Decomposition of Difficulties in Complex Optimization Problems Using a Bilevel Approach
Ankur Sinha, Dhaval Pujara, Hemant Kumar Singh
TL;DR
The paper addresses the challenge of solving complex optimization problems that exhibit multiple, coexisting difficulties by introducing a bilevel optimization-based decomposition (BOBD). BO BD partitions the problem into an upper-level (handled by an evolutionary algorithm) and a lower-level (solved by a classical method), leveraging mappings that connect the two levels to improve efficiency. A 10-problem test suite, including scalable extensions, demonstrates that BO BD yields feasible, high-quality solutions consistently and often outperforms standalone mathematical programming or evolutionary approaches, especially as problem size grows. The work provides a practical framework for integrating complementary optimization techniques and suggests avenues for automatic, dynamic level classification in future research.
Abstract
Practical optimization problems may contain different kinds of difficulties that are often not tractable if one relies on a particular optimization method. Different optimization approaches offer different strengths that are good at tackling one or more difficulty in an optimization problem. For instance, evolutionary algorithms have a niche in handling complexities like discontinuity, non-differentiability, discreteness and non-convexity. However, evolutionary algorithms may get computationally expensive for mathematically well behaved problems with large number of variables for which classical mathematical programming approaches are better suited. In this paper, we demonstrate a decomposition strategy that allows us to synergistically apply two complementary approaches at the same time on a complex optimization problem. Evolutionary algorithms are useful in this context as their flexibility makes pairing with other solution approaches easy. The decomposition idea is a special case of bilevel optimization that separates the difficulties into two levels and assigns different approaches at each level that is better equipped at handling them. We demonstrate the benefits of the proposed decomposition idea on a wide range of test problems.