1pc 23cm Faux Pearl Chain Design Nail Art Decoration

1pc 23cm Faux Pearl Chain Design Nail Art Decoration


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Reformulating the Lagrange Multiplier Theorem

Let $phi: R^kto R^+$ be a convex function and let $g:Rto R^k$ be a continuous function. Is
$$phi(g(r))-frac{1}{2}r(mathrm{grad}phi)^{-T}(mathrm{grad}phi) rle0$$
correct for any $rin R$. If this is not, what is a counterexample?
The motivation for the above is that if $f$ is a convex function and $g:Rto R^k$ is a non-convex function, then
$$f(s)-frac{1}{2}s(mathrm{grad}f)^{-T}(mathrm{grad}f)le 0,$$
if and only if, $P_k(x)-frac{1}{2}x^TA(x)le0$, where $A$ is the Hessian matrix and $P_k$ is the matrix $ktimes k$ permutation matrix.
The case when $P_k(x)-frac{1}{2}A(x)le0,$ for some $A$ is obvious. However, when $f(x)=frac{x^TA(x)}{2}$ and $g:Rto{xin R^k: P_k(x)-P_k^T(x) = 0}$ is a linear map, I do not see it.

A:

The condition is indeed correct.

Firstly, note that the case of equality can

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