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Neural architecture search (NAS) has been very successful at outperforming human-designed convolutional neural networks (CNN) in accuracy, and when hardware information is present, latency as well. However, NAS-designed CNNs typically have a complicated topology, therefore, it may be difficult to design a custom hardware (HW) accelerator for such CNNs. We automate HW-CNN codesign using NAS by including parameters from both the CNN model and the HW accelerator, and we jointly search for the best model-accelerator pair that boosts accuracy and efficiency. We call this Codesign-NAS. In this paper we focus on defining the Codesign-NAS multiobjective optimization problem, demonstrating its effectiveness, and exploring different ways of navigating the codesign search space. For CIFAR-10 image classification, we enumerate close to 4 billion model-accelerator pairs, and find the Pareto frontier within that large search space. This allows us to evaluate three different reinforcement-learning-based search strategies. Finally, compared to ResNet on its most optimal HW accelerator from within our HW design space, we improve on CIFAR-100 classification accuracy by 1.3% while simultaneously increasing performance/area by 41% in just~1000 GPU-hours of running Codesign-NAS.