Mathematical Foundations

This page collects the core equations driving OptimizR’s Rust kernels, plus short intuition blurbs and micro-checks you can run in a notebook. For visuals and full walkthroughs, see the example notebooks in examples/notebooks/.

Differential Evolution (DE)

We minimize f: \mathbb{R}^d \to \mathbb{R} with a population \{\mathbf{x}_{i,g}\}_{i=1}^N.

Mutation (rand/1):


\mathbf{v}_{i,g} = \mathbf{x}_{r_1,g} + F \cdot (\mathbf{x}_{r_2,g} - \mathbf{x}_{r_3,g}),\quad r_1 \neq r_2 \neq r_3 \neq i.

Intuition: The differential term is a directional finite-difference estimate of the gradient; scaling F sets the step length. Population diversity controls exploration.

Crossover (binomial):


u_{i,j,g} = \begin{cases}
v_{i,j,g} & \text{if } \mathrm{Uniform}(0,1) < CR \text{ or } j = j_{\mathrm{rand}},\\
x_{i,j,g} & \text{otherwise.}
\end{cases}

Selection (greedy):


\mathbf{x}_{i,g+1} = \begin{cases}
\mathbf{u}_{i,g} & \text{if } f(\mathbf{u}_{i,g}) \le f(\mathbf{x}_{i,g}),\\
\mathbf{x}_{i,g} & \text{otherwise.}
\end{cases}

Self-adaptive jDE (used by OptimizR):


F_i^{g+1} = \begin{cases}
F_{\min} + r_1 \cdot F_{\max} & r_2 < \tau_1,\\
F_i^{g} & \text{otherwise,}
\end{cases}
\qquad
CR_i^{g+1} = \begin{cases}
\mathrm{Uniform}(0,1) & r_3 < \tau_2,\\
CR_i^{g} & \text{otherwise.}
\end{cases}

Typical \tau_1, \tau_2 = 0.1. This adaptation reduces manual tuning and improves robustness on multimodal landscapes.

Notebook check: In 05_performance_benchmarks.ipynb, plot F_i and CR_i histograms every 50 generations to verify adaptation is active (expect spread around 0.5–0.9 for CR and 0.5–0.9 for F on hard landscapes).

Optimal Control (HJB)

For dynamics dX_t = b(X_t, u_t)\,dt + \sigma(X_t,u_t)\,dW_t with running cost \ell and terminal cost g, the value function satisfies the Hamilton–Jacobi–Bellman PDE:


-\partial_t V(t,x) = \inf_{u\in\mathcal{U}} \Big[ \ell(x,u) + \nabla_x V(t,x)^{\top} b(x,u) + \tfrac12 \operatorname{Tr}\big(\sigma\sigma^{\top}(x,u) \, \nabla_x^2 V(t,x)\big) \Big],\quad V(T,x) = g(x).

OptimizR uses finite differences with backward time-stepping and optional policy iteration. On a uniform grid (t_n, x_j):


V^{n} = \min_{u}\Big\{ \ell(x_j,u)\,\Delta t + V^{n+1} + \nabla_x V^{n+1}\cdot b\,\Delta t + \tfrac12 \operatorname{Tr}(\sigma\sigma^{\top}\nabla_x^2 V^{n+1})\,\Delta t \Big\}.

The control that attains the minimum yields the feedback policy u^{\star}(x_j, t_n) exported by compute_policy.

Interpretation: HJB is dynamic programming in continuous time; V encodes the optimal cost-to-go. The quadratic example in 03_optimal_control_tutorial.ipynb shows V becoming steeper where volatility is high or costs penalize deviation.

Mean Field Games (1D solver)

OptimizR’s MFG module solves the coupled system for value u and density m:


\begin{aligned}
-\partial_t u(t,x) - \nu\,\partial_{xx} u(t,x) + H\big(x,\partial_x u(t,x), m(t,x)\big) &= 0,\\
\partial_t m(t,x) - \nu\,\partial_{xx} m(t,x) - \operatorname{div}\big(m(t,x) \, \partial_p H(x,\partial_x u, m)\big) &= 0,\\
u(T,x) &= g(x), \qquad m(0,x) = m_0(x).
\end{aligned}

We use fixed-point iterations on the transport term with implicit diffusion (stable for \nu > 0) and normalize m after each step to preserve mass.

Practical tip: Monitor \|m^{k+1}-m^{k}\|_1 and \|u^{k+1}-u^{k}\|_\infty; both appear in the notebook to diagnose non-convergence.

Kalman Filtering

For linear-Gaussian state space models


\begin{aligned}
\mathbf{x}_{t} &= F\,\mathbf{x}_{t-1} + \mathbf{w}_{t}, && \mathbf{w}_t \sim \mathcal{N}(0, Q),\\
\mathbf{y}_{t} &= H\,\mathbf{x}_{t}   + \mathbf{v}_{t}, && \mathbf{v}_t \sim \mathcal{N}(0, R),
\end{aligned}

prediction and update follow:


\begin{aligned}
	ext{Predict: } & \hat{\mathbf{x}}^-_t = F \hat{\mathbf{x}}_{t-1}, && P^-_t = F P_{t-1} F^{\top} + Q,\\
	ext{Update: } & K_t = P^-_t H^{\top} (H P^-_t H^{\top} + R)^{-1},\\
& \hat{\mathbf{x}}_t = \hat{\mathbf{x}}^-_t + K_t(\mathbf{y}_t - H \hat{\mathbf{x}}^-_t),\\
& P_t = (I - K_t H) P^-_t.
\end{aligned}

These steps back the init_kalman_filter, kalman_predict, and kalman_update helpers.

MCMC (Metropolis–Hastings)

For target density \pi(x) and proposal q(x'\mid x):


\alpha(x \to x') = \min\Big(1, \frac{\pi(x')\, q(x \mid x')}{\pi(x)\, q(x' \mid x)}\Big).

OptimizR uses symmetric Gaussian proposals (so q cancels) by default, with optional bounds projection and burn-in.

Heuristic: Tune proposal std so acceptance is ~0.25–0.35 for moderate dimensions; see examples/notebooks/02_mcmc.ipynb for trace plots.

Hidden Markov Models (HMM)

We maximize the likelihood of observations \mathbf{y} under latent states \mathbf{z} using Baum–Welch (EM):


\mathcal{L}(\theta) = \sum_{t} \log \Big( \sum_{z_t} p(y_t \mid z_t, \theta) p(z_t \mid z_{t-1}, \theta) \Big).

Forward–backward computes posteriors, then M-step re-estimates transition and emission parameters; Viterbi gives the MAP state path.

Quality check: Plot log-likelihood per iteration; it should be non-decreasing. The HMM tutorial notebook includes a simple convergence plot and a confusion matrix for decoded states.

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