实用鲁棒优化指南：理论与实践应用

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"实用鲁棒优化指南 - Bram L. Gorissen, İhsan Yanıkoglu, Dick den Hertog" 这篇论文"实用鲁棒优化指南"由Bram L. Gorissen, İhsan Yanıkoglu 和 Dick den Hertog撰写，分别来自荷兰蒂尔堡大学的经济研究中心和土耳其奥泽大学的工业工程部门。该文于2013年11月提交，2014年12月被接受，并于2015年1月在线发表。文章关键词包括鲁棒优化和可调整鲁棒优化。鲁棒优化是一个相对较新的且活跃的研究领域，近15年来得到了快速发展。它主要针对实际操作中的信息条件进行设计，并能生成计算上可行的模型，因此在实践中具有很高的价值。然而，尽管鲁棒优化的优势明显，但其在现实生活中的应用仍然滞后，还有很大的潜力尚未得到充分利用。该论文的目标是帮助实践者理解鲁棒优化的基本概念，并指导如何在实际问题中成功应用。作者首先给出了鲁棒优化的简要介绍，然后详细讨论了在使用鲁棒优化时应遵循的重要原则和注意事项。鲁棒优化的核心思想是考虑决策问题的不确定性，通过设计能够抵御各种可能扰动的解决方案来增强系统的稳健性。在实际应用中，这通常意味着找到一个在各种可能情况下的最优解，而不是仅仅针对预期情况。论文中提到的“可调整鲁棒优化”是一种更为灵活的方法，它允许决策者在问题的实际状态暴露后对初始决策进行微调，从而提高决策的适应性。在应用鲁棒优化时，作者强调了一些关键要点，包括： 1. **理解不确定性来源**：在构建模型之前，需要深入理解问题中的不确定因素，如需求波动、价格变化或生产延误。 2. **确定不确定性范围**：定义不确定性集，这是鲁棒优化的关键步骤，它反映了可能发生的最糟糕的情况。 3. **模型选择**：选择适合的数学模型来表示问题，如线性或非线性规划，以及如何将不确定性纳入模型。 4. **计算复杂性**：考虑到实际应用的限制，必须平衡模型的复杂性和计算效率。 5. **决策规则**：明确在面对不确定性时的决策规则，比如保守策略或灵活调整策略。 6. **性能指标**：定义合适的性能指标来评估不同决策方案在各种情况下的表现。 7. **实施与反馈**：实施鲁棒优化策略后，收集数据并反馈，以便在未来的问题中改进和优化。通过这篇指南，读者将能够更好地理解和利用鲁棒优化技术，解决实际工程、经济管理和其他领域的复杂决策问题，提高系统的稳定性和应对不确定性挑战的能力。

We illustrate the three steps of deriving the RC based on a

polyhedral uncertainty set:

Z ¼f

ζ : Dζ þq Z 0g;

where DA R

mL

; ζ A R

, and qA R

Step 1(worst case reformulation): Notice that (3) is equivalent to

the following worst case reformulation:

x þ max

þq Z 0

ðP

xÞ

ζ r d: ð4Þ

Step 2(duality): We take the dual of the inner maximization

problem in (4). The inner maximization problem and its dual yield

the same optimal objective value by strong duality. Therefore, (4)

is equivalent to

x þmin

w : D

w ¼P

x; w Z 0gr d: ð5Þ

Step 3(RC): It is important to point out that we can omit the

minimization term in (5), since it is sufﬁcient that the constraint

holds for at least one w. Hence, the ﬁnal formulation of the RC

becomes

( w : a

x þq

wr d; D

w ¼P

x; w Z 0: ð6Þ

Note that the constraints in (6) are linear in xA R

and wA R

Table 1 presents the tractable robust counterparts of an

uncertain linear optimization problem for different classes of

uncertainty sets. These robust counterparts are derived using the

three steps that are described above. However, we need conic

duality instead of LP duality in Step 2 to derive the tractable robust

counterparts for the conic uncertainty set; see the fourth row of

Table 1. Similarly, to derive the tractable RC for an uncertainty

region speciﬁed by general convex constraints, i.e., in the ﬁfth row

of Table 1, we need Fenchel duality in Step 2; see Rockafellar [45]

for details on Fenchel duality, and Ben-Tal et al. [11] for the formal

proof of the associated RC reformulation. Notice that each RC

constraint has a positive safeguard in the constraint left-hand side,

e.g., J P

, J P

, and q

w; see the tractable RCs in the third

column of Table 1. These safeguards represent the level of

robustness that we introduce to the constraints. Note the equiva-

lence between the robust counterparts of the polyhedral and the

conic uncertainty set when K is the nonnegative orthant.

Nonlinear problems: Table 1 focuses on uncertain linear opti-

mization problems, i.e., both linear in the decision variables and

the uncertain parameters. Notice that, different than the presen-

ted results, the original uncertain optimization problem can be

nonlinear in the optimization variables and/or the uncertain

parameters; for more detailed treatment of such problems that

are nonlinear in the uncertain parameters, we refer to Ben-Tal

et al. [9, pp. 383–388] and Ben-Tal et al. [11].

Adversarial approach: If the robust counterpart cannot be

written as or approximated by a tractable reformulation, we

advocate to perform the so-called adversarial approach. The

adversarial approach starts with a ﬁnite set of scenarios S

Z

for the uncertain parameter in constraint i. For example, at the

start, S

only contains the nominal scenario. Then, the robust

optimization problem, which has a ﬁnite number of constraints

since Z

has been replaced by S

, is solved. If the resulting solution

is robust feasible, we have found the robust optimal solution. If

that is not the case, we can ﬁnd a scenario for the uncertain

parameter that makes the last found solution infeasible, e.g., we

can search for the scenario that maximizes the infeasibility. We

add this scenario to S

, and solve the resulting robust optimization

problem, and so on. For a more detailed description, we refer to

Bienstock and Özbay [21]. It appeared that this simple approach

often converges to optimality in a few number of iterations. The

advantage of this approach is that solving the robust optimization

problem with S

instead of Z

in each iteration, preserves the

structure of the original optimization problem. Only constraints of

the same type are added, since constraint i should hold for all

scenarios in S

. This approach could be faster than reformulating,

even for polyhedral uncertainty sets. See Bertsimas et al. [20] for a

comparison. Alternatively, if the probability distribution of the

uncertain parameter is known, one may also use the randomized

sampling of the uncertainty set proposed by Calaﬁore and Campi

[23]. The randomized approach substitutes the inﬁnitely many

robust constraints with a ﬁnite set of constraints that are

randomly sampled. It is shown that such a randomized approach

is an accurate approximation of the original uncertain problem

provided that a sufﬁcient number of samples is drawn; see Campi

and Garatti [24, Theorem 1].

Pareto efﬁciency: Iancu and Trichakis [38] discovered that “the

inherent focus of RO on optimizing performance only under worst

case outcomes might leave decisions un-optimized in case a non-

worst case scenario materialized”. Therefore, the “classical” RO

framework might lead to a Pareto inefﬁcient solution; i.e., an

alternative robust optimal solution may guarantee an improve-

ment in the objective or slack size for (at least) one scenario

without deteriorating it in other scenarios. Given a robust optimal

solution, Iancu and Trichakis propose optimizing a new problem to

Table 1

Tractable reformulations for the uncertain constraint [ðaþP

xr d 8

A Z] for different types of uncertainty sets.

Uncertainty Z Robust counterpart Tractability

Box

r 1

xþ J P

r d

Ellipsoidal

r 1

xþ J P

r d

CQP

Polyhedral

þqZ 0

xþq

wr d

w ¼P

wZ 0

Cone (closed, convex, pointed)

þqA K

xþq

wr d

w ¼P

wA K

Conic opt.

Convex cons.

Þr 0 8k

xþ



r d

¼P

uZ 0

Convex opt.

is the convex conjugate function, i.e, h

ðxÞ¼sup

yhðyÞg; and K

is the dual cone of K. Source: Ben-Tal et al. [11].

B.L. Gorissen et al. / Omega 53 (2015) 124–137126

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