# Questions tagged [convex-analysis]

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### Is this function concave? If it is, can we show it in a theoretical way?

Suppose we have a function: \begin{equation} \begin{aligned} f(x_1,x_2,\cdots,x_n)&=\sum\limits_{i=0}^n\frac{(x_i e^{-x_i}-x_{i+1}e^{-x_{i+1}})^2}{e^{-x_i}-e^{-x_{i+1}}}\\ &=\frac{(x_0 e^{-x_0}...
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### Can we substitute this KKT condition into this optimization problem to reformulate the optimization problem?

Suppose I have the following optimization problem $$\min\limits_{\mathbf{x},\mathbf{y}} f(\mathbf{x},\mathbf{y}) \tag{1}$$ It is already known that the target function $f$ is continuous and ...
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While thinking about convex functions, I managed to put together the following proof which I find a bit too good to be true. $X$ is a topological vector space that is also a Baire space. Lemma: Let $f ... 1 vote 1 answer 119 views ### Convex/concave points of a differentiable function I am wondering about the following question: A strictly convex (concave) differentiable function$f:\mathcal{R}\to\mathcal{R}$has the geometrical property that its graph lies completely above (below) ... 0 votes 1 answer 94 views ### Estimation via projecting onto a convex body Assume that$\theta$is in a convex body$K \in \mathbb{R}^n$and we observe$y = \theta + z$, where$z$is a noise term (following, say, the normal distribution). Consider an estimator of$\theta$by ... 8 votes 1 answer 545 views ### Is the square root of the Kullback-Leibler divergence a convex map?$\newcommand{\KL}{\operatorname{KL}}$Let$X$be a Polish metric space and$P(X)$the space of probability measures on$X$. Given$\mu, \nu\in P(X)$, recall that $$\KL(\mu\parallel\nu) = \begin{cases}\... 3 votes 1 answer 124 views ### On the convexity of certain set of random vectors Let {\cal X} be the set of pairs of random variables (X,Y) with finite expectations. For constant c\in[0,1], define set$$ \{(X,Y)\in{\cal X}:\exists a\geq 0, \, b\geq 0 \text{ such that } E[\... 4 votes 1 answer 232 views ### Which set of functions admits the existence of the minimizer? Let$a,b \in \mathbb R$and consider the functional$J$on$X$: $$J[u] = \int_0^1 \left( (u'(x))^2 -a)^2 + b \ln (1+ u^2(x))\right) dx$$ Providing reasons specify if the$\inf J$over$X$is attained ... 0 votes 0 answers 51 views ### Projection onto a cone followed by a Schur-convex function Let$Proj_C(x)$denote the projection of a point$x$onto a cone$C$. Let$f$be a Schur-convex function. I'm considering$f(Proj_C(x))$as a function of$x$. Are there any conditions on the cone$C$... 1 vote 0 answers 24 views ### Sufficient condition for an$n$-tuple to be a convex conjugate We say$(f_1,f_2,\dotsc,f_N)$is a convex conjugate if for any$i=1,2,\dotsc,N$and any$x_i\in\Bbb R^d$, we have: $$f_i(x_i)=\sup\left\{\sum_{k=1}^{N}\sum_{j=k+1}^N x_k x_j - \sum_{j=1,j\neq i}^N f_j(... 2 votes 1 answer 97 views ### Smoothness of Minkowski functional is equivalent to smoothness of boundary Let C\subseteq \mathbb{R}^n be a convex body containing 0 in its interior. I recently read that Minkowski functional of C,$$ f_C(x):=\inf\Big\{t>0:\frac1{t}\cdot x\in C\Big\}, $$is C^1 ... 1 vote 1 answer 174 views ### Proof of extended version of non-random "almost supermartingale" In this question, a non-random version of "almost supermartingale" theorem is proved. Here, I would like to extend/apply the non-random version to the slightly different situation. I wonder ... 1 vote 1 answer 67 views ### Sufficient conditions for the boundary unit-normal vector field of a closed convex set to be Lipschitz continuous This is followup to the following question: On the Lipschitz continuity of the unit-normal vector field of a polytope Let C be a (nonempty) closed convex subset of \mathbb R^n. Note that to every ... 1 vote 1 answer 162 views ### Can we invoke "almost supermartingale" Theorem for deterministic sequences? Perhaps stupid question. Question: Can "almost supermartingale" theorem be equally applicable to prove the convergence of some algorithms solving non-random optimization problems? Attempt ... 2 votes 1 answer 99 views ### On the Lipschitz continuity of x \mapsto \arg\min_{c \in C}d(x,c) w.r.t Hausdorff distance Let C be a (nonempty) compact subset of euclidean \mathbb R^n, and consider the set-valued map p_C:\mathbb R^n \to 2^C defined by$$ p_C(x) = \{c \in C \mid \|x-c\| = \mbox{dist}(x,C)\}, $$... 1 vote 1 answer 68 views ### On the Lipschitz continuity of the unit-normal vector field of a polytope Let \mu be a probability measure on \mathbb R^n and let P be a compact polytope in \mathbb R^n. For any x \in \mathbb R^n \setminus P, let p(x) \in P be (unique!) point in P which is ... 1 vote 1 answer 96 views ### Conditions for Lipschitzness of boundary normal vector, almost everywhere Let C be a nonempty closed subset of \mathbb R^n. It is known that any such set satisfies the following condition (Unique CPP a.e). For almost every x \in \mathbb R^n, there exists a unique ... 2 votes 0 answers 60 views ### A truncated Frobenius norm of a matrix is convex or not? Given a positive integer k and a matrix X\in \mathbb{R}^{m\times n}. A truncated frobenius norm of a matrix X is defined by$$\Vert X \Vert_{k,F} = \sqrt{\sum_{i=k+1}^{m} \sigma_i^2(X)},$$where ... 3 votes 1 answer 180 views ### Well-behaved trajectories Call trajectory any continuous function f: \mathbb{R}_{\geq 0} \to \mathbb{R}^n (here, \mathbb{R}_{\geq 0} is interpreted as time). A polyhedral partition of \mathbb{R}^n is a finite set of ... 4 votes 0 answers 151 views ### Legendre-Fenchel transform Suppose F:\mathbb R^n\to \mathbb R is a convex continuous function. Moreover, for any x\in \mathbb R^n,$$ \limsup_{\lambda\to\infty} \frac {|F(\lambda x)|}{\lambda}<\infty. $$I would like to ... 3 votes 0 answers 65 views ### Projection onto level set of convex functional Fix a probability space (\Omega,\mathcal{F},\mathbb{P}) and let F:L^2_{\mathbb{P}}(\mathcal{F})\rightarrow (-\infty,\infty] be bounded-blow, convex, lower semi-continuous, and not identically ... 4 votes 1 answer 179 views ### On some convergence theorems by Felix E. Browder (1967) I have been reading Felix E. Browder's Convergence Theorems for Sequence of Nonlinear Operators in Banach Space and I was hoping I could find answers to a couple of questions I have about the paper. ... 1 vote 1 answer 49 views ### Measure of intersection of convex set with hyperplane is concave function Let \Omega \subset \mathbb{R}^n be convex. We write points of \mathbb{R}^n as (x_1, x_2, \dots, x_n). Set p(x) = m(\Omega \cap \{x_1 = x\}), where m is the n-1 dimensional Lebesgue measure ... 4 votes 2 answers 263 views ### Convexity of (X, y) \mapsto y^T X^{-1} y [closed] Let y \in \mathbb{R}^n, X \in \mathcal{S}^n_{++}(\mathbb{R}). Why would function f : (X, y) \mapsto y^T X^{-1} y be convex? I tried with (X, x) + t.(Y, y) with no result. Also, I thought ... 2 votes 1 answer 157 views ### Bounded variation of the partial derivatives of a convex function Let f:\mathbb R^2\to\mathbb R be a convex function. For simplicity, assume that f\in C^1. A general theorem which can be found in the book of Evans and Gariepy says that the gradient \nabla f is ... 1 vote 1 answer 44 views ### Elementary inequality about integrals of exponentials of concave functions (possibly connected to log concave distributions) I think the following inequality might be true and was hoping somebody might spot it or know a proof: Suppose f:\mathbb R\to \mathbb R is convex and suitably nice so that$$\int_{\mathbb R} e^{-f(x)}... 0 votes 0 answers 47 views ### An inequality regarding operator concave function Crossposted from math.SE Let$\mathbb P_n$be the space of all$n \times n$self-adjoint positive definite matrices. Consider the function$\varphi: \mathbb P_n \longrightarrow \mathbb R$defined by$...
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The Hilbert-Haar theory says that functionals $\mathcal{F}(u,B)=\int_{B} F(\nabla u)\,dx$, where $F$ is a convex function and $B$ is a bounded domain in $\mathbb{R}^N$, take a minimum in the space of ...
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### Log-concavity of the modified Bessel function of a second kind

I was searching for some results for the log-concavity of the modified Bessel function of a second type, but I failed. Has there been any known work on this? I am not even sure if it is the modified ...
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### Subgradients of $|Du|(\Omega)$ and Hodge decomposition in dual spaces of measures

Let $\Omega \subset \mathbb R^d$ be open and bounded. I have a function $u\in BV(\Omega)$ and a vector field $\mu \in L^\infty(\Omega)^d$ such that $-div\ \mu\in L^2(\Omega)$, and $-div\ \mu$ is in ...
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### Biconjugate of a quasiconvex lower semi-continuous function

Let $f:\mathbb{R}^d \to [0,\infty]$ be a quasiconvex lower semi-continuous function whose effective domain $C:=\{x \in \mathbb{R}^d:f(x) < \infty\}$ is nonempty and bounded (and convex since $f$ is ...
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### A question on regularity of the Legendre transform

Let $f(x)$ be a strictly convex real-valued $C^{\infty}$ function on an open neighborhood of the origin in $\mathbb R^n$ with $f (0, \ldots , 0)= \partial_j f(0, \ldots , 0)=0$ for all $j$. If the ...
Consider a lower-semicontinuous convex function $f\colon \mathbb{R}^n \to \mathbb{R}$ with domain $C = \{x \in \mathbb{R}^d: f(x) < \infty\}$. I am interested in understanding under what conditions ...