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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Proving a function is convex

From the Defintion of convex: Theorem to be proven: If $f$ is differentiable and $f'$ is increasing, then $f$ is convex. Use Proof by Contradiction. Consider, $I = (a, b)$ with $a < x < ...
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How to proof that a straight line can split a convex to at most two regions?

I am self-studying the book "Concrete Mathematics". The authors state the statement: "A straight line can split a convex region into at most two new regions, which will also be convex" 1) How can one ...
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Multivariate Normal Density Concavity

For this variance compunent model $Y$~$N(X\beta, \Omega)$, where $\Omega=\sum_{i=1}^m\sigma_i^2V_i$, the log likelihood function is $(\beta, \sigma_1^1, ..., \sigma_m^2)=C+\frac12\log ...
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Convexity and equality in Jensen inequality

Theorem 3.3 from W. Rudin, Real and complex analysis, says: Let $\mu$ be a probabilistic measure on a $\sigma$-algebra of subsets of a given set $\Omega$. If a function $f:X \rightarrow \mathbb R$ ...
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Does having positive second derivative at a point imply convexity in some neighborhood?

Suppose that I have a real valued function of a single variable $f(x)$ which is twice differentiable in some open interval $I$. Then, I know from calculus that if $f''(x) >0 $ on $I$, then $f$ is ...
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is this a convex optimization problem?

Can someone clarify is this a convex optimization problem or not. $min \| X-UV\|_{F}\quad $ s.t $ \quad U \geq ,V\geq0$ . If not , what makes the problem non-convex?
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Elements of a Convex Set

Let $ S \subset \mathbb R^n $ be a convex set. Given $ \vec x, \vec y, \vec z \in S $ and three positive numbers such that $ a+b+c=1 $, show that $a\vec x+b\vec y+c\vec z$ is in $S$ also. Ok, so, I ...
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Proof sketch for a convex function, help. [duplicate]

Assume that $f(x)$ has two derivatives in $(0,2)$ and $0<a<b<a+b<2$. Prove that if $f(a)\ge f(a+b)$ and $f″(x)\le 0$ $\forall x \in (0, 2)$, then: $$\frac{af(a)+bf(b)}{a+b} \ge f(a+b) ...
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Convexity of a certain function connected to the norm

Suppose that we are given two vectors $x,y$ in a normed space $X$. Can we prove in general that the function $$t\mapsto \|x-ty\|$$ is convex? It is certainly the case if the normed space has ...
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Non-linear analysis

1) I am looking for a book which would give the proof of the following theorem(see below). I didn't find any book who does it: in infinite dimension (in Rockafellar Convex analysis book we are in ...
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Preimage of Legendre-Fenchel transform

Let $X$ be a Banach space with dual $X'$, and let $f : X'\to (-\infty,+\infty]$ be a convex lower semicontinuous function. Does there exist some characterization or some nontrivial results concerning ...
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When are the extreme points of a set the bondary?

Let $X$ be a convex compact set. When is the set of extreme points equal to the boundary of $X$? NOTE: by boundary I mean $\overline{X} \setminus \mbox{Int}(X)$.
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sup is bounded or not?

The sup is as following: $c_f = sup_{x,s\in D} \ f(y) - f(x) - (y-x)^Tb$ where $y=x+\alpha(s-x)$, $\alpha \in (0,1 )$ is constant and $b$ is a constant vector. $D$ is a convex compact set and ...
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Proof sketch for a convex function, help.

Assume that $f(x)$ has two derivatives in $(0,2)$ and $0<a<b<a+b<2$. Prove that if $f(a)\ge f(a+b)$ and $f″(x)\le 0$ $\forall x \in (0, 2)$, then: $$\frac{af(a)+bf(b)}{a+b} \ge f(a+b) ...
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Proving a function of matrix is convex

I have a function of a matrix and a vector $f(A,b)=y^\top (I-A)^{-1} b$ and I want to know the conditions under which it is convex. For functions of a vector, the positive definiteness of the Hessian ...
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Term for a Convex Function whose derivative is also convex

Let $f(x)$ be a monotone non-decreasing convex function such that its derivative $\frac{d}{dx}f(x) = f'(x)$ is also a convex function. Is there a term in literature that is used to refer to such ...
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Confusion on convex functions

I got a problem while solving a problem regarding convex functions on an interval $(a,b)$. What I had to show is if $f$ is convex then $f'$ exists except possibly at countably many points and is ...
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uniform convergences of convex functions

Let $f_n(\cdot)$ be a sequence of continuous and convex function on $\mathbb{R}^d$, and be supported on a full dimensional compact convex set $D$. If $f_n(\cdot)$ converges point-wise to $f$ in the ...
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Prove that the Cartesian Product of two Convex Sets is a Convex Subset

Here's the problem: Suppose that $S\subset \mathbb R^m$ is a convex set and $T\subset \mathbb R^n$ is a convex set. Show that the set $$S \times T = \{ (x_1 ,...,x_{m+n}\in \mathbb R^{m+n}):(x_1 ...
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Proving Convexity of an Open Disk

I need to prove that the following set is convex: $$ \{(x,y):x^2 +y^2 \lt 2\} $$ Obviously, this an open disk of radius $\sqrt2$. My intuition is to use triangle inequality for this proof because a ...
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How to derive the solution in quadratic optimization

I'm reading the book "Convex Analysis and Optimization" written by Prof. Bertsekas. In Example 2.2.1, there are the following description: I don't know how to ...
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Example of maximal monotone operators in non-reflexive Banach spaces with applications in PDE

My question is about examples of maximal monotone operators that are defined in non-reflexive Banach spaces and have applications in PDEs, variational inequalities, etc (any application actually)? If ...
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Proving disjoint sets and finding separating hyperplane

Consider \begin{equation} \mathbf{h}=\begin{bmatrix} h_0 & h_1 & \cdots & h_p \end{bmatrix} \end{equation} where $h_i \in \mathbb{R}\forall i=1:p$ and is known. \begin{equation} ...
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Projection on Epigraph of a convex function

Given a convex function $h:\mathbb{R}^n \mapsto \mathbb{R}$, and a point $(x,\alpha) \in \mathbb{R}^n \times \mathbb{R}$, how can I find a closed formula to compute the projection of $(x,\alpha)$ in ...
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Is the norm of a convex function convex?

I know that the norm of $x\in R^n$, $(\sum\limits_{i=1}^n|x_i|^2)^{0.5}$ is a convex function. Also, not any composition of two convex functions is convex. So my question is: Lets say we have a real ...
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Why is a local min also a global min for convex functions?

As the title states, for an unconstrained minimizaton problem, of a convex function, why is it that the local minimum is also the global solution?
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the continuity of argmin on convex funtion

Define $$x'=\text{argmin}_{x_1}f(x_1,\lambda),$$ where $f$ is a strictly convex function on $x_1$ and $\lambda$. I would like to ask if there is any theorem about the continuity of $x'$ w.r.t ...
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Lotka-Volterra First Integral and Fixed Point

I have the following problem that I am dealing with, quite a long time, I must say. Let us assume that we have a predator-prey, Lotka-Volterra system given to us by: \begin{align} & ...
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intuitive meaning of sphericity

i interested in the following definition but i don't understand it because i don't understand what mean by "flat space generated by C" . the same definition is given by i have also the same ...
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How to prove that convex function has an increasing slope?

A function $f(x)$ in some domain $a\leq x \leq b$ is convex if and only if for any $x_1 < x_2 < x_3$ from domain $[a,b]$, $$\frac{(f(x_2)-f(x_1))}{(x_2-x_1)} \leq ...
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Singularities in convex functions of more than one variable

I've heard that a convex function of a single variable is continuous in the interior of its domain, and is differentiable everywhere with the possible exception of a countable number of points. (I ...
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Question composition rules in standard deviation

Standard deviation is defined as following: $f(x) = (\frac{1}{n} \sum_{i=1}^nx_i^2-(\frac{1}{n} \sum_{i=1}^nx_i)^2)^{1/2}$ Obviously, $(.)^{1/2}$ is not convex, so can I say $f(x)$ is not ...
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convex function - global minimum

Suppose that $f(x):R^p \rightarrow R$ is a convex function with global minimum, say 0. Let $C=(x: f(x)=0)$, i.e. the set of the global minimum. Suppose that there exist at least one point $y$ such ...
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Does $\log(f(X))$ concave implies $\log(f(X^{-1}))$ convex?

One of my professor claims that $\log f(X)$ concave implies that $\log(f(X^{-1}))$ convex where $X$ is symmetric positive definite matrix. $\log(f(X))$ is a function defined on symmetric positive ...
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Subspace of tangent feasible directions

If $h: \mathbb{R}^3 \to \mathbb{R}^3$ defined by $h(x_1,x_2,x_3) = (x_1^3 - x_2 + x_3^2, x_2,x_1+x_2+x_3)$ I define $V(x)$ as a subspace formed by all directions tangent to some constrained set at ...
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Subdifferential of a convex function

How would I find a convex function $f: \mathbb{R} \to \mathbb{R}$ such that $\partial f(0) = [0,1]$ A subdifferential is just the collection of vectors $w \in \mathbb{R}^n$ such that $f(y) \geq ...
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Why is this set a subset of its polyhedral approximation - contradicting the gradient inequality?

Say we have a set $C:= \{y\in \mathbb{R}^n : g_i(y) \leq 0, \space i=1,...,m\}$ where $g_i : \mathbb{R}^n \to \mathbb{R}$ are convex and differentiable functions, then we have $\tilde C : = \{y: ...
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Convexity Proof with constraints on the gradient

Consider a minimization problem $(P)$ : minimize $f(x)$ subject to $\delta_C(x) \leq 0$ Now assume that $\emptyset \neq C \subset \mathbb{R}^n$ is convex and let $f: \mathbb{R}^n \to \mathbb{R}$ be ...
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Show the following statements are equivalent - convexity

Let $C \subset \mathbb{R}^n$ be a set. Show the following are equivalent: (a) The set $C$ is convex. (b) The function $\delta_C : \mathbb{R}^n \to \mathbb{R} \cup \infty$ defined as: ...
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About the alternating optimization

The problem is defined as follows: $$ min_{A,B,C} f(A,B,C) $$ and the problem couldn't solve by gradient descent or close-form solution. Thus, the usual way is to use the alternating optimization: ...
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Minimum volume covering ellipse

Given a convex polygon in the plane, consider the smallest-area ellipse which contains this polygon. This is the "minimal volume covering ellipsoid" or "minimal volume enclosing ellipsoid" or in short ...
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Convexity proof - can I get some pointers?

Prove that $C \subset \mathbb{R}^n$ is convex iff $\forall m \in \mathbb{N}$ and every set of $m$ points $\{x_1,...,x_m\} \subset C$ we have that $\sum_{i=1}^m \lambda_i x_i \in C$ Where ...
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Steepest Descent Sequence

How can I compute the first three iterates for the steepest descent sequence $f(x_1,x_2) = \frac{(x_1^2+3x_2^2)}{2}$ beginning at $x_0 = (\frac{\sqrt{3}}{2}, \frac{1}{2 \sqrt{3}})^T$ $\nabla ...
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Constructing a newton sequence

How may I construct the newton sequence for the following: $(1) f(x_1,x_2) = x_1^4 + 2x_1^2x_2^2 + x_2^4$ with $x_0 = (1,1)$ and $x_0 = (1,0)$ $(2) f(t) = t^4 - 32t^2$ and $t_0 = 1$ To find ...
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Different ways to prove convexity of quadratic form associated to rank 1 matrix

Let $v \in \Bbb R^n$, and $f:\Bbb R^n \to \Bbb R^n$ with $f(x)=\langle x,(vv^T)x\rangle$. Show that $f$ is convex. I'm looking for different approaches to solve this (rather simple) problem. ...
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A property similar to the convexity, but actually different?

If we know there exists a point $x_0$, such that $f(x_0)$ lies below the tangent line at $0$ of $f(x)$, which means below the line $y=f(0)+f'(0)x$. My question is how to prove this violate $f(0)\le ...
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Is the function $f(A)=-\log(tr(A^{-1}))-\log(\det(A))$ convex?

I am trying to show the following function is convex or not $$f(A)=-\log(\text{trace}(A^{-1}))-\log(\det(A)),$$ where $ A$ is positive definite. I know $\text{trace}(A^{-1}), -\log(\cdot)$ and ...
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Property of probability density function (pdf)

If $X$ is a random variable with a log-concave pdf. And suppose $Z = h(X)$ If $h(X)$ is convex, can we say $Z$ has a log-concave pdf? If $h(X)$ is affine, can we say $Z$ has a log-concave pdf? ...
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Legendre Transform of this function

Is the Legendre-Fenchel transform of $$f(x)=1-\sqrt{1-|x|^2}, x\in B(0,1)\subset\mathbb{R}^n$$ just $$f^*(x^*)=-1+\sqrt{1+|x^*|^2}?$$ I calculated this using the table here ...
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Chebyshev sets in finite dimension are closed and convex

Prove a finite-dimensional converse to the “best approximation theorem”: Let $K$ be a subset of a finite-dimensional Hilbert space $H$ which satisfies the following property: for each $x \in H$ there ...