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In this paper, we present a novel control scheme for feedback optimization. That is, we propose a discrete-time controller that can steer the steady state of a physical plant to the solution of a constrained optimization problem without numerically solving the problem. Our controller can be interpreted as a discretization of a continuous-time projected gradient flow. Compared to other schemes used for feedback optimization, such as saddle-point flows or inexact penalty methods, our algorithm combines several desirable properties: It asymptotically enforces constraints on the plant steady-state outputs, and temporary constraint violations can be easily quantified. Our algorithm requires only reduced model information in the form of steady-state input-output sensitivities of the plant. Further, as we prove in this paper, global convergence is guaranteed even for non-convex problems. Finally, our algorithm is straightforward to tune, since the step-size is the only tuning parameter.