Nonlinear Matroid Optimization and Experimental Design

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.

Minimizing Lipschitz-continuous strongly convex functions over integer points in polytopes

“…More precisely: Theorem 2 Let F = P ∩ Z n be presented by an oracle for solving quadratic minimization problems of the type min c T x + τ 2 x 2 with varying c ∈ Q n and τ ∈ Q + . Let x * denote an optimal solution for the optimization problem (1). There is no polynomial time algorithm that can produce for every F = P ∩ Z n and every convex function f…”

“…If n is constant, and if s designates a bound on the binary encoding length of the data of a subproblem (8), then we can determine an optimal solution for min{ (1) .…”

Minimal average degree aberration and the state polytope for experimental designs

“…That is to say, the set of lower vertices of the state polytope (corresponding to models in the algebraic fan) contains the solution to minimising any concave aberration function. This can be understood as minimisation over a matroid, which was studied further in Berstein et al (2008). A further development is to consider aberration A(w, S(w)) with respect to other distributions rather than the uniform.…”

A short history of algebraic statistics

“…This is a new concept in experimental designs for quantitative factors which is an alternative to aberration and is strongly related to corner cut models and state polytopes. Algorithmic and computational aspects are related to non-linear matroid optimization [11]. Another novel idea in design of experiment is to interpret an algebraic variety, not necessarily a zero-dimensional one, as a (repository of potential) designs of experiments [9,50].…”