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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Convex functions on real vector spaces

So I'm trying to solve the following problem, Suppose that $f$ is a non-zero convex real-valued function on a real vector space $V$ with $f(0) = 0$ Show that there is a linear functional $g$ on $V$ ...
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indirectly convex

Let g:$\mathbb{R^2}$$\rightarrow$$\mathbb{R}$ be defined by g(x)=Max{$x_1$,$x_2$} at each x=($x_1$,$x_2$)$\in$ $\mathbb{R}$. Determine whether or not g is indirectly convex on $\mathbb{R^2}$. ...
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Example on Correspondences

Giva an example of correspondence $F : \mathbb{R} \rightarrow \mathbb{R}$ such that the closure of $F$ is $ \overline{F}: \mathbb{R} \rightarrow \mathbb{R}$, upper semi continuous on $\mathbb{R}$, ...
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Finding convex conjugate of a bounded function

The convex conjugate of a function $f:\mathcal{X}\mapsto \mathbb{R}$ is formally defined as $$f^\star\left(y\right)=\sup_{x\in\mathcal{X}}\ \left\langle x,y\right\rangle-f\left(x\right).$$ In cases ...
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Linear Difference Equations

Let $g_1, g_2 \in \mathbb{R}^{[0,5]}$ be defined by $g_1(x) = x$ and $g_2(x) = 1-x$ for each $x \in [0,5]$. Find a second-order homogeneous linear difference equation on $[0,3]$ such that {$g_1, g_2 ...
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Indirect Concavity

Let $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined by $g(x_1,x_2) = e^{\min{(x_1,x_2)}}$ at each $ x_1, x_2 \in \mathbb{R}^2$. Find whether or not $g$ is indirect concave on $\mathbb{R}$.
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How can i show this inequality?

Let $n>1$ and $a_1,...,a_n \in \mathbb{R}^+$ be such that $\sum a_i=1$. For evey $i$, define $b_i=\sum_{j=1,j\neq i}a_j$. Show that $\sum_{k=1}^n \dfrac{a_k}{1+b_k}\ge \dfrac{n}{2n-1}$ Thanks a ...
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convexity of norm

I want to show that $f(v)=\|v\|^p$ for $1\leq p<\infty$ is strictly convex. In the simplest case when $p=2$, we have: ...
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Does convex and radially open imply open?

I want to show that a convex set $A$ is radially open iff $A\cap W$ is open in W, for every finite dimensional linear subspace. Here the 'openness' we are talking about is from any normed space. ...
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secant method twice on convex decreasing function

I have a continuous, decreasing and convex function $f$. Given an interval $[a, b]$ such that $f (a)>0 $and $f (b)<0$, if I apply the secant method twice, where the outcome point will be ...
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Suggestions for a reference-level text on optimization theory?

I'd be interested in knowing if anybody has suggestions on an advanced but still self-contained reference on optimization theory, centered around linear and convex problems. The key feature of my ...
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246 views

Maximum of quasi-convex functions

A function $f$ is quasiconvex if all its sub-level sets are convex (i.e., $\{ x: f(x) \le \alpha\}$ is convex for all $\alpha$.) For a convex function $f$, it is true that $f$ acheives its maximum ...
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How to prove the property of convex function in higher dimension

Suppose $f:\mathbb{R}^n\mapsto \mathbb{R}$ is convex, could anyone tell me how to prove the following fact? (1) If $f\in C^1$, then for any $u,v$ $$ f(v)\geqslant \langle\nabla f(u),v-u\rangle $$ ...
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On a version of gradient descent

I am trying to read this paper and have gotten stuck. The author considers the problem of minimizing a convex function whose gradient has coordinate-wise Lipschitz constant $M$ (meaning that for all ...
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KKT conditions of this convex optimization problem

Consider a continuous function $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ such that $\forall y \in \mathbb{R}^m$ $f(\cdot,y)$ is convex, and $\forall x \in \mathbb{R}^n$ $f(x,\cdot)$ ...
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Convex analysis: relative interior in finite and infinite dimension

Let $X$ be a normed space. Given a nonempty convex set $C \subset X$, the relative interior of $C$, denoted by $\text{ri} C$, is the set of the interior points of $C$, considered as a subset of ...
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Convexity of polylogarithms

I want to prove the following proposition: The function $w\to (-Li_{5/2}(-e^w))^{2/5}$ is convex on $\mathbb R$. And, as I think, the same is true for the function $w\to (-Li_{p}(-e^w))^{1/p}$ for ...
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Space of solutions to $f(x+y) = f(x) + f(y)$ when $f$ is convex

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a nonlinear convex function, and let $x\in\mathbb{R}^n$ be an arbitrary vector. Define the set $\Omega_f(x)$ as $\Omega_f(x) \triangleq ...
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Where the gradient of a convex function approaches zero

Does there exists a differentiable convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ with a unique global minimum which we denote by $x^*$ such that there exists a sequence $x_k$, not ...
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Find function with given properties

Find a smooth function $g: \mathbb{R} \to \mathbb{R}$ that domain $g$ is $\mathbb{R}$ range of $g$ is a subset of $\mathbb{R^+}$ $g$ is concave.
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Are there logarithm functions for arbitrary rings?

The logarithm function for $\mathbb{R}$ satisfies $\log xy = \log x + \log y$ whenever both $\log x$ and $\log y$ are defined. Are their conditions for a ring $R$ which guarantee the existence of a ...
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Jensen's inequality and $L^p$ norms

Let $(X,\Sigma,\mu)$ be a probability space; in particular, $\mu(X)=1$. The integral form of Jensen's inequality can be phrased in terms of permuting a convex function $\varphi$ (say, with the ...
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A robust convex optimization problem

Consider a function $ f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R} $ such that $\forall x \in \mathbb{R}^n$ the map $f(x,\cdot)$ is convex, and $\forall y\in \mathbb{R}^m$ the map ...
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How to prove that this function is convex

My problem is that: The domain is $\mathbb{R} ^n _{++}$ . I need to prove that $f(x_1,...,x_n)=\sum_{i=1}^{n}x_i\cdot ln(x_i) -(\sum_{i=1}^{n}x_i)\cdot ln(\sum_{i=1}^{n}x_i) $ is convex. I tried to ...
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Every exposed point is a extreme point

Let $C$ be a non-empty convex subset of $\mathbb{R}^n$. We say that $x\in C$ is a extreme point of $C$ if for every $z,y\in C$ and $t\in [0,1]$ such that $x=ty+(1-t)z$ we have $x=z$ or $x=y$. Or ...
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Special type of convexity

Does anyone know if there is a name for a function $f:W\rightarrow\mathbb{R}$ ($W$ is a vector space) that satisfies $$ f\left(\theta x + \left(1-\theta\right) y\right) \leq \theta^\alpha ...
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47 views

Relation about Gateaux differentiable and differentiable

Assume there is a multivariable function $f:\mathbb{R}^n\to\mathbb{R}$. I want to know the relation between Gateaux differentiable and differentiable. That is, if $f$ is differentiable, is it Gateaux ...
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Extreme points and positive linear combinations

Let $C$ be a non-empty convex subset of $\mathbb{R}^n$. We say that $x\in C$ is a extreme point of $C$ if for every $z,y\in C$ and $t\in [0,1]$ such that $x=ty+(1-t)z$ we have $x=z$ or $x=y$. Or ...
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Using $AM \ge GM$ which is greater among $1+\dfrac{1}{n}$ and $2^{1/n}$.

From expansion I see that $1+\dfrac{1}{n} \ge2^{1/n}$.But can't solve it using $AM \ge GM$. Please help.
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Lipschitz functions in $\mathbb{R}^n, \ \ \mathbb{R}^m$, extension

I've found the following lemma : Let $\{x_1, . . . , x_k\}$ be a finite collection of points in $\mathbb{R}^n$ , and let $\{y_1, . . . , y_k\}$ be a collection of points in $\mathbb{R}^m$, such that ...
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Prove that for every $f$ in $H$, the sequence $u_n$ which is the projection of $f$ on $K_n$ converges to a limit

$K_n$ is non-increasing sequence of closed convex sets in Hilbert space $H$ such that the intersection of $K_n$ is different from emptiness. Prove that for every $f \in H$, the sequence $u_n$ which is ...
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KKT and Slater's condition

I was studying Stephen Boyd's text book and got confused in the KKT part. The book says the following: "For any convex optimization problem with differentiable objective and constraint function, any ...
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Does the Border (Boundary) Points of a convex shape in the positive quadrant make a convex function?

Let $\mathbb{S}$ be a convex body in 2-D with some non-zero intersection with positive quadrant and let it also contain origin. Let $c>0$ be the right-most point on the x-axis such that $(c,y)\in ...
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Does convexity of a function guarantee tractability of finding its minimum?

Formulating a problem as a convex optimization problem usually implicitly considered to imply being able to find global minimizer of the objective. My question is that if it is true or not. ...
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Solution of a Quadratic Optimization Problem

Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two given $N\times N$ hermitian matrices. Then how do I solve the problem, \begin{align} ...
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Robust feasibility with halfspace?

Consider a convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, such that for all $x \in \mathbb{R}^n$ we have $$ a_1^\top x + b_1 \leq f(x) \leq a_2^\top x + b_2 $$ for some given $a_1, a_2 ...
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strict convexity with a measure theoretic property

Suppose $(x_n)$ is a positive sequence of reals converging to $x$. Furthermore we have a measure space $(E,\mathcal{E},\mu)$ given, with finite measure $\mu$. There are a measurables nonnegative and ...
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About Balanced-Convex Hull of a Set

Definition. $\newcommand{\bco}{\operatorname{bco}}$Let $X$ be a real vector space and let $A\subseteq X$. The balanced-convex hull of $A$, deonoted $\bco A$, is the intersection of all balanced-convex ...
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Explain about convexity in geometry and in optimization.

My question is 'what is a difference between convexity in geometry and optimization?'
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Given some points in the Euclidean space, find a plane satisfying some restrictions

In a 3-D Cartesian coordinates, suppose we are given $n$ points and their coordinate values in the form $(x,y,z)$. Obviously there are uncountable planes which divide those points into two groups, ...
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How to prove that is a cone

I have to prove that the set $$K=\lbrace u\in C[0,1]\mid u(t)\geq a(t)||u|| \text{ on } [0,1]\rbrace,$$ where $a(t)=\displaystyle\frac{t(2p-t)}{p^2}$ with $\frac12<p<1$, and ...
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Show that the maximum of a set of convex functions is again convex

Let $f_1(x), f_2(x), \ldots, f_n(x)$ be a set of convex functions. We define $f(x)$ as $$ f(x) = \underset{i}{\text{max}} \left\{ f_i(x) \right\}. $$ How do I show that $f(x)$ is also convex, and ...
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Proving function is convex

How do you show that $c + max(0,1-x)^{2}$ is convex where $c$ is a constant? I can graph it and observe that the function is below any line segment between any two points but I am not sure how to ...
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Epigraph of a function f: D $\rightarrow$ R is convex iff epif(f) is a subset of D*R which is a convex set

As in the topic, how to show that $epi(f)$ is convex iff $epi(f)$ belongs to D*R which is a convex set.
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Can one define the derivative of a function using tangent cones? Does such a notion already exist?

I'm interested in finding an analogue of a derivative that applies to functions which are defined more general subsets of $\mathbb{R}^n$ than open subsets. In particularly, I'm looking at functions ...
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Coding Distributions as a Convex Constraint

In convex optimization, how can we impose a constraint that a variable has certain distribution? e.g. elements of vector $v$ have power law distribution?
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Does $\inf_{(a,b)\in A} ax+by=\inf_{(a,b)\in B} ax+by$ imply $A=B$?

Let $A,B\subseteq\Bbb [0,1]\times [0,1]$, and for all $(x,y)\in \Bbb R^2$ with $x^2+y^2=1$, $$\inf_{(a,b)\in A} ax+by=\inf_{(a,b)\in B} ax+by$$ Can we deduce $\overline A=\overline B$.
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Convex cone question.

Hoi, let $V$ be finite dimensional real vector space with inner product $\left\langle . \right\rangle$ and let $\Gamma \neq \{0\}$ be a closed convex cone. Let $$\Gamma_0^{\perp}:=\{v\in ...
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Is permutation-invariance of an objective a problem in convex optimization?

I have difficulty understanding how permutation-invariance and convexity are related in an optimization problem. Let $f(.)$ be a convex function defined on $d$-dimensional vectors. Suppose, also that ...
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Find a point in a polytope that always cuts off a constant fraction of the polytope.

I have $k$ linear inequalities in $\mathbb{R}^n$, which we can express as $Ax \ge c$ (where $A \in \mathbb{R}^{k \times n}$ and $c \in \mathbb{R}^k$). Assume that the set of $x \in \mathbb{R}^n$ that ...