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.

In this article we establish an exponential lower bound on the Graver complexity of integer programs. This provides new type of evidence supporting the presumable intractability of integer programming. Specifically, we show that the Graver complexity of the incidence matrix of the complete bipartite graph K 3, m satisfies g(m) = Ω (2 m ), with g(m) ≥ 17 · 2 m−3 − 7 for every m > 3.

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We study the problem of optimizing nonlinear objective functions over bipartite matchings. While the problem is generally intractable, we provide several efficient algorithms for it, including a deterministic algorithm for maximizing convex objectives, approximative algorithms for norm minimization and maximization, and a randomized algorithm for optimizing arbitrary objectives.

Corner cut, Design ideal, Factorial design, Latin hypercube sampling, Linear aberration, State polytope,

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