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Results tagged with convex-optimization
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user 8799
Optimization with convex constraints and convex objectives; notions related to convex optimization such as sub-gradients, normal cones, separating hyperplanes
6
votes
Matrix function as gradient
This is true if $M=mI_n$ for some $m>0$, and false otherwise.
The function $f$ is a gradient if its differential is symmetric, that is if
$$\langle \nabla_Yf(X),Z\rangle=\langle \nabla_Zf(X),Y\rangle$ …
- 52.3k
3
votes
Bounding the spectral gap of a simple symmetric matrix
Here is an elementary bound. The second eigenvalue of $A$ satisfies
$$\lambda_2(A)> \max_k\min_{i\ne k}a_ia_k=(\max_ka_k)(\min_ia_i).$$
To prove it, let $A_k$ be the principal submatrix obtained by de …
- 52.3k
8
votes
Accepted
On the convexity of element-wise norm 1 of the inverse
The answer is Yes when $n=2$,but No when $n\ge3$. Here is the analysis.
The differential $L_A$ of $A\mapsto A^{-1}$ is $L_A=-A^{-1}BA^{-1}$. Likewise, the Hessian is
$$H_A[B]=2A^{-1}BA^{-1}BA^{-1}=\f …
- 52.3k
3
votes
Accepted
Simultaneous extensions of strongly convex functions
Here is an elementary proof that the answer is positive, at least if you relax the condition that the extension be $C^2$.
Lemma. Given $x_0\in C_M$, there exists a linear form $\lambda$ such that …
- 52.3k