Convex Discrete Optimization

Abstract

We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension. We discuss some of the many applications of this theory including to quadratic programming, matroids, bin packing and cutting-stock problems, vector partitioning and clustering, multiway transportation problems, and privacy and confidential statistical data disclosure. Highlights of our work include a strongly polynomial time algorithm for convex and linear combinatorial optimization over any family presented by a membership oracle when the underlying polytope has few edge-directions; a new theory of so-termed n-fold integer programming, yielding polynomial time solution of important and natural classes of convex and linear integer programming problems in variable dimension; and a complete complexity classification of high dimensional transportation problems, with practical applications to fundamental problems in privacy and confidential statistical data disclosure.

Graver basis and proximity techniques for block-structured separable convex integer minimization problems

“…We now briefly explain the Graver basis techniques; we refer the reader to the survey paper [18] or the monograph [17] for more details. Let E ∈ Z d×n be a matrix.…”

On the Complexity of Nonlinear Mixed-Integer Optimization

“…For an excellent recent survey focusing on other aspects of the complexity of nonlinear optimization, including the performance of oracle-based models and combinatorial settings such as nonlinear network flows, we refer to Hochbaum [34]. We also do not cover the recent developments by Onn et al [11][12][13][21][22][23]32,44,45] in the context of discrete convex optimization, for which we refer to the monograph [53]. Other excellent sources are [16] and [55].…”

Theory and Applications of n-Fold Integer Programming

“…We conclude in Section 4 with a discussion of the universality of n-fold integer programming and of a new (di)-graph invariant, about which very little is known, that is important in understanding the complexity of our algorithms. Further discussion of n-fold integer programming within the broader context of nonlinear discrete optimization can be found in [21] and [22].…”