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Computing MultiHomogeneous Bézout Numbers is Hard
, 2004
"... The multihomogeneous Bézout number is a bound for the number of solutions of a system of multihomogeneous polynomial equations, in a suitable product of projective spaces. Given an arbitrary, not necessarily multihomogeneous system, one can ask for the optimal multihomogenization that would mini ..."
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The multihomogeneous Bézout number is a bound for the number of solutions of a system of multihomogeneous polynomial equations, in a suitable product of projective spaces. Given an arbitrary, not necessarily multihomogeneous system, one can ask for the optimal multihomogenization that would minimize the Bézout number. In this paper, it is proved that the problem of computing, or even estimating the optimal multihomogeneous Bézout number is actually NPhard. In terms of approximation theory for combinatorial optimization, the problem of computing the best multihomogeneous structure does not belong to APX, unless P = NP. Moreover, polynomial time algorithms for estimating the minimal multihomogeneous Bézout number up to a fixed factor cannot exist even in a randomized setting, unless BPP ⊇ NP. 1
OPTIMIZATION PROBLEM IN MULTIHOMOGENEOUS HOMOTOPY METHOD
"... Multihomogeneous homotopy continuation method is one of the most efficient approaches in solving all isolated solutions of a polynomial system of equations. Finding the optimal partition of variables with the minimal multihomogeneous Bézout number is clearly an optimization problem. Multihomogeneo ..."
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Multihomogeneous homotopy continuation method is one of the most efficient approaches in solving all isolated solutions of a polynomial system of equations. Finding the optimal partition of variables with the minimal multihomogeneous Bézout number is clearly an optimization problem. Multihomogeneous continuation using the optimal partition of variables reduces the computational cost in path tracking to the minimum. In this article, a heuristic method for finding the optimal variable partition is constructed based on backward greedy idea. Numerical examples show the efficiency of the method. Keywords: Multihomogeneous Bézout number; Homotopy continuation method; Partition of variables; Backward greedy 1
Computing the optimal partition of variables . . .
, 2004
"... partition of variables reduces the computational cost in curve following to the minimum. However, finding the optimal variable partition is likely an NP hard problem. An Consider a system of polynomial equations Applied Mathematics and Computation xxx (2004) xxx–xxx www.elsevier.com/locate/amc ARTI ..."
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partition of variables reduces the computational cost in curve following to the minimum. However, finding the optimal variable partition is likely an NP hard problem. An Consider a system of polynomial equations Applied Mathematics and Computation xxx (2004) xxx–xxx www.elsevier.com/locate/amc ARTICLE IN PRESS * Corresponding author.approximate algorithm is introduced in this paper to avoid exhaustive search in finding the (approximate) optimal variable partition. The global convergence of this algorithm is proved with Markov chain theory. Numerical comparisons with algorithms existed show the efficiency of the new method.
A Tabu Search Method for Finding Minimal MultiHomogeneous Bézout Number
"... Abstract: Problem statement: A homotopy method has proven to be reliable for computing all of the isolated solutions of a multivariate polynomial system. The multihomogeneous Bézout number of a polynomial system is the number of paths that one has to trace in order to compute all of its isolated so ..."
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Abstract: Problem statement: A homotopy method has proven to be reliable for computing all of the isolated solutions of a multivariate polynomial system. The multihomogeneous Bézout number of a polynomial system is the number of paths that one has to trace in order to compute all of its isolated solutions. Each partition of the variables corresponds to a multihomogeneous Bézout number. It is a crucial problem to find a partition with the minimum multihomogeneous Bézout number since the size of the space of all the partitions increases exponentially. Approach: This study presented a new method by producing the Tabu Search Method (TSM) as a powerful technique for finding minimum multihomogeneous Bézout number. Results: A comparison is made between the new method and some recent methods. It is shown that our algorithm is superior to the latter, besides being simple and efficient in the implementation. Conclusion: Furthermore the present study extended the applicability of the Tabu search method.