Combinatorial optimization problems such as Maximum Independent Set (MIS) and Maximum Cut (MaxCut) arise across a wide range of applications, yet solving large-scale instances remains computationally challenging. While discrete local-search procedures can produce high-quality solutions, their inherently sequential and iterative nature can become increasingly expensive as problem size and density grow. In this paper, we present a differentiable perspective on combinatorial optimization that leverages continuous relaxation with parallelizable gradient-based updates to improve scalability while reducing dependence on discrete local search. More specifically, for MIS, we show that solutions that remain repairable through a discrete (1,2)-swap can nevertheless be fixed points of projected gradient ascent, revealing a fundamental limitation of purely local gradient dynamics. For MaxCut, we develop a new quadratic formulation whose landscape provably allows projected gradient ascent to escape 1-flip-repairable solutions while ensuring binary local maximizers. Building on these insights, and inspired by discrete perturb-and-improve methods, we introduce a framework that combines projected gradient ascent with differentiable mutations (partial resets) to efficiently explore new regions of the solution space}. Experimental results on MIS and MaxCut demonstrate the potential of this approach for scalable combinatorial optimization, particularly in regimes where conventional discrete local search becomes computationally prohibitive.

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