We develop and analyze an optimization-based approach for the robust solution of PDE problems comprised of multiple physics operators with fundamentally different mathematical properties. Our approach relies on three essential steps: additive decomposition of the original problem into subproblems for which robust solution algorithms are available; integration of the subproblems into an equivalent PDE-constrained optimization problem; and solution of the resulting optimization problem. This strategy allows us to design robust solvers for the original problem by utilizing the available solver technology for its subproblems. An application to a scalar advection-diffusion PDE illustrates the new approach. In particular, we derive a robust iterative solver for advection-dominated problems using standard multilevel solvers for the Poisson equation.