Classical frameworks in mechanism design often specify an objective function and maximize it by choosing allocation. We extend these frameworks by allowing maximizing an objective function (such as expected revenue in an auction) subject to additional constraints (such as lower bounds on efficiency or welfare). The additional complexity arising due to each additional constraint manifests in the reduced form of the optimal mechanism as at most one jump discontinuity in an "ironed" interval. We apply our results to demonstrate the simplicity of optimal mechanisms despite the presence of a side constraint in common economic applications such as contract and auction design. We also introduce a regularity condition under which the general structure of optimal mechanisms bears no additional complexity due to the presence of a side constraint. The analysis builds on the findings of Kleiner et al. (2021) by considering optimal mechanisms as extreme points of function spaces.