Skip to main content

All Questions

Filter by
Sorted by
Tagged with
4 votes
5 answers
2k views

Reference request: importance of Lipschitz continuity

I see that Lipschitz continuity is a common assumption used in optimisation, statistics, machine learning, etc. Could you point me in the direction of some literature that discusses why Lipschitz ...
12345's user avatar
  • 161
3 votes
1 answer
189 views

Sensitivity of the solution of QP with respect to parameters

Given a quadratic program, $$\begin{array}{ll} \text{minimize} & \displaystyle \frac12 x^TAx + b^Tx \\ \text{subject to} & Cx \le d \end{array}$$ Suppose $A \succ 0$, so the program strongly ...
gcy's user avatar
  • 33
2 votes
2 answers
323 views

Reference request on computational schemes for $\inf_{x\in\Omega^n}\sup_{y\in\mathbb R^n}F(x,y)$

Let $\Omega\subset \mathbb R^d$ be compact, $\rho$ be a density function on $\Omega$ and $p_1,\ldots, p_n\in (0,1)$ be weights satisfying $\int_{\Omega}\rho(z)dz=1=\sum_{k=1}^n p_k$. We consider the ...
user avatar
1 vote
1 answer
309 views

Numerical estimation of partial derivatives of convolved functions when closed forms do not exist

Summary: Some peak functions are convolutions which may not have a closed form solution. A classical example can that of a Voigt which is a convolution of a Lorentzian and a Gaussian, followed by ...
ACR's user avatar
  • 879
1 vote
0 answers
267 views

Minimum Preserving Transformations [closed]

If $f:X\rightarrow Y$, $g:Y\rightarrow Y$ are functions and $g$ is monotone increasing function then $$ \operatorname{argmin}_{x \in X} f(x) = \operatorname{argmin}_{x \in X} g\circ f(x) . $$ X and Y ...
ABIM's user avatar
  • 5,405
0 votes
1 answer
329 views

Gradient-descent "type" Methods for non-convex and non-smooth functions

Most (stochastic) "gradient descent" type algorithms (such as Nesterov-accelerated gradient-descent or ADAM) seem to be well-defined only for functions which are either: lower semi-...
ABIM's user avatar
  • 5,405
0 votes
1 answer
291 views

Quasiconvexity property of quasinorms

Schatten $p$ norm is convex when $p\geq1$ holds and if $p\in(0,1)$ it is quasinorm. If $p\in(0,1)$ then is Schatten $p$ norm quasi convex? I am interested in definition of quasi convexity here https:...
Turbo's user avatar
  • 13.9k