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Suppose I have an optimization problem of the form $$ \inf_{\{x \in \mathbb{R}^d: g(x)=0\}} f(x), $$ for some convex function $f$ and non-convex l.s.c. function $g$.

Can we reinterpret the Lagrange multiplier $\mu$, as the reciprocal of smallest $\lambda \in [0,1]$ such that the set $$ \operatorname{arginf} \left[ (1-\lambda)f(x) + \lambda g(x) \right] \cap \{ x \in \mathbb{R}^d : g(x)=0 \} \neq \emptyset \text{?} $$

(sort of a first point of contact)

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    $\begingroup$ I changed $\displaystyle inf_{\{x \in \mathbb{R}^d: g(x)=0\}} f(x)$ to $\displaystyle \inf_{\{x \in \mathbb{R}^d: g(x)=0\}} f(x). \vphantom{\frac 1 {\displaystyle\int}}$ The backslash does not only prevent italicization but also affects the format in the way that you see in this example. It also affects spacing in things like $a\inf b$ and $a\inf(b). \qquad$ $\endgroup$
    – Michael Hardy
    Commented Jan 28, 2017 at 0:19

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