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Nonlinear integer programming
 DISC. OPTIM
, 2009
"... Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapt ..."
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Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations
Convex Integer Maximization via Graver Bases
, 2008
"... We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R d and w1x,..., wdx are linear forms on R n, max {c(w1x,..., wdx) : Ax = b, x ∈ N n}. This method works for arbitrary input data A, b, d, w1,..., wd, c. ..."
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Cited by 4 (2 self)
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We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R d and w1x,..., wdx are linear forms on R n, max {c(w1x,..., wdx) : Ax = b, x ∈ N n}. This method works for arbitrary input data A, b, d, w1,..., wd, c. Moreover, for fixed d and several important classes of programs in variable dimension, we prove that our algorithm runs in polynomial time. As a consequence, we obtain polynomial time algorithms for various types of multiway transportation problems, packing problems, and partitioning problems in variable dimension.
FINITE GRÖBNER BASES IN INFINITE DIMENSIONAL POLYNOMIAL RINGS AND APPLICATIONS
"... Abstract. We introduce the theory of monoidal Gröbner bases, a concept which generalizes the familiar notion in a polynomial ring and allows for a description of Gröbner bases of ideals that are stable under the action of a monoid. The main motivation for developing this theory is to prove finitenes ..."
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Cited by 4 (1 self)
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Abstract. We introduce the theory of monoidal Gröbner bases, a concept which generalizes the familiar notion in a polynomial ring and allows for a description of Gröbner bases of ideals that are stable under the action of a monoid. The main motivation for developing this theory is to prove finiteness results in commutative algebra and applications. A basic theorem of this type is that ideals in infinitely many indeterminates stable under the action of the symmetric group are finitely generated up to symmetry. Using this machinery, we give new streamlined proofs of some classical finiteness theorems in algebraic statistics as well as a proof of the independent set conjecture of Ho¸sten and the second author. 1.
Convex Integer Maximization via Graver Bases
, 2008
"... We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R d and w1x,..., wdx are linear forms on R n, max {c(w1x,..., wdx) : Ax = b, x ∈ N n}. This method works for arbitrary input data A, b, d, w1,..., wd, c. ..."
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Cited by 2 (2 self)
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We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R d and w1x,..., wdx are linear forms on R n, max {c(w1x,..., wdx) : Ax = b, x ∈ N n}. This method works for arbitrary input data A, b, d, w1,..., wd, c. Moreover, for fixed d and several important classes of programs in variable dimension, we prove that our algorithm runs in polynomial time. As a consequence, we obtain polynomial time algorithms for various types of multiway transportation problems, packing problems, and partitioning problems in variable dimension.
Convex Discrete Optimization
"... 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 variab ..."
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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 cuttingstock 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 edgedirections; a new theory of sotermed nfold 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.
A Lower Bound for the Graver Complexity of the Incidence Matrix of a Complete Bipartite Graph
, 2011
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LogLinear Models, Toric Varieties, and Their Markov Bases
"... My goal in this chapter of lecture notes is to introduce the class of loglinear models and study them from the algebraic perspective. There are at least three different ways to introduce loglinear models: in statistics they are also called discrete exponential families and in algebraic geometry the ..."
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My goal in this chapter of lecture notes is to introduce the class of loglinear models and study them from the algebraic perspective. There are at least three different ways to introduce loglinear models: in statistics they are also called discrete exponential families and in algebraic geometry they are known as toric varieties. Our goal is to introduce these models from all three perspective and show that they are equivalent. Then we take the approach suggested in the first chapter of lecture notes and study the ideals defining the loglinear model. They are known as toric ideals. Extensive further information about toric ideals can be found in [15]. In Section 3 of this chapter, we explain how the generators of the toric ideals are useful for performing conditional inference. In particular, these generating sets can be used to perform certain random walks to generate random samples from the hypergeometric distribution, and compute pvalues of various statistical tests. In this context, generating sets for toric ideals are known as Markov bases. This connection was first made in [5]. In Section