Are Deterministic Algorithms Invariant Under Coordinate Transforms?

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I am currently using Matlab's "fmincon" command with the "sqp" optimization algorithm in a problem formulation with lower and upper bounds and inequality constraints. As part of my work, I am attempting to conduct a sanity check and show that after a coordinate transform the algorithm converges to the same solution (i.e. the original problem was , and the new problem is , where C is invertible). The function f is nonconvex, but since I transformed the initial guess as well and (as I understand it) "sqp" is a deterministic algorithm, I figured that both problem formulations would converge to the same local minima. However, this is not the behavior I am seeing in practice. Is there a good explaination for this, or is there necessarily a bug in my code?

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