Robust Optimization VS Stochastic Optimization

本文探讨了鲁棒优化和随机规划两种处理不确定性的优化方法。鲁棒优化针对不确定参数不使用概率分布描述的情况,寻求在不确定性集合内所有可能实现下都可行且最优化的解决方案。对比之下,随机规划通过定义可能发生的情景及其概率来解决不确定性问题,并分为两阶段问题,第一阶段决策不依赖于随机变量的实际值,而第二阶段决策则基于已知的不确定性参数。文章还讨论了这两种方法的优点与局限。

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1. Robust optimization addresses optimization problems with uncertain parameters that are not described using probability distributions but uncertainty sets.


2. A robust optimization problem seeks to determine a solution to an optimization problem that is feasible for any realization of the uncertain parameters within the uncertainty set, and optimal for the worst-case realization of these uncertain parameters.


a stochastic programming problem 的缺点:

(1) the number of scenarios太多,导致问题规模增大

(2) distribution functions很难描绘不确定变量的特性


Robust Optimization优点:

computationally efficient and modelling flexibility

代价:At the cost of reduced flexibility


可以用对偶变换来变成 single-level的问题来求解,但一个重要前提是最里层的优化问题是convex的


对比:

Stochastic:

Random vector λ is then modeled as a set Ω of plausible outcomes or scenarios
ω, where each ω ∈ Ω has an associated probability of occurrence πω such that
ω∈Ω πω = 1.



Stochastic Programming Problems with Recourse

The simplest recourse problem is the two-stage stochastic programming problem,
in which decisions are divided into two groups, namely:
• Decisions that have to be made before the realization of uncertain parameters:

first-stage or here-and-now decisions and do not depend on the realization of the random parameters. 联系动态经济调度中,t=1时的优化叫first-stage,变量叫 here-and-now decision variables。

• Decisions that are made after the actual values of uncertain parameters are
disclosed.  

“second-stage, wait-and-see, or recourse decisions”



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