Planning primary electric power distribution involves solving an optimization problem using nonlinear components, which makes it difficult to obtain the optimum solution when the problem has dimensions that are found ...
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Planning primary electric power distribution involves solving an optimization problem using nonlinear components, which makes it difficult to obtain the optimum solution when the problem has dimensions that are found in reality, in terms of both the installation cost and the power loss cost. To tackle this problem, heuristic methods have been used, but even when sacrificing quality, finding the optimum solution still represents a computational challenge. In this paper, we study this problem using genetic algorithms. With the help of a coding scheme based on the dandelion code, these genetic algorithms allow larger instances of the problem to be solved. With the stated approach, we have solved instances of up to 40,000 consumer nodes when considering 20 substations;the total cost deviates 3.1% with respect to a lower bound that considers only the construction costs of the network.
For a vector space Fn over a field F, an (η, β)-dimension expander of degree d is a collection of d linear maps Γj : Fn → Fn such that for every subspace U of Fn of dimension at most ηn, the image of U under all ...
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ISBN:
(纸本)9783959770699
For a vector space Fn over a field F, an (η, β)-dimension expander of degree d is a collection of d linear maps Γj : Fn → Fn such that for every subspace U of Fn of dimension at most ηn, the image of U under all the maps, Σj=1dΓj(U), has dimension at least β dim(U). Over a finite field, a random collection of d = O(1) maps Γj offers excellent "lossless" expansion whp: β ≈ d for η ≥ ω(1/d). When it comes to a family of explicit constructions (for growing n), however, achieving even modest expansion factor β = 1 + ϵ with constant degree is a non-trivial *** present an explicit construction of dimension expanders over finite fields based on linearized polynomials and subspace designs, drawing inspiration from recent progress on list-decoding in the rank-metric. Our approach yields the following:• Lossless expansion over large fields; more precisely β ≥ (1 − ϵ)d and [EQUATION] with d = Oϵ(1), when |F| ≥ ω(n).• Optimal up to constant factors expansion over fields of arbitrarily small polynomial size; more precisely β ≥ ω(δd) and η ≥ ω(1/(δd)) with d = Oδ(1), when |F| ≥ nδ.Previously, an approach reducing to monotone expanders (a form of vertex expansion that is highly non-trivial to establish) gave (ω(1), 1 + ω(1))-dimension expanders of constant degree over all fields. An approach based on "rank condensing via subspace designs" led to dimension expanders with [EQUATION] over large fields. Ours is the first construction to achieve lossless dimension expansion, or even expansion proportional to the degree.
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