Lagrangian Relaxation Method for Unit Commitment Problem Solving

To address the unit commitment problem, we employ the Lagrangian Relaxation method which constructs a modified objective function by incorporating constraints through Lagrangian multipliers. This approach effectively handles complex constraints by relaxing them into the objective function, making the problem more computationally tractable. The implementation typically involves initializing multiplier values and iteratively updating them using subgradient methods until convergence criteria are met.

This methodology particularly accommodates the stochastic nature of domain search, which is crucial for navigating complex solution spaces. By systematically exploring different regions of the search domain through randomized initial conditions and neighborhood exploration strategies, we enhance the probability of discovering high-quality solutions. The algorithm can be implemented with adaptive step sizes and penalty parameters to balance exploration and exploitation phases.

Furthermore, we integrate heuristic search algorithms such as genetic algorithms or simulated annealing to refine solutions. These metaheuristics work synergistically with Lagrangian Relaxation by providing local search capabilities to improve solution quality. Key functions in implementation include constraint handling through dual variable updates, solution feasibility restoration, and objective function optimization using linear programming solvers for subproblems.