New Open-Source MATLAB Toolkit Brings Derivative-Free Optimization to Experimental Design

A team of statisticians and software engineers has released a compact, open-source MATLAB toolkit designed to make one of the most stubborn tasks in modern statistics far easier to reproduce: optimizing experiments when the mathematics of the problem refuses to cooperate with traditional calculus. The package, called CSOMA, is described in a paper published in the journal SoftwareX and is built around a nature-inspired algorithm known as the competitive swarm optimizer with mutated agents, or CSO-MA. Rather than requiring derivatives, Hessians, or smoothness assumptions about the objective function being minimized, the method learns where to search by staging random head-to-head competitions among candidate solutions. The developers, led by Elvis Han Cui of the University of California, Los Angeles, together with Zizhao Zhang, Yuhan Evelyn Pan, and Weng Kee Wong, position the toolkit not as a new statistical theory but as a reusable, inspectable software layer that researchers can attach to their own model-specific criteria and rerun with confidence.

The motivation behind CSOMA stems from a recurring bottleneck in statistical modeling, data science, and experimental design. In many realistic applications, the objective function a researcher needs to minimize is nonlinear, nonconvex, nonsmooth, simulation-based, or available only through a numerical approximation. These awkward features appear constantly in model-based optimal experimental design, in likelihood-based estimation with complex dependence structures, in regression with distribution-valued responses, in design problems posed on curved manifolds, and in continuous reformulations of combinatorial tasks. When derivatives are unreliable or algebraically intractable, gradient-based solvers stumble, and researchers are often forced to write one-off scripts tied to a single example. CSOMA addresses this gap by packaging a derivative-free optimizer behind a deliberately narrow interface: users write a scalar objective function, specify lower and upper bounds, choose a swarm size and iteration budget, and call a single function named csoma().

The algorithmic heart of the toolkit is an evolutionary twist on particle-swarm thinking. Conventional particle swarm optimizers track personal best positions and a global best, updating every particle toward these remembered landmarks. CSO-MA takes a leaner approach: at each iteration, the swarm is randomly permuted and paired off into head-to-head competitions. In each pair, the particle with the lower objective value is declared the winner and stays put, while the loser is updated using a velocity rule that blends three influences. The first term carries over the loser’s previous velocity, preserving momentum. The second term moves the loser toward the winner of its own pairing, extracting local order information from objective evaluations without any gradient calculation. The third term pulls the loser toward the empirical center of the entire swarm, with a tunable parameter phi controlling the strength of this population-level attraction. If the swarm contains an odd number of particles, one simply sits out the round.

After the loser updates, CSOMA adds a distinctive step that gives the method its name: the mutated-agent mechanism. One updated loser is selected at random, one of its coordinates is chosen, and that coordinate is reset to the corresponding lower bound, upper bound, or interval midpoint. This deliberately simple mutation probes boundary regions that are common in bounded design parameterizations and helps the stochastic search recover from premature concentration, where all particles have clustered in one region of the search space. A clipping step then ensures every position remains inside the user-specified box, so feasibility is enforced by construction. The developers note that each pairwise competition extracts useful ordering information from objective evaluations alone, making the method a natural match for design criteria whose values can be computed through information matrices, determinants, likelihood approximations, or simulation summaries, but whose derivatives may be numerically unstable or impossible to write down cleanly.

The software architecture keeps model-specific code strictly outside the optimizer. The core function, src/csoma.m, uses only base MATLAB operations, requires no compilation, and depends on no external data. A setup helper exposes the package folders, while bundled examples demonstrate how statistical criteria can be encoded as function handles and optimized. The examples span an impressive range: finding locally D-optimal designs for logistic regression, constructing Bayesian designs for nonlinear mixed-effects models of HIV dynamics, optimizing Gaussian copula designs and maximum pseudo-likelihood estimates, fitting Wasserstein regressions for distribution-valued data, designing experiments on the surface of a sphere using Riemannian geometry, and even solving a ten-city traveling salesman problem by sorting continuous random keys. Heavier optional examples, such as the Bayesian HIV workflow, require additional MATLAB toolboxes, but the core optimizer and fast replication checks run on a standard laptop in seconds.

Reproducibility is where the package makes its most forceful statement. The repository includes a reviewer-facing script, replication/run_all.m, that executes the fast workflows with fixed random seeds, and a release-verification wrapper, replication/verify_round2.m, that records the full MATLAB version, operating-system details, installed products, and exact Git commit before running the automated suites. Numerical success is defined by explicit assertions rather than visual inspection of console output: a run passes only when every workflow satisfies finite-value, feasibility, normalization, and problem-specific criteria. The final retained verification was executed on September 18, 2026, using MATLAB R2025b Update 7 on GitHub-hosted Ubuntu 24.04, with the public v0.1.5 tag resolving to a specific forty-character commit and a clean worktree. Deterministic seeds make the stochastic workflow auditable, and the core smoke test requires two seeded optimizer calls to produce exactly equal results.

The flagship statistical application is illustrated by a two-factor logistic regression design problem on the square from minus one to one in each coordinate. Assuming a nominal parameter vector of (2, 3, minus 1), CSOMA searches continuously for optimal support points and their weights under the D-optimality criterion, which seeks designs that maximize the determinant of the Fisher information matrix. Crucially, the computed design can be checked against the classical equivalence theorem of Kiefer and Wolfowitz, which characterizes global optimality through a directional-derivative inequality. Under the centered convention used in the paper, the sensitivity function must be no greater than zero throughout the design space and must equal zero at the support points of an optimal design. Evaluating the sensitivity on a fine 201-by-201 grid, the authors found a maximum centered sensitivity of just 1.62 times ten to the minus six, essentially zero, confirming that the swarm-found design is D-optimal for practical purposes.

The Wasserstein regression example reveals the geometric depth the toolkit can reach. Distribution-valued responses, where each observation is an entire probability distribution rather than a single number, can be analyzed through the geometry of optimal transport. In one dimension, the squared Wasserstein distance between two distributions equals the integral of the squared difference of their quantile functions, and the bundled example exploits this by fitting a normal-quantile model and minimizing mean squared quantile error over an eighty-point probability grid. The paper visualizes this as linear motion in quantile coordinates, with the monotone transport map and its associated displacement geodesics providing an elegant interpretation of the loss being minimized. Meanwhile, the Gaussian copula examples show that the same optimizer layer works both upstream, choosing informative experimental settings by maximizing information, and downstream, estimating dependence parameters when a closed-form optimizer is inconvenient.

The developers are candid about limitations. As a stochastic metaheuristic, CSOMA guarantees no recovery of a global optimum in a finite number of iterations, and performance depends on swarm size, iteration budget, random seed, variable scaling, and the numerical stability of the user-supplied objective. The interface directly supports only box constraints; other constraints must be handled through transformations or penalties, and objective functions must return finite scalar values, with infeasible settings managed through penalties or large finite values. The team recommends that users examine stability across multiple seeds and increase the swarm size or iteration budget when results remain unstable. When reliable derivatives are available, mature gradient-based solvers may still be preferable, and the authors explicitly frame CSOMA as a narrow, inspectable tool rather than a replacement for comprehensive optimization suites.

Nevertheless, the potential impact on day-to-day research practice is considerable. By turning a method that previously existed mainly in methodological papers into an accessible, MIT-licensed tool distributed on GitHub and the MATLAB File Exchange, the team hopes to lower the cost of reproducible derivative-free optimization across statistics and data science. The immediate beneficiaries are experimental designers working with nonlinear models, integer run sizes, and simulation-based utilities, where CSO-MA has already demonstrated practical efficiency gains, including applications in developmental toxicology using sea urchin embryos. But the broader pattern matters more: any researcher who can formulate a scalar loss can now attach it to a shared, versioned optimizer, rerun published examples, compare alternative criteria, and hand colleagues a workflow they can inspect and execute. In a field where irreproducible optimization scripts have long been the quiet weak point of published analyses, that kind of transparency may prove to be the most valuable result of all.

Subject of Research: A reusable MATLAB toolkit for derivative-free optimization in optimal experimental design

Article Title: CSOMA : A reusable and reproducible MATLAB toolkit for derivative-free optimization in optimal experimental design

Article References: Cui, E. H., Zhang, Z., Pan, Y. E., & Wong, W. K. (2026). CSOMA: A reusable and reproducible MATLAB toolkit for derivative-free optimization in optimal experimental design. SoftwareX, 36, Article 103121. https://doi.org/10.1016/j.softx.2026.103121

Image Credits: AI Generated

DOI: 10.1016/j.softx.2026.103121

Keywords: CSOMA, derivative-free optimization, optimal experimental design, competitive swarm optimizer, MATLAB, reproducibility, D-optimal designs, metaheuristics, Wasserstein regression, copula models, statistical computing, open-source software