@article{Vidal2014a,
abstract = {We propose an exact polynomial algorithm for a resource allocation problem with convex costs and constraints on partial sums of resource consumptions, in the presence of either continuous or integer variables. No assumption of strict convexity or differentiability is needed. The method solves a hierarchy of resource allocation subproblems, whose solutions are used to convert constraints on sums of resources into bounds for separate variables at higher levels. The resulting time complexity for the integer problem is {\$}O(n \backslashlog m \backslashlog (B/n)){\$}, and the complexity of obtaining an {\$}\backslashepsilon{\$}-approximate solution for the continuous case is {\$}O(n \backslashlog m \backslashlog (B/\backslashepsilon)){\$}, {\$}n{\$} being the number of variables, {\$}m{\$} the number of ascending constraints (such that {\$}m {\textless} n{\$}), {\$}\backslashepsilon{\$} a desired precision, and {\$}B{\$} the total resource. This algorithm attains the best-known complexity when {\$}m = n{\$}, and improves it when {\$}\backslashlog m = o(\backslashlog n){\$}. Extensive experimental analyses are conducted with four recent algorithms on various continuous problems issued from theory and practice. The proposed method achieves a higher performance than previous algorithms, addressing all problems with up to one million variables in less than one minute on a modern computer.},
author = {Vidal, T. and Jaillet, P. and Maculan, N.},
doi = {10.1137/140965119},
file = {:C$\backslash$:/Users/Thibaut/Documents/Mendeley-Articles/Vidal, Jaillet, Maculan/Vidal, Jaillet, Maculan - 2016 - A decomposition algorithm for nested resource allocation problems.pdf:pdf},
journal = {SIAM Journal on Optimization},
mendeley-groups = {Joao-Paper},
number = {2},
pages = {1322--1340},
title = {{A decomposition algorithm for nested resource allocation problems}},
url = {https://arxiv.org/pdf/1404.6694.pdf},
volume = {26},
year = {2016}
}

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The resulting time complexity for the integer problem is $}O(n \\log m \\log (B/n)){$, and the complexity of obtaining an $}\\epsilon{$-approximate solution for the continuous case is $}O(n \\log m \\log (B/\\epsilon)){$, $}n{$ being the number of variables, $}m{$ the number of ascending constraints (such that $}m {\\textless} n{$), $}\\epsilon{$ a desired precision, and $}B{$ the total resource. This algorithm attains the best-known complexity when $}m = n{$, and improves it when $}\\log m = o(\\log n){$. Extensive experimental analyses are conducted with four recent algorithms on various continuous problems issued from theory and practice. 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No assumption of strict convexity or differentiability is needed. The method solves a hierarchy of resource allocation subproblems, whose solutions are used to convert constraints on sums of resources into bounds for separate variables at higher levels. The resulting time complexity for the integer problem is {\\$}O(n \\backslashlog m \\backslashlog (B/n)){\\$}, and the complexity of obtaining an {\\$}\\backslashepsilon{\\$}-approximate solution for the continuous case is {\\$}O(n \\backslashlog m \\backslashlog (B/\\backslashepsilon)){\\$}, {\\$}n{\\$} being the number of variables, {\\$}m{\\$} the number of ascending constraints (such that {\\$}m {\\textless} n{\\$}), {\\$}\\backslashepsilon{\\$} a desired precision, and {\\$}B{\\$} the total resource. This algorithm attains the best-known complexity when {\\$}m = n{\\$}, and improves it when {\\$}\\backslashlog m = o(\\backslashlog n){\\$}. Extensive experimental analyses are conducted with four recent algorithms on various continuous problems issued from theory and practice. 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