New PDF release: Single Facility Location Problems with Barriers

# New PDF release: Single Facility Location Problems with Barriers

By Kathrin Klamroth

This article develops the mathematical implications of boundaries to the geometrical and analytical features of constant place difficulties. The publication will entice these operating in operations learn and administration technology, and mathematicians attracted to optimization idea and its functions.

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Additional info for Single Facility Location Problems with Barriers

Sample text

N. 7)), this is equivalent to Fpi (t, p, p ) − d Fp (t, p, p ) + λ(t)Gpi (p) = 0, i = 1, . . , for a curve p(t) that satisﬁes the side constraint G(p(t)) = 0. 1. A d-shortest permitted path in Rn \ int(B) connecting two feasible points X and Y and given by the parameterization p consists of pieces of arcs where 1. 16), or 2. 18). 1 gives a necessary optimality condition for d-shortest permitted paths in Rn . Note that in the unconstrained case where no barriers are given in Rn , and under some convexity assumptions, suﬃcient optimality conditions for d-shortest paths could also be proven (see, for example, Clarke, 1983; Elsgolc, 1962).

Additionally, we assume that the boundary ∂(B) of B is given by an (n − 1)-dimensional smooth manifold in Rn , which for all X ∈ ∂(B) can be represented relative to an open neighborhood O ⊆ Rn of X as the set of solutions of GX (X) = 0, where GX : O → R is a twice continuously diﬀerentiable mapping with ∇GX (X) = 0. To avoid lengthy discussions involving diﬀerent representations of ∂(B) we assume furthermore that ∂(B) can be represented with respect to only one functional G : Rn → R satisfying the above requirements as ∂(B) = {X ∈ Rn : G(X) = 0}.

1. 1. An example in Ê2 where no l2 -shortest permitted X-Y path exists. The upper boundary of the barrier set B has an inﬁnite number of extreme points at (1/i, 1−1/i2 )T and (−1/i, 1−1/i2 )T , i ∈ {2, 4, 8, 16, . . 1. 18 2. Shortest Paths in the Presence of Barriers F. This is, for example, the case if a ﬁnite number of smooth or polyhedral barrier sets is given in Rn . 1) can be replaced by a minimum. The barrier distance dB is in general not positively homogeneous, which implies that there does not exist a norm inducing the metric dB .