Robust Optimization VS Stochastic Optimization

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”