If I have a function i have to optimize subject to one constraint, are all feasible points automatically regular points since theres only one constraint?
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No. Given $m$ constraints $g_1,g_2,\ldots,g_m:\mathbb R^n\to\mathbb R$, a point $x\in\mathbb R^n$ is called regular if the set of vectors $\{\nabla g_1(x),\nabla g_2(x),\ldots,\nabla g_m(x)\}$ is linearly independent. If there is only one constraint, the set is a singleton $\{\nabla g_1(x)\}$, but it may still not be linearly independent if $\nabla g_1(x)=0$. |
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