Neur2BiLO: Neural Bilevel Optimization (2402.02552v2)

Abstract: Bilevel optimization deals with nested problems in which a leader takes the first decision to minimize their objective function while accounting for a follower's best-response reaction. Constrained bilevel problems with integer variables are particularly notorious for their hardness. While exact solvers have been proposed for mixed-integer linear bilevel optimization, they tend to scale poorly with problem size and are hard to generalize to the non-linear case. On the other hand, problem-specific algorithms (exact and heuristic) are limited in scope. Under a data-driven setting in which similar instances of a bilevel problem are solved routinely, our proposed framework, Neur2BiLO, embeds a neural network approximation of the leader's or follower's value function, trained via supervised regression, into an easy-to-solve mixed-integer program. Neur2BiLO serves as a heuristic that produces high-quality solutions extremely fast for four applications with linear and non-linear objectives and pure and mixed-integer variables.

• The paper introduces NEUR2BILO, a framework that embeds neural network approximations into bilevel optimization to deliver rapid, near-optimal solutions.
• It collects data samples and trains a regressor for the follower’s value function before integrating it into a single-level mixed-integer program formulation.
• Experiments on interdiction and critical node problems demonstrate significant computation time reductions with a bounded optimality gap.

Overview of Neural Bilevel Optimization (NEUR2BILO)

Bilevel optimization (BiLO) is an intricate area of paper that deals with hierarchical decision-making processes where a leader's decision influences a follower's response. Such problems naturally arise in various real-world applications, from transportation planning to network security. Historically, researchers have found mixed-integer bilevel optimization problems to be particularly challenging due to their computational complexity, especially when involving non-linear terms and integer variables.

Methodological Contributions

The proposed framework, NEUR2BILO, addresses some of the significant challenges in bilevel optimization by incorporating neural network approximations of value functions within a mixed-integer programming context. The innovation lies in three key steps: i) collecting data samples for supervised regression; ii) offline training of a regressor depicting the follower’s or leader’s value function; iii) embedding the trained neural network model within a mixed-integer program formulation to approximate the bilevel problem as a single-level one.

Experimentation and Performance

Through extensive trials on problems like the bilevel knapsack interdiction and critical node problem from network security, NEUR2BILO demonstrated its proficiency in generating high-quality solutions in rapid succession. Remarkably, the efficacy of NEUR2BILO is illuminated in scenarios requiring fast execution, such as the knapsack interdiction problem, where it substantially reduced computation time while maintaining near-optimal solution quality.

Theoretical Results

Beyond the empirical success, the paper posits a theoretical underpinning to the NEUR2BILO approach, particularly for classes of interdiction problems. The framework's solutions are claimed to have a bounded optimality gap, primarily dependent on the accuracy of the regression model used to approximate the follower's value function.

Conclusion

NEUR2BILO stands as a significant advancement in solving mixed-integer nonlinear bilevel optimization problems. Its versatility across various problem dynamics, combined with speed and accuracy, makes it a vital tool for optimization practitioners. The data-driven, learning-based approach heralds a paradigm shift, where historical data and neural network approximations converge to solve otherwise intractable optimization problems. The research opens avenues for further exploration into more generalized bilevel problems and other settings where nested structures are central.