# Algorithms for separable convex optimization with linear ascending constraints

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## Abstract

The paper considers the minimization of a separable convex function subject to linear ascending constraints. The problem arises as the core optimization in several resource allocation scenarios, and is a special case of an optimization of a separable convex function over the bases of a polymatroid with a certain structure. The paper generalizes a prior algorithm to a wider class of separable convex objective functions that need not be smooth or strictly convex. The paper also summarizes the state-of-the-art algorithms that solve this optimization problem. When the objective function is a so-called \(d{\text {-}}\)separable function, a simpler linear time algorithm solves the problem.

## Keywords

Convex programming OR in telecommunications ascending constraints linear constraints polymatroid## Notes

### Acknowledgements

This work was supported by the Indo-French Centre for Promotion of Advanced Research under Project Number 5100-IT1.

## References

- 1.Wang Zizhuo 2014 On solving convex optimization problems with linear ascending constraints.
*Optimization Letters*Oct. 2014Google Scholar - 2.Clark A and Scarf H 1960 Optimal policies for a multi-echelon inventory problem.
*Mangement Science*6: 475–490CrossRefGoogle Scholar - 3.Padakandla A and Sundaresan R 2009 Power minimization for CDMA under colored noise.
*IEEE Transactions on Communications*, Oct. 2009Google Scholar - 4.Viswanath P and Anantharam V 2002 Optimal sequences for CDMA with colored noise: A Schur-saddle function property.
*IEEE Trans. Inf. Theory*IT-48: 1295–1318MathSciNetCrossRefzbMATHGoogle Scholar - 5.Palomar D, Lagunas M A and Cioffi J 2004 Optimum linear joint transmit-receive processing for MIMO channels.
*IEEE Transactions on Signal Processing*52(5): 1179–1197MathSciNetCrossRefzbMATHGoogle Scholar - 6.Sanguinetti L and D’Amico A 2012 Power allocation in two-hop amplify and forward mimo systems with QoS requirements.
*IEEE Transactions on Signal Processing*60(5): 2494–2507. OPTMathSciNetCrossRefzbMATHGoogle Scholar - 7.Hanson D, Brunk H, Franck W and Hogg R 1966 Maximum likelihood estimation of the distributions of two stochastically ordered random variables.
*Journal of the American Statistical Association*16: 1067–1080MathSciNetzbMATHGoogle Scholar - 8.Patriksson Michael 2008 A survey on the continuous nonlinear resource allocation problem.
*European Journal of Operational Research*185: 17–42MathSciNetGoogle Scholar - 9.Morton G, von Randow R and Ringwald K (1985) A greedy algorithm for solving a class of convex programming problems and its connection with polymatroid theory.
*Mathematical Programming*32: 238–241MathSciNetCrossRefzbMATHGoogle Scholar - 10.Federgruen A and Groenevelt H 1986 The greedy procedure for resource allocation problems: Necessary and sufficient conditions for optimality.
*Operations Research*34(6): 909–918MathSciNetCrossRefzbMATHGoogle Scholar - 11.Hochbaum D S 1994 Lower and upper bounds for the allocation problem and other nonlinear optimization problems.
*Mathematics of Operations Research*19(2): 390–409MathSciNetCrossRefzbMATHGoogle Scholar - 12.Groenevelt H 1991 Two algorithms for maximizing a separable concave function over a polymatroid feasible region.
*European Journal of Operational Research*54: 227–236CrossRefzbMATHGoogle Scholar - 13.Fujishige S 2003 Submodular functions and optimization, 2nd edition. Elsevier, New YorkGoogle Scholar
- 14.Padakandla A and Sundaresan R 2009 Separable convex optimization problems with linear ascending constraints.
*SIAM Journal on Optimization*20(3): 1185–1204MathSciNetCrossRefzbMATHGoogle Scholar - 15.Akhil P T, Singh R and Sundaresan R 2014 A polymatroid approach to separable convex optimization with linear ascending constraints. In:
*Proceedings of National Conference on Communications*, pages 1–5, Kanpur, India, Feb. 2014Google Scholar - 16.D’Amico Antonio A, Sanguinetti Luca and Palomar Daniel P 2014 Convex separable problems with linear constraints in signal processing and communications.
*IEEE Transactions on Signal Processing*62(22): 2014Google Scholar - 17.Vidal Thibaut, Jaillet Patrick and Maculan Nelson (2016) A decomposition algorithm for nested resource allocation problems.
*SIAM Journal on Optimization*26(22): 1322–1340MathSciNetzbMATHGoogle Scholar - 18.Fujishige S 1980 Lexicographically optimal base of a polymatroid with respect to a weight vector.
*Mathematics of Operations Research*5(2): 186–196MathSciNetCrossRefzbMATHGoogle Scholar - 19.Veinott Jr A F 1971 Least d-majorized network flows with inventory and statistical applications.
*Management Science*17: 547–567MathSciNetCrossRefzbMATHGoogle Scholar - 20.Muckstadt John A and Sapra Amar 2010
*Principles of Inventory Management*, 2nd edition. Springer, BerlinGoogle Scholar - 21.Moriguchi S and Shioura A 2004 On Hochbaum’s proximity-scaling algorithm for the general resource allocation problem.
*Mathematics of Operations Research*29: 394–397MathSciNetCrossRefzbMATHGoogle Scholar - 22.Gilrich E, Kovalev M and Zaporozhets A 1996 A polynomial algorithm for resource allocation problems with polymatroid constraints.
*Optimization*37: 73–86MathSciNetCrossRefzbMATHGoogle Scholar - 23.Gabow H N and Tarjan R E 1985 A linear-time algorithm for a special case of disjoint set union method.
*Journal of Computer and System Sciences*30: 209–221MathSciNetCrossRefzbMATHGoogle Scholar - 24.Moriguchi S, Shioura A and Tsuchimura N 2011 M-convex function minimization by continuous relaxation approach: Proximity theorem and algorithm.
*SIAM Journal on Optimization*21(3): 633–668MathSciNetCrossRefzbMATHGoogle Scholar - 25.Frederickson G N and Johnson D B 1982 The complexity of selection and ranking in x+y and matrices with sorted columns.
*Journal of Computer and System Sciences*24(2): 197–208. OPTMathSciNetCrossRefzbMATHGoogle Scholar