Nonlinear Matroid Optimization and Experimental Design

TL;DR

This paper introduces polynomial-time algorithms for nonlinear optimization over matroids, with applications to experimental design, balancing multi-criteria objectives efficiently.

Contribution

It provides the first combinatorial and algebraic algorithms for nonlinear matroid optimization, expanding computational tools for complex optimization problems.

Findings

Polynomial-time algorithm for oracle-presented matroids

Algebraic algorithm for vectorial matroids

Application to experimental design and model-fitting

Abstract

We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection, and an algebraic algorithm for vectorial matroids. Our work is partly motivated by applications to minimum-aberration model-fitting in experimental design in statistics, which we discuss and demonstrate in detail.