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首页论robust优化理论与广泛应用:计算吸引力与跨领域实践
论robust优化理论与广泛应用:计算吸引力与跨领域实践
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更新于2024-07-15
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"《稳健优化理论与应用》是一篇由Dimitris Bertsimas、David B. Brown和Constantine Caramanis合著的论文,发表在2011年的《SIAM评论》(SIAM Review)上,卷53,第3期,页码464-501。该研究论文聚焦于稳健优化(Robust Optimization, RO)领域的主要理论和应用,特别是强调其计算吸引力、建模能力和广泛的应用性。 论文首先梳理了稳健优化的核心理论成果,包括如何处理不确定性、不完全信息和对最优解决方案的鲁棒性需求。作者深入探讨了RO方法如何通过考虑最坏情况或概率分布的宽泛范围来提供决策的稳定性,这在面对复杂经济环境、市场波动等不确定因素时尤为重要。 除了理论层面,文章还关注了将稳健优化与多阶段决策问题中的适应性模型相结合的研究新进展。这些模型允许在面临不断变化的条件时,动态调整策略,提高了决策的灵活性和实用性。 论文进一步展示了稳健优化在多个领域的实际应用,涵盖了金融、统计学、学习以及工程的各个子领域。在金融领域,RO被用于风险管理和投资组合优化,以应对市场不确定性;在统计学中,它应用于数据挖掘和预测模型的鲁棒估计;在学习方面,RO可以用于机器学习算法中的异常值检测和抗干扰能力提升;而在工程领域,RO有助于设计能够抵抗各种环境变化的产品和服务。 关键词:稳健优化、鲁棒性、适应性优化、稳健优化的应用。三位作者分别来自麻省理工学院的斯隆管理学院和运营研究中心,杜克大学的福夸商学院,以及某个部门(原文未详述),他们的研究旨在推动该领域的发展,并鼓励跨学科的合作,以便在实际问题解决中更好地利用稳健优化的优势。"
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Here x ∈ R
n
is a vector of decision variables, f
0
, f
i
: R
n
→ R are functions, and the uncertainty
parameters u
i
∈ R
k
are assumed to take arbitrary values in the uncertainty sets U
i
⊆ R
k
, which, for
our purposes, will always be closed. The goal of (2.3) is to compute minimum cost solutions x
∗
among
all those solutions which are feasible for all realizations of the disturbances u
i
within U
i
. Thus, if some
of the U
i
are continuous sets, (2.3), as stated, has an infinite number of constraints. Intuitively, this
problem offers some measure of feasibility protection for optimization problems containing parameters
which are not known exactly.
It is worthwhile to notice the following, straightforward facts about the problem statement of (2.3):
• The fact that the objective function is unaffected by parameter uncertainty is without loss of
generality; we may always introduce an auxiliary variable, call it t, and minimize t subject to the
additional constraint max
u
0
∈U
0
f
0
(x, u
0
) ≤ t.
• It is also without loss of generality to assume that the uncertainty set U has the form U =
U
1
× . . . × U
m
. If we have a single uncertainty set U for which we require (u
1
, . . . , u
m
) ∈ U, then
the constraint-wise feasibility requirement implies an equivalent problem is (2.3) with the U
i
taken
as the projection of U along the corresponding dimensions (see Ben-Tal and Nemirovski, [14] for
more on this).
• Constraints without uncertainty are also captured in this framework by assuming the correspond-
ing U
i
to be singletons.
• Problem (2.3) also contains the instances when the decision or disturbance vectors are contained
in more general vector spaces than R
n
or R
k
(e.g., S
n
in the case of semidefinite optimization)
with the definitions modified accordingly.
Robust Optimization is distinctly different than sensitivity analysis, which is typically applied as a
post-optimization tool for quantifying the change in cost of the associated optimal solution with small
perturbations in the underlying problem data. Here, our goal is to compute fixed solutions that ensure
feasibility independent of the data. In other words, such solutions have a priori ensured feasibility when
the problem parameters vary within the prescribed uncertainty set, which may be large. We refer the
reader to some of the standard optimization literature (e.g., Bertsimas and Tsitsiklis, [37], Boyd and
Vandenberghe, [43]) and works on perturbation theory (e.g., Freund, [75], Renegar, [123]) for more on
sensitivity analysis.
8
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It is not at all clear when (2.3) is efficiently solvable. One might imagine that the addition of robust-
ness to a general optimization problem comes at the expense of significantly increased computational
complexity. It turns out that this is true: the robust counterpart to an arbitrary convex optimization
problem is in general intractable ([13]; some approximation results for robust convex problems with
a conic structure are discussed in [35]). Despite this, there are many robust problems that may be
handled in a tractable manner, and much of the literature has focused on specifying classes of functions
f
i
, coupled with the types of uncertainty sets U
i
, that yield tractable robust counterparts. If we define
the robust feasible set to be
X(U) = {x | f
i
(x, u
i
) ≤ 0 ∀ u
i
∈ U
i
, i = 1, . . . , m}, (2.4)
then for the most part,
2
tractability is tantamount to X(U) being convex in x, with an efficiently
computable separation test. More precisely, in the next section we show that this is related to the
structure of a particular subproblem. We now present an abridged taxonomy of some of the main
results related to this issue.
2.2 Robust linear optimization
The robust counterpart of a linear optimization problem is written, without loss of generality, as
minimize c
>
x
subject to Ax ≤ b, ∀ a
1
∈ U
1
, . . . , a
m
∈ U
m
, (2.5)
where a
i
represents the i
th
row of the uncertain matrix A, and takes values in the uncertainty set
U
i
⊆ R
n
. Then, a
>
i
x ≤ b
i
, ∀a
i
∈ U
i
, if and only if max
{a
i
∈U
i
}
a
>
i
x ≤ b
i
, ∀i. We refer to this as the
subproblem which must be solved. Ben-Tal and Nemirovski [14] show that the robust LP is essentially
always tractable for most practical uncertainty sets of interest. Of course, the resulting robust problem
may no longer be an LP. We now provide some more detailed examples.
Ellipsoidal Uncertainty: Ben-Tal and Nemirovski [14], as well as El Ghaoui et al. [77, 80],
consider ellipsoidal uncertainty sets. Controlling the size of these ellipsoidal sets, as in the theorem
below, has the interpretation of a budget of uncertainty that the decision-maker selects in order to
easily trade off robustness and performance.
2
i.e., subject to a Slater condition.
9
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