Causal effect estimation is important for numerous tasks in the natural and social sciences. However, identifying effects is impossible from observational data without making strong, often untestable assumptions which might not be applicable to real-world data. We consider algorithms for the partial identification problem, bounding the effects of multivariate, continuous treatments over multiple possible causal models when unmeasured confounding makes identification impossible. Even in the partial identification setting, most current work is only applicable in the discrete setting. We propose a framework which is applicable to continuous high-dimensional data. The observable evidence is matched to the implications of constraints encoded in a causal model by norm-based criteria. In particular, for the IV setting, we present ways by which such constrained optimization problems can be parameterized without likelihood functions for the causal or the observed data model, reducing the computational and statistical complexity of the task.