Simplexbased

Simplexbased is an umbrella term used to describe algorithms, systems, or approaches that are built around the geometric primitive known as a simplex. A simplex is the simplest convex polytope in a given dimension, such as a triangle in two dimensions or a tetrahedron in three. The term is not standardized in the literature and may appear informally in documentation or codebases to indicate reliance on simplex geometry as a core building block.

In optimization, simplex-based methods include both the classical simplex method for linear programming and derivative-free search techniques that operate on a simplex in the decision space. The linear programming simplex method traverses vertices of the feasible region via pivot operations, while revised and primal/dual variants optimize efficiency for large problems. In unconstrained optimization, the Nelder-Mead simplex method maintains and iteratively updates a simplex to search for a minimum, using operations such as reflection, expansion, contraction, and shrinkage.

Simplexbased approaches are common in fields where the objective function is expensive to evaluate, nondifferentiable, or noisy, making derivative-based methods less suitable. They are also used in geometric computation, sampling, and certain machine learning workflows where a simplex geometry provides a natural scaffold for exploration or approximation. Limitations include potential slow convergence in high dimensions and the absence of guarantees for global optimality in non-convex problems. Hybrid strategies that combine simplex steps with gradient information are sometimes employed to improve robustness and speed.

See also: simplex method, Nelder-Mead method, direct search methods, convex optimization.