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Nonlinear Matroid Optimization and Experimental Design
Yael Berstein,Jon Lee,Hugo Maruri-Aguilar,Shmuel Onn,Eva Riccomagno,Robert Weismantel,Henry P. Wynn +6 more
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TLDR
This work provides a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection and an algebraic algorithm for vectorialMatroids.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.read more
Citations
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Book ChapterDOI
Nonlinear Integer Programming
TL;DR: This chapter is a study of a simple version of general nonlinear integer problems, where all constraints are still linear, and focuses on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure.
Book ChapterDOI
On the Complexity of Nonlinear Mixed-Integer Optimization
TL;DR: This is a survey on the computational complexity of nonlinear mixedinteger optimization, highlighting a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained fully polynomial time approximation schemes in fixed dimension, and to stronglyPolynomialtime algorithms for special cases.
Journal ArticleDOI
Parametric nonlinear discrete optimization over well-described sets and matroid intersections
TL;DR: This work addresses optimization of parametric nonlinear functions of the form f(Wx), where f is a nonlinear function, W is a d × n matrix, and feasible x are in some large finite set of integer points in \mathbb {R}^n}$$.
Journal ArticleDOI
A short history of algebraic statistics
TL;DR: In algebraic statistics, computational techniques from algebraic geometry become tools to address statistical problems as discussed by the authors, and this, in turn, may prompt research in algebraic geometrical geometry.
References
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Book
The Design and Analysis of Computer Algorithms
Alfred V. Aho,John E. Hopcroft +1 more
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
Book
Optimum Experimental Designs, with SAS
TL;DR: This book presents the theory and methods of optimum experimental design, making them available through the use of SAS programs, and stresses the importance of models in the analysis of data and introduces least squares fitting and simple optimum experimental designs.
Journal ArticleDOI
Minimum Aberration 2 k–p Designs
Arthur Fries,William G. Hunter +1 more
TL;DR: In this article, the concept of aberration is proposed as a way of selecting the best designs from those with maximum resolution, and algorithms are presented for constructing these minimum aberration designs.
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