Causal inference necessarily relies upon untestable identification assumptions; hence, it is crucial to assess the robustness of obtained results to potential violations. However, such sensitivity analysis is only occasionally undertaken in practice as many existing methods only apply to relatively simple models and their results are often difficult to interpret. We take a more flexible approach to sensitivity analysis and view it as a constrained stochastic optimization problem. This work focuses on linear models with an unmeasured confounder and a potential instrument. In this setting, the $R^2$-calculus – a set of algebraic rules that relates different (partial) $R^2$-values and correlations – emerges as the key tool for sensitivity analysis. It can be applied to identify the bias of the family of k-class estimators, which includes the OLS and TSLS estimators, as well as construct sensitivity models flexibly. For instance, practitioners can specify their assumptions on the unmeasured confounder by comparing its influence on treatment/outcome with an observed variable. We further address the problem of constructing sensitivity intervals which generalize the concept of confidence intervals for partially identified models. Since the heuristic ‘plug-in’ sensitivity interval may not have any confidence guarantees, this work instead follows a bootstrap approach. We illustrate the proposed methods with a real data example and provide user-friendly visualization tools.