To deploy compute-intensive neural networks on resource-constrained edge systems, developers use model optimization techniques that reduce model size and computational cost. Existing optimization tools are application-agnostic -- they optimize model parameters solely in view of the neural network accuracy -- and can thus miss optimization opportunities. We propose ApproxCaliper, the first programmable framework for application-aware neural network optimization. By incorporating application-specific goals, ApproxCaliper facilitates more aggressive optimization of the neural networks compared to application-agnostic techniques. We perform experiments on five different neural networks used in two real-world robotics systems: a commercial agriculture robot and a simulation of an autonomous electric cart. Compared to Learning Rate Rewinding (LRR), a state-of-the-art structured pruning tool used in an application-agnostic setting, ApproxCaliper achieves 5.3x higher speedup and 2.9x lower GPU resource utilization, and 36x and 6.1x additional model size reduction, respectively.