Mathematical Optimization & Calibration Solvers

Implementation: packages/simulate/src/optimizer/solvers/

---

1. Limited-Memory BFGS with Bounds (L-BFGS-B)

Academic Citations

• Byrd, R. H., Lu, P., Nocedal, J., & Zhu, C. (1995). "A limited memory algorithm for bound constrained optimization."
  SIAM Journal on Scientific Computing, 16(5), pp. 1190–1208. DOI: 10.1137/0916069
• Zhu, C., Byrd, R. H., Lu, P., & Nocedal, J. (1997). "Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization."
  ACM Transactions on Mathematical Software, 23(4), pp. 550–560. DOI: 10.1145/279232.279236

ModelScript Rationale & Modifications

Solves large-scale bound-constrained non-linear minimization:

$$\underset{x\in {\mathbb{R}}^n}{min}f(x)\text{subject to }l\le x\le u$$

Crucial for parameter identification and direct collocation optimal control where thousands of discretized states and control bounds must be optimized.

• Modifications: Employs compact representation of the inverse Hessian approximation $H_k$ using history buffers $m\in [5,20]$ in typed arrays; Cauchy point computation with piecewise linear path search; exact line search with strong Wolfe conditions.

---

2. Covariance Matrix Adaptation Evolution Strategy (CMA-ES)

Academic Citations

• Hansen, N., & Ostermeier, A. (2001). "Completely derandomized self-adaptation in evolution strategies."
  Evolutionary Computation, 9(2), pp. 159–195. DOI: 10.1162/106365601750190398
• Hansen, N. (2016). "The CMA Evolution Strategy: A Tutorial."
  arXiv preprint: arXiv:1604.00772.

ModelScript Rationale & Modifications

Derivative-free black-box optimization on rugged, non-convex, non-separable objective functions (e.g. tuning PID controller gains over simulations with discontinuous switching events).

• Modifications: Tracks full covariance matrix $C\in {\mathbb{R}}^{n\times n}$ updates with rank-$\mu$ and rank-one update mechanisms; vectorized population sampling using WebAssembly-accelerated Cholesky decomposition; step-size control using cumulative path lengths (CSA).

---

3. Differential Evolution (DE)

Academic Citations

• Storn, R., & Price, K. (1997). "Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces."
  Journal of Global Optimization, 11(4), pp. 341–359. DOI: 10.1023/A:1008202821328

ModelScript Rationale & Modifications

Stochastic population-based global search algorithm utilizing vector differences for directional perturbations. Highly effective for parameter estimation against experimental CSV test bench data.

• Modifications: Supports rand/1/bin, best/1/bin, and current-to-best/1 mutation strategies with adaptive crossover probability ($CR$) and scaling factor ($F$).

---

4. Particle Swarm Optimization (PSO)

Academic Citations

• Kennedy, J., & Eberhart, R. (1995). "Particle swarm optimization."
  Proceedings of IEEE ICNN'95, Vol. 4, pp. 1942–1948. DOI: 10.1109/ICNN.1995.488968
• Shi, Y., & Eberhart, R. C. (1998). "A modified particle swarm optimizer."
  IEEE Congress on Evolutionary Computation, pp. 69–73.

ModelScript Rationale & Modifications

Simulates social swarm dynamics for continuous global search:

$$v_i(t+1)=w{v}_i(t)+c_1{r}_1(p_i-x_i)+c_2{r}_2(g-x_i)$$

• Modifications: Implements constriction factors and dynamically declining inertia weights $w(t)$ to transition smoothly from global exploration to local exploitation.

---

5. Non-Dominated Sorting Genetic Algorithm II (NSGA-II)

Academic Citations

• Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. (2002). "A fast and elitist multiobjective genetic algorithm: NSGA-II."
  IEEE Transactions on Evolutionary Computation, 6(2), pp. 182–197. DOI: 10.1109/4235.996017

ModelScript Rationale & Modifications

Solves multi-objective engineering trade-offs (e.g. simultaneously minimizing electric vehicle energy consumption while maximizing battery longevity and top speed).

• Modifications: Fast non-dominated sorting ($\mathcal{O}(M{N}^2)$ complexity) paired with crowding-distance assignment in continuous parameter spaces; outputs full Pareto fronts directly.

---

Upstream & Downstream Pipeline Connections

• Upstream Inputs:
  • Trajectory residuals evaluated by Simulation Solvers against CSV Datasets.
  • Collocation constraint formulations from msx optimize.
• Downstream Consumers:
  • Powers msx falsify adversarial requirement falsification across SysML v2 / Modelica / CAD.
  • Powers msx cosim optimize for black-box multi-FMU tuning across SSP boundaries.
  • Powers ModelicaCalibrator with hybrid-cmaes-lm (CMA-ES global exploration + Levenberg-Marquardt quadratic polishing).
  • Powers TradeStudyEngine.evolve for generative multi-objective Pareto design trade studies.

---

Programmatic TypeScript API

import { cmaesSolve, deSolve, psoSolve, nsga2Solve, type BlackBoxProblem } from "@modelscript/simulate/optimizer";

// 1. Single-Objective Black-Box Optimization (CMA-ES, DE, PSO)
const problem: BlackBoxProblem = {
  dimension: 2,
  bounds: {
    min: new Float64Array([-5, -5]),
    max: new Float64Array([5, 5]),
  },
  fitness: (x) => Math.pow(1 - x[0]!, 2) + 100 * Math.pow(x[1]! - x[0]! * x[0]!, 2),
};

const result = await cmaesSolve(problem, { maxGenerations: 100 });
console.log("Optimum:", result.bestSolution, "Cost:", result.bestFitness);

// 2. Hybrid Global-Local Model Calibration
import { ModelicaCalibrator } from "@modelscript/simulate/optimizer";

const calibrator = new ModelicaCalibrator(arena, sim, {
  parameters: ["k", "c"],
  parameterBounds: new Map([
    ["k", { min: 1, max: 30 }],
    ["c", { min: 0.1, max: 10 }],
  ]),
  measurements,
  method: "hybrid-cmaes-lm", // Global CMA-ES basin exploration + Levenberg-Marquardt refinement
});
const calResult = calibrator.calibrate();

---

CLI Command Examples

# 1. Adversarial requirement falsification with CMA-ES
msx falsify --formula "always[0,10] (stress <= 180)" --algorithm cma-es --population 20

# 2. Black-box co-simulation parameter tuning with Differential Evolution
msx cosim optimize ssp-archive.ssp --params "kp:0.1:10.0,kd:0.01:1.0" --objective "error_integral" --algorithm de