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I’m a little confused how this differs from standard constraint optimization.

In standard constraint optimization you know the constraints at compile time. In MDO many constraints are generated at runtime and constantly change as you search for solutions.

In MDAO you definitely know all the constraints at compile time, but inequality constraints can be active or inactive as the optimizer progresses.

That's how SNOPT, IPOPT, presumably KNITRO and nonlinear programming optimizers work.

Yes MDAO is "just" constrained nonlinear optimization.

You come up with a model for your thing, which often involves multiple "disciplines" like mass, propulsion, aerodynamics, loads, trajectory/equations of motion, and then usually use some framework to calculate the constraints values and objective value and their gradients

Constraints can very much be dynamic and emergent from complex event chains.

It doesn't, they just want to use a buzzword to make it look like they are doing new and advanced things.

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