Abstract: Instrumental variables (IVs) are widely used to estimate causal effects in the presence of unobserved confounding between an exposure $X$ and outcome $Y$. An IV must affect $Y$ exclusively through $X$ and be unconfounded with $Y$. We present a framework for relaxing these assumptions with tuneable and interpretable "budget constraints". Our algorithm returns a feasible set of causal effects that can be identified exactly given perfect knowledge of observable covariance statistics. This feasible set might contain disconnected sets of possible solutions for the causal effect. We discuss conditions under which this set is sharp, i.e., contains all and only effects consistent with the background assumptions and the joint distribution of observable variables. Our method applies to a wide class of semiparametric models, and we demonstrate how its ability to select specific subsets of instruments confers an advantage over convex relaxations in both linear and nonlinear settings. We adapt our algorithm to form confidence sets that are asymptotically valid under a common statistical assumption from the Mendelian randomization literature. An accompanying R package, budgetIVr, is available from CRAN.