A Continuous Variable Optimization method for the Quadratic Assignment Problem
Keywords: Non-convex optimization, Manifold optimization, Dynamical systems, Differential equations, Combinatorial Optimization, Chaos theory, Boolean Satisfyability, SAT, Quadratic Assignment problem, QAP, Traveling Salesmen problem, TSP, Continuous variable computing
TL;DR: Continuous-time dynamical system solving the quadratic assignment problem, using boolean satisfiability constraint satisfaction for the permutation constraint, and (Riemannian) gradient descent for the objective optimization.
Abstract: We present a novel continuous algorithm for solving the Quadratic Assignment Problem (QAP), leveraging auxiliary dynamics to enforce permutation constraints without the need for post-processing. This approach outperforms traditional continuous methods in terms of constraint enforcement and demonstrates faster convergence compared to branch-and-bound techniques. Despite the algorithm's effectiveness, the number of auxiliary variables currently scales cubically with the problem size, posing a limitation. However, our analysis suggests that associating auxiliary variables with correlators of clause functions could significantly improve efficiency. Additionally, the algorithm encounters challenges with local minima due to the geodesically non-convex QAP potential. We propose future research directions to address this issue, including alternative formulations of the potential landscape and strategies for escaping local minima.
Submission Number: 115
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