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 :doc:`whatis` 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. .. list-table:: Reference problems and surrogate outputs :header-rows: 1 :widths: 25 35 40 * - 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 :ref:`simploid overview `. More output variables do not require more domain factors. Use :doc:`quickstart` for the fitting API, :doc:`advanced/normalization` for units and transforms, and :doc:`advanced/model_selection` for model selection. For an original differentiable objective without reference target values, use :doc:`advanced/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. .. toctree:: :maxdepth: 1 :caption: Regularization paths applications/elastic_net applications/medical_imaging .. toctree:: :maxdepth: 1 :caption: Weighted objective trade-offs applications/facility_location applications/portfolio applications/smart_grids applications/communications applications/federated_learning .. toctree:: :maxdepth: 1 :caption: Planning and control applications/model_predictive_control applications/supply_chain