BNB20 Algorithm for Mixed-Integer Nonlinear Programming

BNB20 stands as a classical solver for Mixed-Integer Nonlinear Programming (MINLP), employing a branch-and-bound algorithmic framework specifically designed for optimization problems containing both discrete and continuous variables. Its source code implementation reveals several critical design features: systematic partitioning of the problem space, derivation of lower bounds through relaxation techniques, and pruning strategies to reduce computational overhead. Key algorithm implementation details include recursive tree traversal methods for handling discrete variables and convex relaxation procedures for bound calculations. The code typically incorporates node management systems to track active subproblems and bounding information. When compared with MATLAB's fmincon function, BNB20 exhibits three distinctive characteristics: Discrete Variable Handling: While fmincon exclusively optimizes continuous variables, BNB20 processes integer constraints through branching strategies, implementing depth-first search or best-first search to explore discrete combinations. The algorithm maintains separate variable type handling in its data structures. Global Optimality Guarantee: fmincon as a local optimizer may converge to local solutions, whereas BNB20 ensures global optimality through convex relaxations and interval reduction techniques, albeit at higher computational cost. The implementation includes rigorous bound tightening procedures and optimality gap calculations. Heuristic Acceleration: BNB20's source code demonstrates optimized node selection rules, such as prioritizing subtrees with tighter bounds, contrasting sharply with fmincon's gradient-based descent mechanisms. The algorithm often incorporates custom heuristics for variable selection and node processing order. Practical application requires careful consideration of problem characteristics - BNB20 proves more suitable for problems with significant integer variables, while fmincon maintains advantages for purely continuous problems or scenarios demanding high computational efficiency. The choice involves trade-offs between solution quality and computational resources.