I have a negative log-likelihood function $\ell(\mathbf{x}) = \ell(x_1, x_2, \dots)$, and I would like to find the parameter values that minimise $\ell$. I am wondering when it would be quicker to first find the value of $x_1 = \hat{x}_1$ that minimises $g(x_1) = \ell(x_1, 0, 0, \dots)$, and use the vector $(\hat{x}_1, 0, 0, \dots)$ as the start for the minimisation of $\ell(\mathbf{x})$. If the optimisation procedure is important, I am using the 'quasi-Newton' method in Matlab's fminunc.
For example, would this method help when we believe that $x_1$ is the regression coefficient of a covariate that has the biggest effect on the response?