Closing the Gap: A Theoretical and Practical Exploration of Branch-and-Bound Procedure for Mixed-Integer Programs
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| I whakaputaina i: | ProQuest Dissertations and Theses (2025) |
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ProQuest Dissertations & Theses
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| Urunga tuihono: | Citation/Abstract Full Text - PDF |
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| Whakarāpopotonga: | The branch-and-bound scheme, invented by Land and Doig, is the method of choice for solving mixed integer linear programs (MILP) by all modern state-of-the-art MILP solvers. While the branch-and-bound algorithm has been studied for a long time, atheoretical understanding of its performance guarantees is lacking. Furthermore, in practice, its efficiency relies on the performance of several heuristics whose interactions are not well understood. This thesis is motivated by the goal to improve both theoretical as well as practical understanding of branch-and-bound for MILPs. Given that node selection rules are well-understood, this thesis focuses primarily on branching rules.Branch-and-bound shares similarity with dynamic programming in the sense that both employ recursions to solve the problem. This raises the question of whether there is a correspondence between the two algorithms. Previous works in literature have proved branching-and-bound with general disjunctions to be exponentially more efficient than dynamic programming on some classes of problems. However, it was unclear if general branch-and-bound is always superior to dynamic programming. In Chapter 2, we show that to be untrue. We consider the lot-sizing problem for which there exists a polynomial time dynamic programming algorithm and prove an (worst case) exponential lower bound on the size of branch-and-bound trees even when branching is considered on general disjunctions.State-of-the-art mixed integer programming (MIP) solvers are based on the branch and-cut framework where cutting planes are added to strengthen the linear relaxation before branching is initiated. This is done with the hope that improved bounds may lead to more efficient pruning and therefore a smaller branch-and-bound tree. However, in practice branching rules typically use local node LP information to decide on which variable to branch. Therefore, the addition of cuts may also potentially change the branching decisions whose impact on tree size is difficult to model. In Chapter 3, we consider the question of whether adding cuts will always lead to smaller trees for a given fixed branching rule. We formally call such a property of a branching rule monotonicity. We prove that most of the standard branching rules used in practice are non-monotonic. We also empirically attempt to estimate the prevalence of non-monotonicity in practice while using full strong branching and discover that non-monotonicity is surprisingly prevalent.Full strong-branching (henceforth referred to as strong-branching) is experimentally known to produce significantly smaller branch-and-bound trees in comparison to all other known variable selection rules and is often used as a benchmark to compare other rules. However, there is no objective metric to evaluate the performance of strong-branching itself. In Chapter 4, we introduce the notion of an ”optimal branch-and-bound tree”, that is, the smallest branch-and-bound tree that solves a given instance. We present a dynamic programming algorithm for generating the optimal trees and use it to evaluate the efficiency of strong-branching on (small) instances from a range of classical problems.With the goal of bridging the gap between the strong-branching trees and optimal trees, we seek to better understand the properties and limitations of FSB in Chapter 5. Strong branching guides branching decisions based exclusively on the information regarding local gains in the linear programming (LP) bounds, but we identify two additional factors that significantly impact its tree sizes. We incorporate these into the strong-branching score to design new rules that demonstrate remarkable performance benefits on benchmark instances. |
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| ISBN: | 9798263341862 |
| Puna: | ProQuest Dissertations & Theses Global |