Branch and Bound Method for Integer Programming

The Branch and Bound method for integer programming is a classical algorithm for solving discrete optimization problems, particularly suitable for scenarios where variables must be integers, such as production scheduling and resource allocation. This method progressively approaches the optimal solution by continuously partitioning the feasible solution space (branching) and calculating upper and lower bounds (bounding), thereby avoiding exhaustive enumeration of all possibilities and improving computational efficiency.

In MATLAB, the Branch and Bound method can be implemented using the built-in Optimization Toolbox, such as the `intlinprog` function specifically designed for mixed-integer linear programming. This function internally integrates Branch and Bound logic. Users need to define the objective function coefficients, constraint matrices, and variable types (e.g., which variables are integers). The algorithm automatically handles branching strategies, such as selecting fractional variables for partitioning, and prunes obviously non-optimal subproblems.

For custom problems, users can manually implement Branch and Bound using recursion or queue structures: - Relaxation Solving: First solve the linear relaxation of the integer programming problem (ignoring integer constraints). If the solution happens to be integer, terminate; otherwise, proceed to branching. - Branching Operation: Select a non-integer variable and create two subproblems (e.g., floor and ceiling branches). - Bounding and Pruning: Update global upper and lower bounds using the objective values of subproblems, and discard branches with objective values worse than the current bounds.

MATLAB’s matrix operation advantages can accelerate constraint processing and objective calculations, while script flexibility allows adjustments to branching strategies (e.g., depth-first or best-first). In practical applications, combining cutting-plane methods or heuristic rules can further enhance performance.