You might be referring to lectures or publications by Alexander Shapiro, a prominent researcher in optimization and stochastic programming. Shapiro has authored numerous papers and books, and it's possible he has given lectures on stochastic programming. His work often focuses on theoretical aspects as well as practical applications of stochastic programming.
Shapiro emphasizes that (Q(x, \xi)) is often:
This is where his lectures diverge from naive Monte Carlo approaches. He stresses: The expectation doesn't smooth the function enough to guarantee differentiability.
Let’s be honest. We’ve all been there.
You’re deep into your PhD, or maybe you’re a quant trying to level up. You hear the name Alexander Shapiro whispered in the same breath as Birge, Louveaux, and Rockafellar. You know that if you don’t understand Stochastic Programming, you’re basically using a flip phone in the age of smart phones.
So you do what any desperate, caffeine-fueled researcher does. You type into Google:
"Shapiro A lectures on stochastic programming cracked" shapiro a lectures on stochastic programming cracked
I know. I did it too.
Here is what I found, why I stopped looking for the crack, and how you can actually master the material without the guilt (or the malware).
One of the most valuable "unlocked" insights: Stochastic programs are inherently unstable w.r.t. small changes in the distribution of (\xi). Shapiro proves that if you solve an SAA with (N) samples, your solution may be far from the true optimum unless (N) grows with the problem’s complexity (e.g., dimension of (x), number of constraints).
He introduces epi-convergence and empirical process theory to quantify this. For practitioners: Do not trust SAA solutions without stability analysis — e.g., perturb the sample set and re-solve.
The "cracked" (i.e., practical breakthrough) method in his lectures is SAA: Progressive Hedging (PH):
[ \min_x \in X ; \hatfN(x) = f(x) + \frac1N \sumj=1^N Q(x, \xi^j) ]
Where (\xi^j) are i.i.d. samples.
Shapiro’s critical theoretical results (often misused in practice):
Practical "crack": Choose (N) large enough that the variance of (\hatf_N(x^*)) is small, then solve via deterministic optimization (e.g., Benders decomposition, progressive hedging). But Shapiro warns: Don't oversmooth — validate with out-of-sample testing.
Treat “cracked” as a study plan. Here’s a step-by-step approach to mastering the core ideas from Shapiro’s lectures. SDDP (multi-stage linear):
Shapiro frames stochastic programming not as a single model, but as a family of optimization problems under uncertainty. The two-stage recourse model is central:
[ \min_x \in X ; f(x) + \mathbbE_\xi[Q(x, \xi)] ]
Where:
Key insight from Shapiro: The expectation makes this an infinite-dimensional problem if (\xi) is continuous. No closed form — hence the need for sampling methods.