# Questions tagged [convex-analysis]

Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

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### Conditions of strong convexity for non-smooth functions

I was reading this paper for some results on the strong convexity for non-smooth functions but I'm not getting this proposition at all: Lema II (i) $f$ is strongly convex with parameter $\mu$. (ii) ...
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### Number of iterations needed for the method of steepest descent

The function $f(x,y) = 4x^2 + 2y^2 + 2xy -4x + 6y$ has a unique global minimizer at $(x,y) = (1, -2)$ Starting at $(5,2)$ how many iterations of the steepest descent method would it take, at least, to ...
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### Hessian matrix negative definite except in a finite number of points

I'm dealing with optimization problems and I have found myself wondering on this statement: If $f$ is twice differenciable on $\mathcal{D}$ and its hessian matrix $\mathbf{H}_f$ is negative definite ...
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### Sum of two-variable and single-variable convex functions to be convex [closed]

It holds that if $f(x)$ and $g(x)$ are convex functions, then $h(x)=f(x)+g(x)$ is also convex function. While I meet a problem in my research. I have a three functions $f_1(x)$, $f_2(x,y)$, $f_3(y,z)$,...
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### Convexity of zero of partial derivative of a convex fonction. [closed]

Is there an integer d, an integer $1 \le i \le d$ and a convex function $f : \mathbb{R}^d \rightarrow \mathbb{R}$. Such that the set $\{x \in \mathbb{R}^d / \frac{\partial f}{\partial x_i}(x) =0 \}$ ...
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### Unit ball with dual set [closed]

Let $X\subseteq \mathbb{R}^d,$ we define the set dual to $X$, denoted by $X^*$, as follows: $$X^*=\{y\in \mathbb{R}^d \mid \langle y,x\rangle\leq 1, \forall x\in X\}.$$ Geometrically, $X^*$ is the ...
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### Do three points of inequality between convex functions imply inequality over an interval? [closed]

Say I have 2 convex functions, $f:\mathbb{R}\rightarrow\mathbb{R}$, and $g:\mathbb{R}\rightarrow\mathbb{R}$. I want to prove that $f(x) > g(x), \forall x \in [l, u]$. I know that $f(l) > g(l)$ ...
1 vote
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### Prove $\frac{d}{dx} \frac{f(x)-f(a)}{x-a} \geq 0$ if $f(x)$ is convex without twice differentiability.

I've recently been trying to understand some proofs about convex functions. The definition of convex I'm using is: Let $f(x)$ be a once differentiable function defined on $[a, b]$. $f$ is convex iff ...
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### Does the set where a convex function fails to be second derivative have some geometric properties? [closed]

I know it is of Hausdorff measure 0 by Alexsandrov's theorem. And the second derivative of a convex function can be viewed as a Radon measure. Does it have some other geometic properties?
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### Set dual with half-spaces

Let $X\subseteq \mathbb{R}^d,$ we define the set dual to $X$, denoted by $X^*$, as follows: $$X^*=\{y\in \mathbb{R}^d \mid \langle y,x\rangle\leq 1, \forall x\in X\}.$$ Geometrically, $X^*$ is the ...
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### Are the Products of Positive Semi-Definite Matrices Always Semi-Definite? [closed]

Is $AT'KTATK^2T + AT'K^2TATKT$ semi-definite when $K$ and $K^2$ are positive? All the entries in A,T,K are real and positive, I know the the product of positive semi-definite matrices are not ...
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### Number of cells in hyperplanes [closed]

I follow up from this question1, question2.. An arrangements of hyperplanes in $\mathbb{R}^d$ is simple if the hyperplanes are in general position (for every $1\leq k\leq d+1,$ the intersection of k ...
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### Degrees of freedom in mirror descent

The mirror descent algorithm makes certain choices which appear somewhat arbitrary, and so for each of them I am hoping for an understanding (maybe a proof) of why they are necessary, or at least some ...
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### Which condition implies that $f(s)/s$ is increasing?

I'm trying to find a class of functions $f$ that has this property: $f(s)/s$ is increasing in some interval $(0, \lambda)$. The function $f(s)= s^p$ with $p>1$ has this property, for example. My ...
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### Curvature measure for convex functions in $\mathbb R^d$
TLDR: convex functions in high dimensions are much weirder than in 1D: do you please have insights or references to share? Let $c(x)$ be a convex function on the real line. There is an obvious ...
How to efficiently solve: \begin{align*} \arg \min_{\boldsymbol{X}} \quad & \frac{1}{2} {\left\| \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} \\ \text{subject to} \quad & \begin{aligned}...