Integer and combinatorial optimization
Integer and combinatorial optimization
Mathematical Programming: Series A and B
Surrogate gradient algorithm for Lagrangian relaxation
Journal of Optimization Theory and Applications - Special issue in honor of Yu-Chi Ho
Quasi-Newton Bundle-Type Methods for Nondifferentiable Convex Optimization
SIAM Journal on Optimization
Dynamic Programming
Convergent Lagrangian and domain cut method for nonlinear knapsack problems
Computational Optimization and Applications
Computers and Operations Research
Combining arc-consistency and dual lagrangean relaxation for filtering CSPs
CPAIOR'05 Proceedings of the Second international conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems
Maximising the net present value of large resource-constrained projects
CP'12 Proceedings of the 18th international conference on Principles and Practice of Constraint Programming
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Bundle methods have been used frequently to solve nonsmooth optimization problems. In these methods, subgradient directions from past iterations are accumulated in a bundle, and a trial direction is obtained by performing quadratic programming based on the information contained in the bundle. A line search is then performed along the trial direction, generating a serious step if the function value is improved by ε or a null step otherwise. Bundle methods have been used to maximize the nonsmooth dual function in Lagrangian relaxation for integer optimization problems, where the subgradients are obtained by minimizing the performance index of the relaxed problem. This paper improves bundle methods by making good use of near-minimum solutions that are obtained while solving the relaxed problem. The bundle information is thus enriched, leading to better search directions and less number of null steps. Furthermore, a simplified bundle method is developed, where a fuzzy rule is used to combine linearly directions from near-minimum solutions, replacing quadratic programming and line search. When the simplified bundle method is specialized to an important class of problems where the relaxed problem can be solved by using dynamic programming, fuzzy dynamic programming is developed to obtain efficiently near-optimal solutions and their weights for the linear combination. This method is then applied to job shop scheduling problems, leading to better performance than previously reported in the literature.