Stochastic Optimization (SO) (随机优化)

Status: Stub — seeded during lint cleanup; pending full treatment.

Stochastic Optimization (SO, 随机优化) is the decision-making framework in which some problem data are modeled as random variables with a known (or assumed) probability distribution, and the decision is chosen to optimise an expectation — classically the expected cost E_{\mathbb{P}}[f(x,\xi)] over decision x and random vector \xi.

It is one of the three modelling paradigms that Distributionally Robust Optimization (DRO unifies, alongside Robust Optimization (RO and coherent risk measures. SO corresponds to the special case of DRO in which the ambiguity set \mathcal{P} collapses to a single distribution \{\mathbb{P}\}.

The canonical forms are:

• Two-stage / multi-stage stochastic programming — a here-and-now decision x made before \xi realises, plus recourse decisions made after.
• Chance-constrained programming — requiring \mathbb{P}[\text{constraint holds}] \geq 1-\epsilon.

In quant trading, SO underlies portfolio construction under return-distribution assumptions and execution / inventory problems under random order flow. Its core weakness — reliance on a known distribution — is precisely what DRO addresses by hedging over an ambiguity set \mathcal{P} instead.

References

• Rahimian, H. & Mehrotra, S. (2022). Distributionally Robust Optimization: A Review. Open Journal of Mathematical Optimization 3, DOI 10.5802/ojmo.15.
• Birge, J.R. & Louveaux, F. (2011). Introduction to Stochastic Programming. Springer, 2nd ed.