Semi-Infinite Optimization for Shape-Constrained Regression

Shape-constrained regression enhances traditional regression by incorporating prior knowledge through shape constraints like monotonicity and convexity. These constraints, often derived from physical laws, are beneficial in engineering fields where data is limited and noisy.

This thesis examines two optimization problems: shape-constrained parametric ridge regression and shape-constrained kernel ridge regression. By rigorously enforcing various shape constraints, these problems become convex semi-infinite optimization problems. To computationally tackle these problems, two adaptive discretization algorithms - the Core Algorithm and the Composite Algorithm - are developed. These efficiently compute approximate feasible solutions within finite iterations while controlling optimality errors. The research covers parametric regression with polynomial and posynomial models, and kernel methods using Gaussian kernels. Real-world manufacturing case studies demonstrate the practicality of these methods. This work advances the theory of shape-constrained regression and provides algorithms to compute interpretable predictive models in small data settings where shape knowledge is given.