Branch and Bound Method for Solving Mixed Integer Linear Programming

The branch and bound method is employed to solve mixed integer linear programming problems, particularly applicable to power system unit commitment optimization. This algorithm addresses optimization challenges by systematically branching at each node, decomposing the original problem into smaller subproblems. It is widely utilized for combinatorial optimization problems such as the traveling salesman problem, knapsack problem, and unit commitment problems. In power system unit commitment applications, the branch and bound method enables finding optimal unit combination schemes through iterative tree traversal. The algorithm implementation typically involves: 1) Solving linear programming relaxations at each node using solvers like MATLAB's intlinprog or Python's scipy.optimize, 2) Implementing branching strategies based on integer variable fractional solutions, 3) Applying bounds pruning using incumbent solutions and objective function thresholds, and 4) Managing search tree traversal through depth-first or best-first approaches. Key functions in implementation include variable partitioning for integer constraints, constraint handling for operational limits, and objective function formulation for cost minimization. This approach enhances power system efficiency by reducing operational costs while maintaining grid reliability and stability through optimal unit scheduling and resource allocation.