Martins and Lambe 2013 ("Mutldisiciplinary design optimization: a survey of architectures")

This is a great review of architectures. Great reference. Seconded.

For an overview of optimization process itself from a mathematical perspective, but still with consideration towards PDE governed systems, I would point to Nocedal and Wright's Numerical Optimization. The book covers multi-objective optimization insofar as pointing out that the typical approach is to combine multiple functions of interest into a single cost function with appropriate weighting and/or adopt a constrained optimization framework.

Your choice of CFD solver is largely irrelevant to the optimization setup, as long as it gives you the results you need. A more important consideration is how expensive it is to run each CFD analysis; this will determine what sorts of algorithms that end up being practical.

I want to amend this by saying that, from a mathematical perspective, the choice of CFD solver is irrelevant. But unfortunately that is not the case when faced with the nitty gritty implementation details.

I would say with some confidence that gradient-free algorithms are usually unacceptable for any high-fidelity system. The cost of each individual CFD analysis is typically too high. There's some exceptions to this, with very fast, research-grade, highly parallelized solvers running on large clusters/supercomputers. In that case high fidelity single-discipline analysis can be reduced down to seconds, or even a couple of minutes, suddenly making it acceptable to iterate thousands of design points. But not very many have access to those resources.

The problem then emerges that the CFD solver used in conjunction with a gradient-based algorithm has to provide the necessary cost function gradients. And more importantly, these gradients have to be provided intelligently -- that is, using the adjoint method, built on top of partials calculated using algorithmic/automatic differentiation. Brute forcing these with finite difference brings us back to the same limitation that turned us away from gradient-free methods in the first place: too many expensive CFD evaluations.

Unfortunately this significantly limits our choice of CFD solver in practice. There's been some recent work in COMSOL and ANSYS providing some cost function gradients via the adjoint method. This is adequate for traditional quasi-Newton algorithms. Unfortunately the lack of stuff like residual Jacobians means that anyone exploring alternative, cutting-edge algorithms (newton-krylov for instance), has to either try to get access to a handful of research grade solvers (Prof. Martins' group at Michigan have a fantastic one) or develop their own in-house.