A generalization of Polyak's convergence result for subgradient optimization

Allen, Ellen P.; Helgason, Richard V.; Kennington, Jeffery L.; Shetty, Bala

doi:10.1007/bf02591740

Cited by 48 publications

(32 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Under certain conditions on the step sizes, the convergence of the subgradient algorithm is well known [1,14,21]. In practice, however, it may be necessary to terminate the subgradient algorithm before optimality.…”

Section: Branch and Boundmentioning

confidence: 99%

See 1 more Smart Citation

Coordinated replenishments from multiple suppliers with price discounts

Syam¹,

Shetty²

1998

Naval Research Logistics

Self Cite

View full text Add to dashboard Cite

Section: Branch and Boundmentioning

confidence: 99%

“…A convergence result for such a step size is given in [1,21]. The updated multipliers at iteration ic / 1 are:…”

Section: Lagrangean Dualmentioning

confidence: 99%

Coordinated replenishments from multiple suppliers with price discounts

Syam¹,

Shetty²

1998

Naval Research Logistics

Self Cite

View full text Add to dashboard Cite

“…In order to simplify the notation we let a denote the vector corresponding to the coefficients in (5); x denote the vector corresponding to the binary decision variables; c denote the vector of costs, and T = {x: (2), (3), and (4)). Then the singly constrained assignment problem can be stated as P1 = min {cx : x E T and ax < b}, and it is well-known that /91 is NP-complete.…”

Section: (Ij)eementioning

confidence: 99%

The singly constrained assignment problem: A Lagrangian relaxation heuristic algorithm

Kennington

Mohammadi

1994

Comput Optim Applic

View full text Add to dashboard Cite

Abstract. This manuscript presents a new heuristic algorithm to find near optimal integer solutions for the singly constrained assignment problem. The method is based on Lagrangian duality theory and involves solving a series of pure assignment problems. The software implementation of this heuristic, ASSIGN+ 1, successfully solved problems having one-half million binary variables (assignment arcs) in tess than 17 minutes of wall clock time on a Sequent Symmetry $81 using a single processor. In computational comparisons with MPSX and OSL on an IBM 3081D, the specialized software was from 100 to 1,000 times faster. In computational comparisons with the specialized code of Mazzola and Neebe, we found that ASSIGN+I was 40 times faster. In computational comparisons with our best alternating path specialized code, we found that ASSIGN+ 1 was more than three times faster than that code. This new software proved to be very robust as well as fast. The robustness is due to an elaborate scheme used to update the Lagrangean multipliers and the speed is due to the fine code used to solve the pure assignment problems. We also present a modification of the algorithm for the case in which the number of jobs exceeds the number of men along with an empirical analysis of the modified software.

show abstract

“…Johnson et al [21] report that exact HK bounds have been computed by a special purposed linear programming code, for instances as large as 33810 cities. For even large scale instances, it is applied the subgradient method proposed on the original Held and Karp papers and speeded up by a number of algorithmic tricks [2,16,34,37,38]. Since for large scale instances the optimal solution is not known, the comparison of the heuristic and HK bounds is common practice.…”

Section: Introductionmentioning

confidence: 99%

Using logical surrogate information in Lagrangean relaxation: An application to symmetric traveling salesman problems

Lorena

Narciso

2002

European Journal of Operational Research

View full text Add to dashboard Cite

. AbstractThe Traveling Salesman Problem (TSP) is a classical Combinatorial Optimization problem, which has been intensively studied. The Lagrangean relaxation was first applied to the TSP in 1970. The Lagrangean relaxation limit approximates what is known today as HK (Held and Karp) bound, a very good bound (less than 1% from optimal) for a large class of symmetric instances. It became a reference bound for new heuristics, mainly for the very large scale instances, where the use of exact methods is prohibitive. A known problem for the Lagrangean relaxation application is the definition of a convenient step size control in subgradient like methods. Even preserving theoretical convergence properties, a wrong defined control can affect the performance and increase computational times. We show in this work how to accelerate a classical subgradient method while conserving good approximations to the HK bounds. The surrogate and Lagrangean relaxation are combined using the local information of the relaxed constraints. It results in a one-dimensional search that corrects the possibly wrong step size and is independent of the used step size control. Comparing with the ordinary subgradient method, and beginning with the same initial multiplier, the computational times are almost twice as fast for medium instances and greatly improved for some large scale TSPLIB instances.

show abstract

A generalization of Polyak's convergence result for subgradient optimization

Cited by 48 publications

References 13 publications

Coordinated replenishments from multiple suppliers with price discounts

Coordinated replenishments from multiple suppliers with price discounts

The singly constrained assignment problem: A Lagrangian relaxation heuristic algorithm

Using logical surrogate information in Lagrangean relaxation: An application to symmetric traveling salesman problems

Contact Info

Product

Resources

About