Suggest edit — SU-CS361 MAY092024

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---
title: "SU-CS361 MAY092024"
source: https://www.jemoka.com/posts/kbhsu_cs361_may092024/
date: 2024-05-09
---

optimization uncertainty irreducible uncertainty: uncertainty inherent to a system epistemic uncertainty: subjective lack of knowledge about a system from our standpoint uncertainty can be presented as a vector of random variables, \(z\), where the designer has no control. Feasibility of a design point, then, depends on \((x, z) \in \mathcal{F}\), where \(\mathcal{F}\) is the feasible set of design points.
set-based uncertainty set-based uncertainty treats uncertainty \(z\) as belonging to some set \(\bold{Z}\). Which means that we typically use minimax to solnve:
\begin{equation} \min_{x \in X} \max_{z \in Z} f(x,z) \end{equation}
we don&rsquo;t assume anything about the distribution of \(z\).
probabilistic uncertainty uncertainty expected value optimization Instead of \(z \in Z\) blindly, we assume some underlying distribution of \(z\). The most natural way to do this is to compute the expectation directly:
\begin{equation} \min_{x \in X} \mathbb{E}_{z \sim P} [f(x,z)] = \min_{x \in X}\int_{Z} f(x,z) p(z) \dd{z} \end{equation}
problem additive noise
For a moment, let&rsquo;s assume that the noise is added directly:
\begin{equation} f(x,z) = f(X) + z \end{equation}
Also, let&rsquo;s consider \(z \sim \mathcal{N}(0, \Sigma)\).
This means that:
\begin{equation} \min_{x \in X} \mathbb{E}_{z \sim P} [f(x,z)] = \min_{x \in X} \qty(\mathbb{E}_{z \sim P} [f(x)] + \mathbb{E}_{z \sim P}[z]) = \min_{x \in X} \qty(f(x) + 0) \end{equation}
meaning, in this specific case, optimizing for expected value is bad.
uncertainty variance optimization \begin{align} \Var[f(x,z)] &amp;= \mathbb{E}_{z \in Z} \qty[\qty(f(x,z) - \mathbb{E}_{z \in Z}\qty[f(x,z)])^{2}] \\ &amp;= \int_{z \in Z} f(x,z)^{2}p(z) \dd{z} - \mathbb{E}_{z \in Z} \qty[f(x,z)]^{2} \end{align}
If you have a covariance matrix and a mean vector, you can formulate:
\begin{equation} \min_{x} x^{\top} u + \lambda x^{\top} \Sigma x \end{equation}
feasible set approaches statistical feasibility
&ldquo;the probability that a design point is feasible&rdquo;
\begin{equation} P((x,z) \in \mathcal{F}) = \int_{z} ((x,z) \in \mathcal{F}) p(z) \dd{z} \end{equation}