Applications

This section explores the practical applications of Bézier simplex fitting. By exploiting the continuous relationships between optimal solutions and competing objectives, Bézier simplices provide a powerful framework for multi-objective optimization and parametric modeling.

The most prominent and elegant application of this technique is the continuous approximation of the multi-parameter regularization path in Elastic Net and other sparse modeling methods. We detail how a Bézier simplex can efficiently capture these paths, enabling comprehensive model selection without repeatedly re-training models over a discrete grid.

The other examples use small quadratic reference problems to explain the same workflow in different settings. The assumptions belong to the stated reference formulation: a realistic network, controller or inverse problem can introduce constraints, nonconvexity or discontinuities absent from the example. Strong convexity can give a unique weighted-sum optimizer, but a one-to-one simplex topology requires further conditions. See What is Bézier simplex fitting? for the precise unconstrained theorem.

What is fitted in each application?

First distinguish the optimization decision from the surrogate input. A reference solver takes priority weights and returns an optimal decision. The Bézier model learns that map, its objective values, or both. It does not learn directly from an untagged cloud of Pareto points: every output row needs its generating parameter row.

Table 22 Reference problems and surrogate outputs

Application

Input parameter

Decision-map output

Elastic Net

Loss and penalty weights on a triangle

Regression coefficients

Medical imaging

Logarithmic regularization coordinate

256 image pixels

Facility location

Weights for two, three or four centers

Two spatial coordinates

Portfolio

Two regularized objective weights

Three unconstrained allocation variables

Smart grids

Cost versus emissions weights

Three generator outputs

Communications

Two or three priority weights

Two allocation variables in the toy problem

Federated learning

Two task weights

Two model parameters in the toy problem

Model predictive control

Priority weights and independent initial state

Five future inputs

Supply chain

Inventory versus ordering-cost weights

Five future orders

The objective-map output instead contains one column per objective. Its low dimensionality makes it convenient to plot, but it does not contain enough information to recover decisions in general. Evaluate original objectives on predicted decisions when checking consistency between two independently fitted maps.

A common reproduction and validation workflow

  1. Define objectives, feasible decisions and parameter bounds. Record all fixed problem data such as demand, initial state and measurements.

  2. Solve at training parameters and retain convergence evidence. Numerical errors in these reference solutions become errors in the training data.

  3. Choose a simplex, product or assembled domain and fit aligned Samples. Inspect numerical success, observation rank and residuals.

  4. Obtain new reference responses at held-out parameters, including important boundaries. Compare each output and check feasibility of decisions.

  5. Change degree, sampling or penalties according to those errors. Retain the parameter transform and problem definition with the saved model.

Figure scripts are named on each application page and run from the repository root. Install the optional dependencies imported by the selected script, typically pytorch-bsf[plot,sklearn,sparse]. Reported numerical values describe those examples and their stated evaluation sets; small training error or a smooth picture alone does not establish generalization.

Choosing a domain and a next guide

Two priorities give a segment and three give a triangle. An independent operating variable requires another factor: two priorities plus initial state give a square, as in the MPC example. This geometric distinction is illustrated in the simploid overview. More output variables do not require more domain factors.

Use Quickstart for the fitting API, Output scaling for units and transforms, and Cross-validation and degree search for model selection. For an original differentiable objective without reference target values, use Objective minimization; that is a different operation from regression on solver-generated observations.

Application guides

Choose a group by the map you want to approximate. The regularization examples trace fitted solutions as penalty weights change. The trade-off examples map objective priorities to decisions, starting with geometric facility location. The planning and control examples use sequences of decisions, with MPC also introducing an independent initial-state input.

Regularization paths

Weighted objective trade-offs

Planning and control