Results 1  10
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19
Most tensor problems are NP hard
 CORR
, 2009
"... The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has ..."
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The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has attracted a lot of attention recently. We examine here the computational tractability of some core problems in numerical multilinear algebra. We show that tensor analogues of several standard problems that are readily computable in the matrix (i.e. 2tensor) case are NP hard. Our list here includes: determining the feasibility of a system of bilinear equations, determining an eigenvalue, a singular value, or the spectral norm of a 3tensor, determining a best rank1 approximation to a 3tensor, determining the rank of a 3tensor over R or C. Hence making tensor computations feasible is likely to be a challenge.
Nonconvex mixedinteger nonlinear programming: A survey
 Surveys in Operations Research and Management Science
, 2012
"... A wide range of problems arising in practical applications can be formulated as MixedInteger Nonlinear Programs (MINLPs). For the case in which the objective and constraint functions are convex, some quite effective exact and heuristic algorithms are available. When nonconvexities are present, how ..."
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Cited by 21 (0 self)
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A wide range of problems arising in practical applications can be formulated as MixedInteger Nonlinear Programs (MINLPs). For the case in which the objective and constraint functions are convex, some quite effective exact and heuristic algorithms are available. When nonconvexities are present, however, things become much more difficult, since then even the continuous relaxation is a global optimisation problem. We survey the literature on nonconvex MINLP, discussing applications, algorithms and software. Special attention is paid to the case in which the objective and constraint functions are quadratic. Key Words: mixedinteger nonlinear programming, global optimisation, quadratic programming, polynomial optimisation.
Computer Algebra, Combinatorics, and Complexity: Hilbert’s Nullstellensatz and NPcomplete Problems
, 2008
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Recognizing Graph Theoretic Properties with Polynomial Ideals
, 2010
"... Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of ..."
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Cited by 6 (1 self)
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Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect kcolorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.
On the connection of the SheraliAdams Closure and Border
 Bases, 2009, Working Paper, Technische Universität Darmstadt / Massachusetts Institute of Technology
"... ABSTRACT. The SheraliAdams liftandproject hierarchy is a fundamental construct in integer programming, which provides successively tighter linear programming relaxations of the integer hull of a polytope. We initiate a new approach to understanding the SheraliAdams procedure by relating it to me ..."
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Cited by 6 (4 self)
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ABSTRACT. The SheraliAdams liftandproject hierarchy is a fundamental construct in integer programming, which provides successively tighter linear programming relaxations of the integer hull of a polytope. We initiate a new approach to understanding the SheraliAdams procedure by relating it to methods from computational algebraic geometry. Our two main results are the equivalence of the SheraliAdams procedure to the computation of a border basis, and a refinement of the SheraliAdams procedure that arises from this new connection. We present a modified version of the border basis algorithm to generate a hierarchy of linear programming relaxations that are tighter than those of Sherali and Adams, and over which one can still optimize in polynomial time (for a fixed number of rounds in the hierarchy). In contrast to the wellknown Gröbner bases approach to integer programming, our procedure does not create primal solutions, but constitutes a novel approach of using computeralgebraic methods to produce dual bounds. 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 5 (2 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.
Approximating amoebas and coamoebas by sums of squares
"... Amoebas and coamoebas are the logarithmic images of algebraic varieties and the images of algebraic varieties under the argmap, respectively. We present new techniques for computational problems on amoebas and coamoebas, thus establishing new connections between (co)amoebas, semialgebraic and con ..."
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Amoebas and coamoebas are the logarithmic images of algebraic varieties and the images of algebraic varieties under the argmap, respectively. We present new techniques for computational problems on amoebas and coamoebas, thus establishing new connections between (co)amoebas, semialgebraic and convex algebraic geometry and semidefinite programming. Our approach is based on formulating the membership problem in amoebas (respectively coamoebas) as a suitable real algebraic feasibility problem. Using the real Nullstellensatz, this allows to tackle the problem by sums of squares techniques and semidefinite programming. Our method yields polynomial identities as certificates of noncontainment of a point in an amoeba or coamoeba. As the main theoretical result, we establish some degree bounds on the polynomial certificates. Moreover, we provide some actual computations of amoebas based on the sums of squares approach.
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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Cited by 3 (3 self)
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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.