Appendix B: Formula Sheet¶
Expected value:
\[
E[X] = \sum_i p_i x_i
\]
Translation:
Variance:
\[
\mathrm{Var}(X)=E[(X-E[X])^2]
\]
Translation:
Monte Carlo estimate:
\[
E[F(x,\omega)] \approx \frac{1}{N} \sum_{n=1}^N F(x,\omega^n)
\]
Translation:
Expected value-variance score:
\[
\text{score}(x)=E[F(x,\omega)]+q\,\mathrm{Var}(F(x,\omega))
\]
Constraint-violation penalty:
\[
\text{penalty}(x,\omega)=p\max\{\text{load}(x,\omega)-\text{capacity},0\}
\]
Hurwicz criterion:
\[
H(x)=(1-\lambda)\cdot\text{worst}(x)+\lambda\cdot\text{best}(x)
\]
Minmax regret:
\[
\min_x\max_s\left(F(x,s)-\min_{x'\in X}F(x',s)\right)
\]
Bellman decision:
\[
x = \arg\min_{x \in X(S_k)} \left[ R(S_k,x) + V(S_k^x) \right]
\]
Maximization variant:
\[
x = \arg\max_{x \in X(S_k)} \left[ R(S_k,x) + V(S_k^x) \right]
\]
Post-decision expected value:
\[
V(S_k^x) = \sum_{S_{k+1}} P(S_{k+1} \mid S_k,x) V(S_{k+1})
\]
State transition:
\[
S_{k+1}=S^M(S_k,x,\omega_{k+1})
\]
Translation:
RHO:
\[
x = \arg\min_{x \in X(S)} R(S,x)
\]
Course interpretation:
CFA:
\[
x = \arg\min_{x \in X(S)} \left[ R(S,x) + I(S,x) \right]
\]
Constraint-shaping variant:
\[
(A+\Theta)x \le b+\theta
\]
VFA:
\[
x = \arg\min_{x \in X(S_k)} \left[ R(S_k,x) + \hat{V}(S_k^x) \right]
\]
Feature / aggregation version:
\[
\hat V(S_k^x)\approx \hat V(A(S_k^x))
\]
where \(A(S_k^x)\) is a smaller feature vector such as \((\text{time},\ \text{remaining capacity})\).
VFA update:
\[
\hat{V}_{new} = \hat{V}_{old} + \alpha \left(\text{observed value} - \hat{V}_{old}\right)
\]
PFA threshold:
\[
\pi_\theta(S_k)=
\begin{cases}
\text{accept}, & \text{if money/space}>\theta\\
\text{reject}, & \text{otherwise}
\end{cases}
\]
Look-ahead score:
\[
\text{score}(x)=R(S_k,x)+\frac{1}{N}\sum_{n=1}^{N}J^n(\text{future after }x)
\]
Supervised learning:
\[
f(z) \approx y
\]
Sample-policy evaluation:
\[
\widehat{J}(\pi) = \frac{1}{N} \sum_{n=1}^N J^n(\pi)
\]