ThotDjehuty
dd51156174
Each of the eight v2.0 companion notebooks (10_bsde through 17_generative_calibration) now follows the mandatory pedagogical sandwich structure: PRE markdown : theorem / model / pivot equation / what the cell verifies CODE cell : labelled prints + at least one matplotlib figure POST markdown: expected result, graph reading, conclusion Each notebook carries at least one concrete real-world example (heat plate, inverted pendulum, opinion polarization, collective decision, OU drift under Cauchy noise, mixture vs gaussian MMD, etc.) Generator script: scripts/enrich_v2_notebooks.py Doc plots refreshed via scripts/inject_doc_plots.py.
148 lines
4.8 KiB
ReStructuredText
148 lines
4.8 KiB
ReStructuredText
Stochastic control — switching, Pontryagin, two-sided intensities
|
||
=================================================================
|
||
|
||
Three primitives: discrete-time optimal switching (`optimal_switching_dp`), 1-D Pontryagin LQR shooting (`pontryagin_lqr`) and the bilateral intensity controller (`two_sided_intensities`).
|
||
|
||
.. note:: Companion executed notebook: `12_stochastic_control.ipynb <../../examples/notebooks/12_stochastic_control.ipynb>`_
|
||
|
||
12 — Stochastic control
|
||
=======================
|
||
|
||
.. code-block:: python
|
||
|
||
import numpy as np
|
||
import matplotlib.pyplot as plt
|
||
from optimizr import _core as opt
|
||
plt.rcParams['figure.figsize'] = (7, 4)
|
||
plt.rcParams['figure.dpi'] = 110
|
||
|
||
Optimal switching (Snell envelope)
|
||
----------------------------------
|
||
|
||
Two modes; only mode 1 pays a unit reward. Free switching should give `V_0(0) = N - 1` and `V_0(1) = N`.
|
||
|
||
.. code-block:: python
|
||
|
||
n_steps, n_modes = 5, 2
|
||
stage = np.zeros((n_steps, n_modes)); stage[:, 1] = 1.0
|
||
cost = [0.0] * (n_modes * n_modes)
|
||
res = opt.optimal_switching_dp(stage.flatten().tolist(),
|
||
[0.0] * n_modes, cost,
|
||
n_modes, n_steps)
|
||
value = np.array(res['value']).reshape(n_steps + 1, n_modes)
|
||
policy = np.array(res['policy']).reshape(n_steps + 1, n_modes)
|
||
print('V_0 =', value[0])
|
||
print('Optimal next mode at each (k, i):'); print(policy)
|
||
|
||
.. code-block:: python
|
||
|
||
fig, ax = plt.subplots()
|
||
ax.step(range(n_steps + 1), value[:, 0], where='post', label='V_k(mode 0)')
|
||
ax.step(range(n_steps + 1), value[:, 1], where='post', label='V_k(mode 1)')
|
||
ax.set_xlabel('k'); ax.set_ylabel('value'); ax.legend(); ax.grid(alpha=0.3)
|
||
ax.set_title('Snell envelope — free switching')
|
||
fig.tight_layout(); plt.show()
|
||
|
||
|
||
|
||
|
||
|
||
|
||
.. AUTO-PLOT-BEGIN
|
||
.. image:: ../_static/auto/algorithms__stochastic_control/block_03_fig_01.png
|
||
:align: center
|
||
:width: 80%
|
||
|
||
.. AUTO-PLOT-END
|
||
.. image:: ../_static/v2/stochastic_control/plot_01.png
|
||
:align: center
|
||
:width: 80%
|
||
|
||
Pontryagin 1-D LQR
|
||
------------------
|
||
|
||
Closed-form Riccati for $a=q=0$, $b=r=s_T=1$, $T=1$ is $P(t) = 1/(1 + (T - t))$, hence $P(0) = 0.5$.
|
||
|
||
.. code-block:: python
|
||
|
||
res = opt.pontryagin_lqr(a=0.0, b=1.0, q=0.0, r=1.0,
|
||
s_terminal=1.0, x0=1.0,
|
||
t_horizon=1.0, n_steps=2000)
|
||
tg = np.array(res['time_grid'])
|
||
P = np.array(res['riccati'])
|
||
x = np.array(res['state']); u = np.array(res['control'])
|
||
P_an = 1.0 / (1.0 + (1.0 - tg))
|
||
print('P(0) =', P[0], ' analytic =', P_an[0])
|
||
print('cost =', res['cost'])
|
||
|
||
.. code-block:: python
|
||
|
||
fig, axes = plt.subplots(1, 3, figsize=(13, 4))
|
||
axes[0].plot(tg, P, label='numeric'); axes[0].plot(tg, P_an, '--', label='analytic')
|
||
axes[0].set_title('Riccati P(t)'); axes[0].set_xlabel('t'); axes[0].legend(); axes[0].grid(alpha=0.3)
|
||
axes[1].plot(tg, x); axes[1].set_title('state x(t)'); axes[1].set_xlabel('t'); axes[1].grid(alpha=0.3)
|
||
axes[2].plot(tg[:-1], u); axes[2].set_title('feedback u(t) = -(b/r) P(t) x(t)'); axes[2].set_xlabel('t'); axes[2].grid(alpha=0.3)
|
||
fig.tight_layout(); plt.show()
|
||
|
||
|
||
|
||
|
||
|
||
|
||
.. AUTO-PLOT-BEGIN
|
||
.. image:: ../_static/auto/algorithms__stochastic_control/block_05_fig_01.png
|
||
:align: center
|
||
:width: 80%
|
||
|
||
.. AUTO-PLOT-END
|
||
.. image:: ../_static/v2/stochastic_control/plot_02.png
|
||
:align: center
|
||
:width: 80%
|
||
|
||
Two-sided intensity control
|
||
---------------------------
|
||
|
||
Affine premium $δ_±(λ) = α_± + κ_± λ$. First-order condition: $\lambda^*_\pm = \max(0, (α_\pm - ΔV_\pm) / (2 κ_\pm))$.
|
||
|
||
.. code-block:: python
|
||
|
||
deltas = np.linspace(-2.0, 2.0, 41)
|
||
lam_plus = []
|
||
for dv in deltas:
|
||
r = opt.two_sided_intensities(1.0, 1.0, 0.5, 0.5, dv, -dv)
|
||
lam_plus.append(r['lambda_plus'])
|
||
lam_plus = np.array(lam_plus)
|
||
fig, ax = plt.subplots()
|
||
ax.plot(deltas, lam_plus, lw=2)
|
||
ax.set_xlabel('ΔV_+'); ax.set_ylabel('λ*_+')
|
||
ax.set_title('Optimal upward intensity vs value-function gradient')
|
||
ax.grid(alpha=0.3); fig.tight_layout(); plt.show()
|
||
|
||
|
||
|
||
|
||
|
||
|
||
.. AUTO-PLOT-BEGIN
|
||
.. image:: ../_static/auto/algorithms__stochastic_control/block_06_fig_01.png
|
||
:align: center
|
||
:width: 80%
|
||
|
||
.. AUTO-PLOT-END
|
||
.. image:: ../_static/v2/stochastic_control/plot_03.png
|
||
:align: center
|
||
:width: 80%
|
||
|
||
**Verified:** switching `V_0` matches analytic recursion exactly; Pontryagin `P(0) = 0.4999` against analytic `0.5`.
|
||
|
||
API
|
||
---
|
||
|
||
.. code-block:: rust
|
||
|
||
pub fn solve_optimal_switching<R, T>(stage_reward: R, terminal_payoff: T, switching_cost: &[f64], cfg: &SwitchingConfig) -> Result<SwitchingResult>
|
||
where R: Fn(usize, usize) -> f64, T: Fn(usize) -> f64;
|
||
|
||
pub fn solve_pontryagin_lqr(cfg: &PontryaginConfig) -> Result<PontryaginResult>;
|
||
pub fn optimal_two_sided_intensities(cfg: &TwoSidedConfig, delta_v_plus: f64, delta_v_minus: f64) -> Result<TwoSidedResult>;
|