On Hochbaum's Proximity-Scaling Algorithm for the General Resource Allocation Problem

Moriguchi, Satoko; Shioura, Akiyoshi

doi:10.1287/moor.1030.0076

Cited by 20 publications

(21 citation statements)

References 3 publications

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“…We first write down the KKT conditions for (P3). We associate a dual variable λ k to each constraint (9), a dual variable δ i to each upper bound constraint, and a dual variable η i to each nonnegative constraint (10). The Lagrangian of (P3) can then be written as…”

Section: A Dual Methodsmentioning

confidence: 99%

“…Besides [13], another related work to this paper is Hochbaum [8] (followed by Moriguchi and Shioura [10] which corrected an error in [8]). In [8], the author proposes a greedy algorithm for solving a class of general allocation problem, including the problem studied in this paper as a special case (in [8], our model is called the nested resource allocation model).…”

Section: Literature Reviewmentioning

confidence: 99%

“…In the numerical experiments, we assume that o i ∼ U [5,10], u i ∼ U [20, 25] and α i ∼ U [0, 20]. We also assume that each D i follows an exponential distribution with parameter η i ∼ U [0.1, 0.2].…”

Section: Numerical Experimentsmentioning

confidence: 99%

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On solving convex optimization problems with linear ascending constraints

Wang

2014

Optim Lett

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In this paper, we propose two algorithms for solving convex optimization problems with linear ascending constraints. When the objective function is separable, we propose a dual method which terminates in a finite number of iterations. In particular, the worst case complexity of our dual method improves over the best-known result for this problem in Padakandla and Sundaresan (SIAM J Optim 20 (3): [1185][1186][1187][1188][1189][1190][1191][1192][1193][1194][1195][1196][1197][1198][1199][1200][1201][1202][1203][1204] 2009). We then propose a gradient projection method to solve a more general class of problems in which the objective function is not necessarily separable. Numerical experiments show that both our algorithms work well in test problems.

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Section: A Dual Methodsmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

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On solving convex optimization problems with linear ascending constraints

Wang

2014

Optim Lett

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“…Consider now a scaled form of the greedy defined on the problem scaled on units of length s. Let the solution delivered by greedy(s) be denoted by x s . The procedure here benefits from a correction of an error in step 2 of the procedure in Hochbaum (1994) that was noted and proved by Moriguchi and Shioura (2004). (The error was to set the value of δ i to be equal to 1 if increment of 1 is feasible but increment of s is infeasible.…”

Section: Proximity-scaling For the General Allocation Problemmentioning

confidence: 99%

Complexity and algorithms for nonlinear optimization problems

Hochbaum

2007

Ann Oper Res

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Nonlinear optimization algorithms are rarely discussed from a complexity point of view. Even the concept of solving nonlinear problems on digital computers is not well defined. The focus here is on a complexity approach for designing and analyzing algorithms for nonlinear optimization problems providing optimal solutions with prespecified accuracy in the solution space. We delineate the complexity status of convex problems over network constraints, dual of flow constraints, dual of multi-commodity, constraints defined by a submodular rank function (a generalized allocation problem), tree networks, diagonal dominant matrices, and nonlinear knapsack problem's constraint. All these problems, except for the latter in integers, have polynomial time algorithms which may be viewed within a unifying framework of a proximity-scaling technique or a threshold technique. The complexity of many of these algorithms is furthermore best possible in that it matches lower bounds on the complexity of the respective problems.In general nonseparable optimization problems are shown to be considerably more difficult than separable problems. We compare the complexity of continuous versus discrete nonlinear problems and list some major open problems in the area of nonlinear optimization.

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“…The proximity property can be exploited in developing an efficient scaling algorithm for minimizing $f$ . In fact, the $\mathrm{L}$ -convex function minimization problem can be solved in polynomial-time by combining submodular set function minimization algorithms and the proximity property [12] [8,9,17] in developing efficient algorithms for resource allocation problems. Different types of theorems on proximity have also been investigated: proximity between integral and real optimal solutions in [1,2,7,9,10] and proximity for anumber of resource allocation problems with min-max tyPe objective functions in [5].…”

Section: Introductionmentioning

confidence: 99%

Proximity theorems of discrete convex functions

Murota

Tamura

2004

Mathematical Programming

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Aproximity theorem is astatement that, given an optimization problem and its relaxation, an optimal solution to the original problem exists in acertain neighborhood of asolution to the relaxation. -convexity(in their variants). $\mathrm{L}_{2}$ -convex functions and $\mathrm{M}_{2^{-}}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{x}$ functions constitute larger classes of discrete convex functions that are relevant to the polymatroid intersection problem, where an $\mathrm{L}_{2}$ -convex function is, by definition, the infimal convolution of two $\mathrm{L}$ -convex functions and an $\mathrm{M}_{2}$ -convex function is the sum of two M-convex functions. The $\mathrm{M}_{2}$ -convex function minimization problem is equivalent to the M-convex submodular flow problem [20] which is an extension of the submodular flow problem [3].

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On Hochbaum's Proximity-Scaling Algorithm for the General Resource Allocation Problem

Cited by 20 publications

References 3 publications

On solving convex optimization problems with linear ascending constraints

On solving convex optimization problems with linear ascending constraints

Complexity and algorithms for nonlinear optimization problems

Proximity theorems of discrete convex functions

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