Surrogate "Level-Based" Lagrangian Relaxation for Mixed-Integer Linear Programming
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Mixed-Integer Linear Programming problems, playing a prominent role in Operations Research, are prone to the curse of dimensionality; specifically, the combinatorial complexity of the associated MILP problems increases exponentially with the increase of the problem size. To efficiently solve the "separable" MILP combinatorial problems, decomposition, and coordination Surrogate "Level-Based" Lagrangian Relaxation method is developed. The new method efficiently exploits the underlying geometric-convergence potential - the best convergence theoretically possible - inherent to Polyak's step-sizing formula in the dual space without the need to know the optimal dual value required for convergence. Unlike all the previous methods to obtain "level" estimates of the optimal dual value by adjusting hyperparameters, the key novel idea to obtain "level" value is through a hyper-parameter-free optimization: a novel "multiplier-convergence-feasibility" Linear Programming constraint satisfaction problem needs to be solved, from which, in conjunction with the Polyak's formula, the sought-for "level" estimate is inferred. Testing results for medium- and large-scale instances of standard Generalized Assignment Problems from OR-library demonstrate that 1. "level" estimates generally decrease and the dual values increase faster than those obtained by using other existing methods, 2. the new method is computationally efficient in terms of the quality of solutions (for several large-scale cases, optimal solutions are obtained) and in terms of the CPU time, 3. the new method is robust with respect to the choice of initial parameters such as step-sizes and multipliers. Moreover, the procedure to estimate the "level" value is independent of linearity and separability of the problem, and thus the new method also paves the way for efficient resolution of general Mixed-Integer Programming (MIP) problems.
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