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REFORMULATIONS IN MATHEMATICAL PROGRAMMING: DEFINITIONS AND SYSTEMATICS
, 2008
"... A reformulation of a mathematical program is a formulation which shares some properties with, but is in some sense better than, the original program. Reformulations are important with respect to the choice and efficiency of the solution algorithms; furthermore, it is desirable that reformulations c ..."
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Cited by 17 (13 self)
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A reformulation of a mathematical program is a formulation which shares some properties with, but is in some sense better than, the original program. Reformulations are important with respect to the choice and efficiency of the solution algorithms; furthermore, it is desirable that reformulations can be carried out automatically. Reformulation techniques are very common in mathematical programming but interestingly they have never been studied under a common framework. This paper attempts to move some steps in this direction. We define a framework for storing and manipulating mathematical programming formulations, give several fundamental definitions categorizing reformulations in essentially four types (optreformulations, narrowings, relaxations and approximations). We establish some theoretical results and give reformulation examples for each type.
Hilbert’s Nullstellensatz and an Algorithm for Proving Combinatorial Infeasibility
"... Systems of polynomial equations over an algebraicallyclosed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution o ..."
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Cited by 10 (5 self)
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Systems of polynomial equations over an algebraicallyclosed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In this paper, we investigate an algorithm aimed at proving combinatorial infeasibility based on the observed low degree of Hilbert’s Nullstellensatz certificates for polynomial systems arising in combinatorics and on largescale linearalgebra computations over K. We report on experiments based on the problem of proving the non3colorability of graphs. We successfully solved graph problem instances having thousands of nodes and tens of thousands of edges.
Hilbert’s Nullstellensatz and an Algorithm for Proving Combinatorial Infeasibility
"... Systems of polynomial equations over an algebraicallyclosed field K can be used to easily model combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In ..."
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Cited by 2 (2 self)
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Systems of polynomial equations over an algebraicallyclosed field K can be used to easily model combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In this paper we investigate an algorithm aimed at proving combinatorial infeasibility based on the low degree of Hilbert’s Nullstellensatz certificates for polynomial systems arising in combinatorics and largescale linear algebra computations over K. We report on experiments based on the problem of proving the non3colorability of graphs. We successfully solved graph problem instances having thousands of nodes and tens of thousands of edges. 1
COMPUTATION WITH POLYNOMIAL EQUATIONS AND INEQUALITIES ARISING IN COMBINATORIAL OPTIMIZATION
"... The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create largescale linear algebra or semidefinite programming relaxations of many kin ..."
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Cited by 2 (1 self)
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The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create largescale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.
Computing Infeasibility Certificates for Combinatorial Problems through Hilbert’s Nullstellensatz
, 2009
"... Systems of polynomial equations with coefficients over a field K can be used to concisely model combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over the a ..."
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Systems of polynomial equations with coefficients over a field K can be used to concisely model combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over the algebraic closure of the field K. In this paper, we investigate an algorithm aimed at proving combinatorial infeasibility based on the observed low degree of Hilbert’s Nullstellensatz certificates for polynomial systems arising in combinatorics, and based on fast largescale linearalgebra computations over K. We also describe several mathematical ideas for optimizing our algorithm, such as using alternative forms of the Nullstellensatz for computation, adding carefully constructed polynomials to our system, branching and exploiting symmetry. We report on experiments based on the problem of proving the non3colorability of graphs. We successfully solved graph instances with almost two thousand nodes and tens of thousands of edges.
unknown title
"... Often in mathematics, a theoretical investigation leads to a system of polynomial equations. Generically, such systems are difficult to solve. In applications, however, the equations come equipped with additional structure that can be exploited. It is crucially important, therefore, to develop techn ..."
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Often in mathematics, a theoretical investigation leads to a system of polynomial equations. Generically, such systems are difficult to solve. In applications, however, the equations come equipped with additional structure that can be exploited. It is crucially important, therefore, to develop techniques for studying structured polynomial systems. I work on a wide range of such problems that arise from other areas of mathematics and the physical sciences. The intellectual merit of this research is twofold: on the one hand, I am advancing the theoretical understanding of fundamental mathematical objects; and on the other, I am developing algorithms for performing computations with them. I have collaborated with 13 researchers, many of whom are near the beginning of their careers. In several cases, I have taken a leadership role with these younger people. These interactions also unite groups in different fields towards common goals. Numerical algorithms from semidefinite programming have become useful in many applications. A guiding open problem is to remove the need for approximations in these methods, while preserving their efficiency. I am working on a solution to this problem, building on recent success. A specific application is the BessisMoussaVillani trace conjecture
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"... I work on a wide range of problems that arise from other areas of mathematics and the physical sciences. Currently, I am focused on using mathematical and computational tools to solve basic problems in theoretical neuroscience, and in this regard, I have begun collaborations with scientists at the R ..."
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I work on a wide range of problems that arise from other areas of mathematics and the physical sciences. Currently, I am focused on using mathematical and computational tools to solve basic problems in theoretical neuroscience, and in this regard, I have begun collaborations with scientists at the Redwood Center for Theoretical Neuroscience and mathematicians at U.C. Berkeley. I am also interested in theoretical questions involving semidefinite programming, optimization, and computational algebra. The following is a description of several interrelated lines of research in which I will actively participate in the coming years. The first three sections contain very brief discussions of topics related to theoretical neuroscience that I have only begun exploring in recent months. The final sections describe more theoretical studies that I have been investigating in recent years and therefore contain more detailed descriptions. 1. Sparse coding and compressed sensing Sparse coding refers to the process of representing a real vector input (such as an image) as a sparse linear combination of an overcomplete set of vectors (called a sparse basis). Here, overcomplete refers to the fact that there are many more vectors in the sparse basis
unknown title
"... Often in mathematics, a theoretical investigation leads to a system of polynomial equations. Generically, such systems are difficult to solve. In applications, however, the equations come equipped with additional structure that can be exploited. It is crucially important, therefore, to develop techn ..."
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Often in mathematics, a theoretical investigation leads to a system of polynomial equations. Generically, such systems are difficult to solve. In applications, however, the equations come equipped with additional structure that can be exploited. It is crucially important, therefore, to develop techniques for studying structured polynomial systems. Hillar proposes to work on a wide range of problems that arise from other areas of mathematics and from the physical sciences. The intellectual merit of this research is twofold: on the one hand, Hillar is advancing the theoretical understanding of fundamental mathematical objects; and on the other, he is developing algorithms for performing computations with them. Hillar has collaborated with 13 researchers, many of whom are near the beginning of their careers. In several cases, he has taken a leadership role with these younger people. These interactions broadly impact mathematics by uniting groups in different fields towards common goals as well as by preparing the next generation of mathematicians. Numerical algorithms from semidefinite programming have become useful in many applications. A guiding open problem is to remove the need for approximations in these methods, while preserving their efficiency. Hillar proposes to solve this problem, building
An Algebraic Exploration of Dominating Sets and Vizing’s Conjecture
"... Systems of polynomial equations are commonly used to model combinatorial problems such as independent set, graph coloring, Hamiltonian path, and others. We formulate the dominating set problem as a system of polynomial equations in two different ways: first, as a single, highdegree polynomial, and ..."
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Systems of polynomial equations are commonly used to model combinatorial problems such as independent set, graph coloring, Hamiltonian path, and others. We formulate the dominating set problem as a system of polynomial equations in two different ways: first, as a single, highdegree polynomial, and second as a collection of polynomials based on the complements of dominationcritical graphs. We then provide a sufficient criterion for demonstrating that a particular ideal representation is already the universal Gröbner bases of an ideal, and show that the second representation of the dominating set ideal in terms of dominationcritical graphs is the universal Gröbner basis for that ideal. We also present the first algebraic formulation of Vizing’s conjecture, and discuss the theoretical and computational ramifications to this conjecture when using either of the two dominating set representations described above. Keywords: dominating sets, Vizing’s conjecture, universal Gröbner bases 1