁned in (2.7).

The set A is called the risk envelope in Rockafellar, Uryasev and Zabarankin [21],

where the above result has been developed in the space X := L

(Ω, F, ¯µ).

3 Continuity and Diﬀerentiability Properties

of Risk Functions

In applications it is usually straightforward to verify assumptions (A1)–(A3). The

assumption of the lower semicontinuity of ρ is more delicate. Suppose that ρ is proper

and convex, and denote by int(dom ρ) the interior of the domain of ρ. We have that

if ρ is bounded from above on a neighborhood of some point

X ∈X, then ρ is

continuous on int(dom ρ) (e.g., [7, p. 170, Theorem 1]). In order to verify continuity

properties of ρ it is technically advantageous to use the strong (rather than weak)

topology of X ,ifX is a Banach space. Therefore when dealing with a Banach space

X we equip it with its strong topology and use Y := X

∗

A linear functional ` : X→R is called an algebraic subgradient of ρ at

X ∈ dom ρ

ρ(X) ≥ ρ(

X)+`(X −

X), ∀X ∈X. (3.1)

Note that the algebraic subgradient functional ` is not required to be continuous.

If, moreover, ` ∈Y, then we say that ` is a subgradient of ρ at

X. The set of

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