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<p>Nonlinear multiobjective optimization presents significant challenges, particularly when addressing problems with three or more objectives. Building on our previous work using Interval Branch and Bound (B&amp;B) techniques for biobjective optimization, we extend these methods to handle the increased complexity of multiobjective scenarios. Interval B&amp;B methods involve dividing the original problem into smaller subproblems and solving them recursively to achieve a desired level of accuracy. These methods offer the advantage of guaranteeing global optimality and providing rigorous bounds on solution quality. We introduce a refined representation of non-dominated regions using two sets of vectors: the non dominated feasible vectors found so far and its associated set of extreme vectors. To accurately define the feasible non-dominated region within each subspace, we adapt an “envelope constraint” from biobjective solvers. Additionally, we propose a novel method for computing the distance from a subspace to the dominated region, and enhance a peeling technique for contracting the subspace based on the dominated region and the envelope constraint. Our approach is evaluated on a set of benchmark problems, and our results show a significant improvement in solution quality and convergence speed compared to a basic approach. Specifically, our enhanced strategy achieves a 42% reduction in relative CPU time, with a remarkable average time reduction of 63% in problems with three objectives. The code of our solver can be found in our git repository (<a href="https://github.com/INFPUCV/ibex-lib/tree/master/plugins/optim-mop">https://github.com/INFPUCV/ibex-lib/tree/master/plugins/optim-mop</a>).</p>

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Extending interval branch-and-bound from two to few objectives in nonlinear multiobjective optimization

  • 期刊论文

  • 发表日期：

    2025年3月17日

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  • Ignacio Araya,
  • Victor Reyes,
  • Javier Montero

摘要

Nonlinear multiobjective optimization presents significant challenges, particularly when addressing problems with three or more objectives. Building on our previous work using Interval Branch and Bound (B&B) techniques for biobjective optimization, we extend these methods to handle the increased complexity of multiobjective scenarios. Interval B&B methods involve dividing the original problem into smaller subproblems and solving them recursively to achieve a desired level of accuracy. These methods offer the advantage of guaranteeing global optimality and providing rigorous bounds on solution quality. We introduce a refined representation of non-dominated regions using two sets of vectors: the non dominated feasible vectors found so far and its associated set of extreme vectors. To accurately define the feasible non-dominated region within each subspace, we adapt an “envelope constraint” from biobjective solvers. Additionally, we propose a novel method for computing the distance from a subspace to the dominated region, and enhance a peeling technique for contracting the subspace based on the dominated region and the envelope constraint. Our approach is evaluated on a set of benchmark problems, and our results show a significant improvement in solution quality and convergence speed compared to a basic approach. Specifically, our enhanced strategy achieves a 42% reduction in relative CPU time, with a remarkable average time reduction of 63% in problems with three objectives. The code of our solver can be found in our git repository (https://github.com/INFPUCV/ibex-lib/tree/master/plugins/optim-mop).