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  • 1
    ISSN: 1573-2878
    Keywords: Convex polyhedral sets ; linear inequalities ; minimal representation ; prime representation ; redundancy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Consider a convex polyhedral set represented by a system of linear inequalities. A prime representation of the polyhedron is one that contains no redundant constraints. We present a sharp upper bound on the difference between the cardinalities of any two primes.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1436-4646
    Keywords: Random methods ; necessary constraints ; redundancy ; coordinate directions ; hypersphere directions ; linear inequalities
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A new approach is given for the analysis of random methods for detecting necessary constraints in systems of linear inequality constraints. This new approach directly accounts for the fact that two constraints are detected as necessary (hit) at each iteration of a random method. The significance of this two-hit analysis is demonstrated by comparing it with the usual one-hit analysis.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1436-4646
    Keywords: Hit and run methods ; redundancy ; linear inequalities ; coupon collector's problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The hit and run methods are probabilistic algorithms that can be used to detect necessary (nonredundant) constraints in systems of linear constraints. These methods construct random sequences of lines that pass through the feasible region. These lines intersect the boundary of the region at twohit-points, each identifying a necessary constraint. In order to study the statistical performance of such methods it is assumed that the probabilities of hitting particular constraints are the same for every iteration. An indication of the best case performance of these methods can be determined by minimizing, with respect to the hit probabilities, the expected value of the number of iterations required to detect all necessary constraints. We give a set of isolated strong local minimizers and prove that for two, three and four necessary constraints the set of local minimizers is the complete set of global minimizers. We conjecture that this is also the case for any number of necessary constraints. The results in this paper also apply to sampling problems (e.g., balls from an urn) and to the coupon collector's problem.
    Type of Medium: Electronic Resource
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