All Questions
Tagged with convex-optimization ca.classical-analysis-and-odes
22 questions
2
votes
1
answer
106
views
Is $f^{-a}$ locally integrable if $f\geq 0$ has a unique stationary point ( a minimum) at which the Hessian is positive definite, $0<a<d/2$
Let $0<a<d/2$, let $B$ be the unit ball in $\mathbb{R}^{d}$ centered at the origin, and let $f:B \to [0,\infty[$ be a a smooth function such that
(1) $f(x)\geq f(0).$
(2) $\nabla f(x)\neq 0,\...
- 852
0
votes
1
answer
103
views
Constrained linear optimization problem on $C^1$
I am dealing with a problem of the form ($a<b$)
$$
\displaystyle \max_{v \in C^1([a, b])} \int_a^b v(x)~\mathrm{d}x, \quad \mathrm{s.t.} \int^b_a \big(-o'(x)v(x)-v'(x)o(x)\big)f(x)~\mathrm{d}x \...
4
votes
5
answers
2k
views
Reference request: importance of Lipschitz continuity
I see that Lipschitz continuity is a common assumption used in optimisation, statistics, machine learning, etc.
Could you point me in the direction of some literature that discusses why Lipschitz ...
- 161
4
votes
1
answer
194
views
How to solve this minimax matrix optimization problem?
Recently, I want to know how well can a $\ell_1$ ball be approximated by the image of a $\ell_2$ ball under a linear transform. I formulate this problem as the following optimization problem.
\begin{...
- 550
4
votes
1
answer
210
views
On some convergence theorems by Felix E. Browder (1967)
I have been reading Felix E. Browder's Convergence Theorems for Sequence of Nonlinear Operators in Banach Space and I was hoping I could find answers to a couple of questions I have about the paper. ...
user452741
3
votes
1
answer
621
views
Strong convexity inequality w.r.t. infinity norm $\lVert\cdot\rVert_{\infty}$
Consider a twice differentiable 1-strongly convex function $f:\mathbb{R}^n \to \mathbb{R}$.
Is it true that there exists $\alpha>0$ independent of $n$ such that, for all $x \in \mathbb{R}^n$:
\...
- 33
2
votes
1
answer
189
views
Analytic expression for the Moreau envelope of $x \mapsto \|Ax\|$
Given an $m \times n$ matrix $A$ and a vector $c \in \mathbb R^n$, define $\eta(A,c) \ge 0$,
$$
\eta(A,c) := \sup_{u \in \mathbb R^n} u^\top c - \frac{1}{2}\|u\|^2 - \|Au\|.
$$
Note that $\eta(A,c) = \...
- 6,853
0
votes
1
answer
199
views
conditions on the boundary of a compact set to ensure the volume of the intersection of a small ball with the set doesn't vanish
Given a compact set $E$ with non-empty interior in $R^d$ and some small positive number $r$, what kind of conditions should we put on the set $E$ so that for all $x\in E$, the volume of the ...
- 11
0
votes
1
answer
220
views
the subdifferential at points of differentiability in infinite dimensional space
Let $ f: X\to (-\infty,+\infty]$ that $ X$ is an infinite dimensional space.
What are the conditions for $f$ and space $X$ to have the following equality correct?
$$\partial f(x)=\{\nabla f(x)\}$$ for ...
- 3
7
votes
1
answer
297
views
When is the log-permanent concave?
Let $\operatorname{PSD}_n$ be the cone of $n\times n$ semidefinite positive matrices. For any $X\in \operatorname{PSD}_n$, define $$f(X)=\log(\det(X)).$$ Then $f$ is a concave function on $\...
- 3,979
9
votes
1
answer
2k
views
Is There an Induction-Free Proof of the 'Be The Leader' Lemma?
This lemma is used in the context of online convex optimisation. It is sometimes called the 'Be the Leader Lemma'.
Lemma:
Suppose $f_1,f_2,\ldots,f_N$ are real valued functions with the same ...
- 1,955
5
votes
0
answers
269
views
Fenchel conjugate on a Hadamard manifold
Let $M$ be a Hadamard manifold and let $F:M\to\mathbb{R}$ be a real-valued convex function on $M$. What would be the Fenchel-Young conjugate of $F$?
In general for a real locally convex vector space $...
- 364
4
votes
2
answers
270
views
Is this function always bounded below?
Is there any global constant $C$ such that $$C<\frac{\sum_{i=1}^{n}x_{i}\log x_{i}-x_{i}+(1-x_{i})\log(1-x_{i})}{\sum_{i=1}^{n}x_{i}}+\log(\sum_{i=1}^{n}x_{i})-\log(\sum_{i=1}^{n}x_{i}^{2})$$ for ...
- 4,049
9
votes
1
answer
749
views
property of convex functions
I am able to give a proof to the following inequality for convex functions. Most likely this is well known, but I am unable to find a reference. I would appreciate if someone more knowledgeable in the ...
- 1,211
3
votes
2
answers
642
views
Simultaneous extensions of strongly convex functions
21/03/2017: I have decided to accept Denis Serre's answer, even though it does not exactly answer my question, however I like its simplicity and I'd say it is close enough to the desired claim. Of ...
- 11.3k
1
vote
0
answers
60
views
Optimizing sum of approximate and exact functions
This is a research question that I had asked in Math.SE about a month ago, but even after putting a bounty on it, I did not get any answers.
I have two real values functions, where one ($g(w;x):\...
- 189
-1
votes
1
answer
518
views
Does element-wise concavity guarantee joint concavity?
I have a function of two variables, and I have checked that along one direction (fixing another variable), it is a monotonically increasing and concave function. Whereas in another direction (fixing ...
- 1
1
vote
0
answers
82
views
Log convexity for the norm of a vector-valued function
Log convexity of various functions defined on the space of Hermitian matrices plays an important role in matrix analysis and probability theory.
Given $v \in \mathbb{C}^n$, $D$ a diagonal matrix with ...
- 31
1
vote
0
answers
87
views
Characterization of the maximizer of a function based on a parameter's value
Consider a smooth, continuously differentiable, and jointly concave function $f(x,y,z;a)$, where $x,y$ and $z$ are decision variables and $a$ is a problem parameter.
I have two optimization problems. ...
- 11
3
votes
0
answers
207
views
proving quasi convexity of multivariable function
Given
an arbitrary $(N \times N)$ square matrix ${\bf X}$
a positive definite $(M\times M)$ matrix ${\bf T}$
a $(Q\times MN), Q< MN$ matrix ${\bf Z}$ consisting of only 1s and 0s where there is
...
2
votes
2
answers
2k
views
Hessian of function of covariance matrices
Suppose we have a typical logdet function $\mathcal{L}$ with respect to a covariance matrix $\mathbf{A}$,
$$
\mathcal{L}(\mathbf{A}) = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\...
- 125
2
votes
1
answer
1k
views
Subgradient of Minimum Eigenvalue
Consider three $N \times N$ Hermitian matrices $A_0$, $A_1$, $A_2$. Consider the function
\begin{align}
f(t_1,t_2)=\lambda_{\text{min}}(A_0+t_1A_1+t_2A_2)
\end{align}
where $\lambda_{\text{min}}$ ...
- 1,421