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My objective is to maximize $f(x,y,z)$ subject to two constraints $g_1(x,y,z)=0$ and $g_2(x,y,z)≤0$. (There is an Econ story in the background).

For simplicity, I use the following sequential optimization method. First I fix some $z$ and solve

$$\max_{x,y}f(x,y;z) \text{ subject to } g_1(x,y; z)=0 \text{ and } g_2(x,y; z)≤0$$ Let's say $(x^{∗},y^{∗})$ is the unique pair that solves the above problem. In the second step, given the solution above, I focus on $f(x^{∗},y^{∗},z)$, which is just a function of $z$. Thus I solve $$\max_{z}f(x^{∗},y^{∗},z) \text{ subject to } g_1(x^{∗},y^{∗},z)=0 \text{ and } g_2(x^{∗},y^{∗},z)≤0.$$

This is a relatively easy problem, and I get a unique $z^{∗}$ that solves it.

If I were to solve the main problem by selecting $(x,y,z)$ simultaneously (instead of sequentially), would I still get the same solution?

I tried some simple problems with two variables and found that both techniques yield the same result. But, years ago, back in grad school, I remember seeing an optimization problem where this sequential approach was leading to a different (and suboptimal) solution, so there must be some caveat around it.

To sum up, my question is: Under what conditions does sequential optimization yield the same solution as simultaneous optimization?

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If the multivariable function $f(x,y,z)$ is Separable, then yes both will give the same results. If there is a coupling between the variable, then only sequential algorithm grantees the convergence to the unique maximum, if exists.

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