Continuous-Time Dynamics of the Difference-of-Convex Algorithm

• URL: http://arxiv.org/abs/2604.06926v1
• Date: Wed, 08 Apr 2026 10:38:07 GMT
• Title: Continuous-Time Dynamics of the Difference-of-Convex Algorithm
• Abstract summary: We induce the continuous-time structure of the difference-of-generate (DCA) for smooth DC decompositions with a convex component.<n>We show that different DC decompositions of the same objective link different continuous dynamics.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the continuous-time structure of the difference-of-convex algorithm (DCA) for smooth DC decompositions with a strongly convex component. In dual coordinates, classical DCA is exactly the full-step explicit Euler discretization of a nonlinear autonomous system. This viewpoint motivates a damped DCA scheme, which is also a Bregman-regularized DCA variant, and whose vanishing-step limit yields a Hessian-Riemannian gradient flow generated by the convex part of the decomposition. For the damped scheme we prove monotone descent, asymptotic criticality, Kurdyka-Lojasiewicz convergence under boundedness, and a global linear rate under a metric DC-PL inequality. For the limiting flow we establish an exact energy identity, asymptotic criticality of bounded trajectories, explicit global rates under metric relative error bounds, finite-length and single-point convergence under a Kurdyka-Lojasiewicz hypothesis, and local exponential convergence near nondegenerate local minima. The analysis also reveals a global-local tradeoff: the half-relaxed scheme gives the best provable global guarantee in our framework, while the full-step scheme is locally fastest near a nondegenerate minimum. Finally, we show that different DC decompositions of the same objective induce different continuous dynamics through the metric generated by the convex component, providing a geometric criterion for decomposition quality and linking DCA with Bregman geometry.

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