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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Approximate model of a convex/concave surface

I have a set of measurements in 3d that yields a concave surface of a function $f(x,y)$ that I don't know its expression. I am thinking to approximate the function to a model where any point from the ...
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Theorem 3.1 in Erwin Kreyszig's “Introductory Functional Analysis With Applications”: Is the notion of convex set valid in complex vector spaces?

Is the notion of convex sets valid and meaningful for complex vector spaces? Or, do we need to restrict ourselves to real vector spaces and normed spaces when we talk about convex sets? The ...
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Prove local minimum of a convex function is a global minumum (using only convexity)

I'm studying for a calculus exam, and have come across this question in the textbook which I have problem solving; Let $C\subseteq \mathbb{R}^d$ a convex set, and let $f:C\rightarrow \mathbb{R}$ ...
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tangent cone to the set

I'm supposed to solve this problem: Let us consider the set $M=\{(x, \sin{x}):x\in\mathbb{R}\}\cup\{\big(\cos(x)-1,x\big):x\in\mathbb{R}\}$ The question is to find the tangent cone to the set $M$ in ...
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Prove $f(\sum^k_{i=1} \alpha_i x_i) \leq \sum^k_{i=1} \alpha_i f(x_i) $ for a convex function f

I'm learning for an exam in calculus and have come across this question which I can't seem to prove: Let $C \subseteq V$ be a convex set. Let there be a function >$f:C\rightarrow \mathbb{R}$ a ...
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Proof that equation is non-convex function

I have a objective function as following $$E(\phi)=\int_{\Omega}(I(x)-m_1)^2H(\phi(x))dx+\int_{\Omega}(I(x)-m_2)^2(1-H(\phi(x)))dx+\int_{\Omega}|\nabla H(\phi(x)|dx$$ where $I$ is an image; $I: \Omega ...
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Difference of linear transformation of convex function

I'm trying to show that for constants $a,b > 0$, and a convex, continuously differentiable function $f$ with $f(0) = 0$ that $x_1 > x_2 > 0$ implies $f(-a-b x_1) - f(-b x_1) \geq f(-a-b x_2) ...
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What is the relation between $\nabla f(\mathbf x), \mathbf y - \mathbf x, \text{ and } f(\mathbf y) - f(\mathbf x)$?

The Problem: Let $f(\mathbf x)$ be a convex function on $\mathbb R^n$. Given two points $\mathbf x$ and $\mathbf y$, what is the relation between $\nabla f(\mathbf x), \mathbf y - \mathbf x, \text{ ...
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Defining functions for connected sets

Let $\Omega \subset \mathbb{R}^n$ an open, bounded and connected set with a $C^2$ boundary and a function $\rho \in C^2(\mathbb{R}^n)$ such that $$ \Omega = \{ x \in \mathbb{R}^n : \rho(x) < 0 ...
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Proving a half-plane is convex

Given a half plane in $\mathbb{R}^2$ described by the equation $2x-3y \leq6$, how would one go about proving this vector space "$S$" is convex? Clearly it is when graphed, but I'm a bit puzzled by how ...
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Understanding extremal Lipschitz functions

I am new to concept of extremal Lipschitz functions and I have several basic question I'm still unsure about. To fix notation let $(X,d)$ be a metric space, $Lip(X)$ Banach space of Lipschitz ...
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171 views

Improvement of an Inequality

It would nice if someone could help me with this problem. I am looking at an improvement to the classical Jensen's Inequality: $$\int_\limits{}^{} \phi(x) \mu \mathrm{d}x \geq ...
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2answers
117 views

Increasing concave function

Let $f:[0,1]\rightarrow\mathbb{R}$ be a concave function with $f(1)=\sup_{t\in[0,1]} f(t)$. Then $f$ is non-decreasing in $[0,1]$. Does someone know how to prove this?
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1answer
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Is the normalized duality mapping symmetric?

In the area of functional analysis, nonlinear operator theory, the normalized duality mapping on a Banach space $X$ is a set valued map from $J:X\rightarrow 2^{X^*}$ given by \begin{align} ...
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Linear programming in Hilbert spaces

Let $H$ be a real Hilbert space. Let $b,c\in H$, $P\subset H$ be a convex cone and a continuous linear mapping $A:H\rightarrow H$. Consider the following sets: $$ B:=\{(Ax, \langle c,x\rangle:x\in ...
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Product of linear and convex function

More specific, how many maxima are there for product of these two functions: $ f(x) = ax + b $, and $ a > 0 $ $ g(x) $ is (strongly) decreasing convex function, $ \lim_{x\rightarrow\infty} g(x) = ...
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Show that F can have at most two fixed points

Consider a function $F:\mathbb{R}^n\to\mathbb{R}^n$, where $F=(F_1,...,F_n)$. Suppose that $F$ is strictly quasi-concave, and that for all $i=1,...,n$ the function $F_i:\mathbb{R}^n\to\mathbb{R}$ is ...
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1answer
12 views

Checking the convexity of a parametric set

Let $r\in\mathbb{R}$ and $|v|\leq \frac{1}{2}$. Prove that $$ \{x\in[0,1]:\sqrt{x}+vx\leq r\} $$ is convex. Thank you for all kind help.
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Concavity for multivariate functions

What ways are there to prove that a function with more than 2 variable is concave?ٍ I know we can check that the associated Hessian matrix is negative (semi)-definite, but are there other ways?
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Upper and lower bound on Hessian

Let $x \mapsto f(x) \in \mathcal{C}^2$ be convex, i.e. $\forall x \in \mathbb{R}^n$, $\nabla^2f(x) \succeq 0$. Let $A \in \mathbb{R}^{m \times n}$ and suppose we have $M I_n \succeq \nabla^2f(x) + ...
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1answer
18 views

Differentiability of the composition of a Lipschitz, convex function and a power function

$f:\mathbb{R}^n\rightarrow \mathbb{R}$ is a positive, convex and Lipschitz function. Is the fuction $|f|^{2+\alpha}$, $\alpha>0$, twice continuously differentiable? How to prove it, or there is ...
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1answer
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Proof of convexity of a quadratic function

I have the next problem: If $f(x)$ is a quadratic function with n variables: $f(x) = 0.5$$\mathbf{x}^T$$A$$\mathbf{x}$$+$$\mathbf{b}$$^T$$\mathbf{x}$$+$$\mathbf{c}$ were $A$ is a symmetric matrix ...
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Minkowski Theorem

Theorem: Let $L$ be an $n$-dimensional lattice in $\mathbb R^n$ with fundamental domain $T$, and let $X$ be a bounded symmetric convex subset of $\mathbb R^n$. If $Vol(X)>2^nVol(T)$ then $X$ ...
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Finding the vertices of a convex set of matrices

I'm a little new here so wasn't sure if this was the right area. I've been trying to figure out how to generate a set of random $K \times N$ (for $K<N$) matrices that are subject to a several ...
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1answer
26 views

Square of a convex non-negative function is still convex

Let $f: \mathbb R \rightarrow \mathbb [0, \infty)$ be a convex function. If $f$ is twice-differentiable, then $$ (f^2)'' = (2ff')' = 2(f')^2 + 2f f'', $$ which is $\geq 0 $ since $f, f'' \geq 0.$ But ...
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Injectivity of a map on a non convex domain

Let $\Omega \subset \mathbb{R}^n$ open, bounded, and connected, a map $f \in C^1(\Omega)$ and $\alpha > 0$ such that $$ \langle \nabla f(x)\xi ; \xi \rangle \geq \alpha |\xi|^2,\quad \forall\, x ...
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Are not all neighborhoods of $0$ in a locally convex space absorbent?

A locally convex space (LCS) can be defined as a topological vector space (i.e. scalar product and sum are continuous) whose topology is generated by translation of a family of balanced and absorbent ...
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Is this expression for the sub-dimensional volume of a convex polytope correct?

Let $\mathbf{S}$ be an $m\times n$ real matrix, with $m\le n$. Let $\vec{a}$, $\vec{b}$ be two real $n$-vectors, such that $a_i < b_i$ for all $i$. Consider the system of equations: ...
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Example of convex subset (unbounded) with $\text {rec} (C) = {0}$

Example of convex subset (unbounded) with $\text {rec} (C) = {0}$ I've proved that for a bounded convex subset $C$ it always holds that $\text {rec} (C) = {0}$. However, now I'm looking for an ...
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Dual norm intuition

The dual of a norm $\|\cdot \|$ is defined as: $$\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$$ Could anybody give me an intuition of this concept? I know the definition, I am using it ...
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Number of local minimums and nonconvexity

I came across the following in my reading, and I like to know why this is true. "$\dots$ but, the fuction $F:\mathbb{R}^n \to \mathbb {R}$ is nonconvex since it has several local minima $\dots$" ...
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1answer
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Lower bound on Hessian, mean-value theorem

Let $x \mapsto f(x) \in \mathcal{C}^2$ be convex, i.e. $\forall x \in \mathbb{R}^n$, $\nabla^2f(x) \succeq 0$. Let $A \in \mathbb{R}^{m \times n}$ and suppose $\nabla^2f(x) + A^\top A \succ 0$. Is it ...
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Expectation of an increasing, bounded concave function of a non-negative random variable

Let $h:[0,\infty)\to [0,1)$ be a strictly increasing and strictly concave function. Let the argument of this function be a random variable $C$ with probability density function (pdf) $f_{C}(c)$ with ...
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What is condition for a convex polyhedron to be uniform?

A uniform polyhedron has all its vertices exactly lying on a spherical surface with a certain radius. Condition: A convex polyhedron will be uniform (i.e. all the vertices will exactly lie on a ...
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Find the edges of a polyhedron P.

Given the polyhedron $P = \{v \in \mathbb R^2 \mid Av \le b\}$ with $A = \begin{bmatrix} -1 & -1 \\ 2 & -1 \\ -1 & 2 \\ 1 & 2 \end{bmatrix}$ and $b = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 2 ...
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39 views

Curvature and convexity of a plane curve

Let $\mathbf{r}:[a,b]\to\mathbb{R}^2$ be a $C^2([a,b])$ regular curve. Is it true that $\mathbf{r}$ is convex if and only if its curvature $\kappa(t)\leq 0, \forall t\in [a,b]$ or $\kappa(t)\geq 0, ...
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Polytopes inside polytopes

I would like to share some facts about polytopes and how to check whether a polytope is a subset of some other polytope. I will provide some methods and I would like to know whether there exist other ...
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1answer
29 views

Definition of a fan of a polytope

In Fulton's book Introduction to Toric varieties (page 25), he says that: A rational convex polytope $K$ in $N_{\mathbb{R}}$ determines a fan $\Delta$ whose cones are the cones over proper faces ...
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Convex open neighborhood of compact convex subset

I'm stuck on what ought to be a straightforward topology problem. Say $X$ is a compact convex subset of a locally convex space (everything in sight is assumed Hausdorff). Say $Y\subseteq X$ is a ...
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Boundedness condition of Minkowski's Theorem

Statement: "Let L be a lattice in $R^n$ and $S\subset R^n$ be a convex, bounded set symmetric about the origin. If $Volume(S) > 2^ndet(L)$, then S contains a nonzero lattice vector. Moreover, if ...
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Weighted least squares with nuclear norm minimizaiton, how to optimize?

Nuclear norm minimization is very popularization and formulation is least squares term with nuclear norm term as following, $$\min\limits_{X} ...
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Proximal Operator of $\ell_{\infty,1}$ norm of a matrix

How can I calculate the proximal operator of mixed norm $\ell_{\infty,1}$ for any general matrix, $X\in R^{m\times n}$ i.e., $X^* = \arg \min_X ||X||_{\infty,1} + \frac{1}{2\tau} ||X-Y||_F^2$ where ...
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Gluing two strongly convex function

Definition: We call $f:\mathbb{R}^n\rightarrow \mathbb{R}$ a $\lambda$-strongly convex function iff for every $x,y\in \mathbb{R}^n$ and $t\in[0,1]$ it follows $$f(tx+(1−t)y)\leq ...
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Triangle Inequality Like Equation [closed]

If we are in $R^2$ and define $d(a,b)$ as the set of points between $a$ and $b$ we can create an equation like this: $$d(x,z) \subseteq d(x,y) \times d(y,z)$$ where the $\subseteq$ is the subset ...
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min max duality

I have been introduced min max duality. When i have a problem : $f:X \to R$ $(Primal) inf (f(x)):x\in A$. Let say that $A=\{x \in X: h(x)=0\}$. I can express my problem : $inf_X sup_Y L(x,y):= ...
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Is a convex salient cone necessarily contained in an open half-space?

A cone $C$ in $\Bbb R^n$ is said to be salient if it does not contain any pair of opposite nonzero vectors; that is, if and only if $C \cap (-C) \subset \{0\}$. Obviously, a cone $C$ such that that ...
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1answer
11 views

Discontinuous semiconcave functions

A function $u: \mathbb{R}^n \to \mathbb{R}$ is defined to be semiconcave if there is a positive constant $c$ such that for all $x,z$ $$ u(x-z) + u(x+z) - 2u(x) \leq c |z|^2. $$ Alternatively, one ...
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Relationships between affine closures and convex closures

Let $V$ denote a vector space. Then the following concepts make sense: affine subset of $V$ affine closure (affine "hull") of a subset of $V$ Suppose $V$ is in fact a real vector space. Then the ...
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About the Affine hull and Span.

I'm learning Linear Algebra and Convex Optimization simultaneously, I notice that the affine hull is, to some extent, analogous to the span, but when I read the lines "We define the affine dimension ...