As we approach the physical limits predicted by Moore's law, a variety of specialized hardware is emerging to tackle specialized tasks in different domains. Within combinatorial optimization, adiabatic quantum computers, CMOS annealers, and optical parametric oscillators are few of the emerging specialized hardware technology aimed at solving optimization problems. In terms of mathematical framework, the Ising optimization model unifies all of these emerging special-purpose hardware. In other words, they are all designed to solve optimization problems expressed in the Ising model. Due to variety of constraints specific to each type of hardware, they usually suffer from two main challenges: (i) number of variables that the hardware can manage to solve; and (ii) precision limitations. Given that large-scale real-world problems, including problems in combinatorial scientific computing, data science and network science require significantly more variables to model than these devices provide, we are likely to witness that cloud-based deployments of these devices will be available for parallel and shared access. Thus hybrid techniques in combination with both hardware and software must be developed to utilize these technologies. Local search as a meta-heuristic seems like a natural approach to tackle this challenge. However, a general optimization step within local search is not traditionally formulated in the Ising model. Thus, straightforward modeling of local search can lead to under-utilization of the hardware. In this work, we propose a new approach to model local search on these special-purpose hardware devices that takes the limitations of the Ising model and current hardware into account. As such, we demonstrate that the proposed approach utilizes a given hardware more efficiently compared to previous approaches.