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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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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1answer
45 views

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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50 views

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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1answer
40 views

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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203 views

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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93 views

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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45 views

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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59 views

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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161 views

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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1answer
74 views

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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123 views

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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25 views

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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56 views

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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30 views

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 ...
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67 views

Pointed Convex cone: one-to-one correspondence extreme rays - extreme points

Hoi, let $V$ be a finite dimensional real vector space with inner product $\left\langle .\right\rangle$. Let $\Gamma\subset V$ and $\Gamma \neq \left\{0\right\}$ a pointed convex cone. (Pointed means ...
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174 views

Prove that if $A, B $ are convex , $B$ is closed, $C$ is bounded and $A+C \subset B+C$ then $A\subset B$

Given the sets $A,B,C \in \mathbb{R}^n$ such that: $$A+C \subset B+C$$ Show that if $A,B$ are convex, $B$ is closed and $C$ is bounded then $A\subset B$. I kind of understand the geometrical ...
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103 views

Trace Constraints and rank-one positive semi-definite matrices.

Let $C_1$,$C_2$...$C_N$ be $M\times M$ hermitian matrices and $c$ be a given positive constant. Let $W$ be a positive (possibly semi) definite matrix such that \begin{align} \text{trace}\{WC_1\}\geq ...
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171 views

Are these convex optimization problems equivalent?

Consider the optimization problem $$ \mathcal{P}_1: \qquad \min_{x \in \mathbb{R}^n} c^\top x \quad \text{sub. to } \ g(x,y_i) \leq 0 \ \ \forall i = 1,2,...,M$$ where $c \in \mathbb{R}^n$, and ...
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1answer
65 views

A property of the minima of a sum of convex functions, take 2

This is a follow-up to my previous question. Let $g_1(x), \ldots, g_k(x)$ be convex functions from $\mathbb{R}^n$ to $\mathbb{R}$, and lets assume that global minimum of each $g_i$ is unique and is ...
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406 views

A property of the minimum of a sum of convex functions

Let $g_1(x), \ldots, g_k(x)$ be convex functions from $\mathbb{R}^n$ to $\mathbb{R}$, and lets assume that global minimum of each $g_i$ is unique and is achieved, denoting $$x_i = \arg \min_{x \in ...
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Distribution of convex combination of i.i.d Gamma random variables

I am wondering what one can say regarding the convex combination of i.i.d Gamma random variables? Specifically, consider $x_{i}$ be $Gamma(\theta,1)$, then would we have the following and if yes, ...
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convexity of the product of two entropy-like functions

Consider the functions $T_p(q)= \sum_i q_i^p$, where p>1 and q is a finite-dimensional vector satisfying $\sum_i q_i = 1, q_i >0$ (ie, a probability mass function). In information-theoretic terms, ...
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Extensions of Carathéodory's theorem

We know about the Carathéodory's theorem which is on the convex bodies of $\mathbb{R}^d$. My question is, how far we can extend it? Is it true for say, any convex object of Banach space, or for convex ...
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585 views

generalized inequalities defined by proper cones

The generalized inequality defined by a proper cone $K$ is that $x \ge_{K} y$ if $x-y \in K$ for $x,y \in K$. Does this means that for any $x \in K$, we have $x \ge_{K} 0$ since $x - 0 = x \in K$ ? ...
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Is there a common name for the property $f(\lambda x+(1-\lambda)y)\leq f(x)+f(y)$?

For $X$ a vector space and $f:X\to\mathbb{R}$, I'm considering the property $$f(\lambda x+(1-\lambda)y)\leq f(x)+f(y) \qquad\forall x,y\in X, \;\forall\lambda\in[0,1].$$ So it's like convexity ...
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Is it a convex function?

Let $f(.)$ be a function. If $f(X)$ is a convex function of $X$, where $X$ is a matrix. Is $f(AXB)$ also a convex function of $X$? ($A$ and $B$ are fixed matrices).
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188 views

Prove the concavity of $F(x,y) =\ln(x)+y$ by arguing the definition of concavity.

I need help with this problem. Prove the concavity of $F(x,y) = \ln(x) + y$ by arguing the definition of concavity. A function $f$ is concave is for any $x_0, x_1 \in \mathbb{R}^2$ and $t \in ...
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87 views

Convexity of product of elements from two convex set

Given two convex set $X\subseteq \mathbb{R}^N$,$Y\subseteq \mathbb{R}^{N\times N}$ Given a $x\in X$, is the set $\{z|z=yx,\forall y \in Y\}$ convex? If no, by adding what can force it to be convex? ...
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What is the name for functions with this property similar to concavity?

What's the name of the class of functions satisfying the following property: Given some convex space $D\subseteq \mathbb{R}^k$, a function $f:D \to \mathbb{R}$ is called $\mathbf{(?)}$ if for all ...
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184 views

Hessian of a function that takes matrix arguments

I have a function that that takes a matrix and returns a scalar, $f : \mathbb{R}^{m\times n} \rightarrow \mathbb{R}$. I know how to calculate the derivative of this function with respect to the matrix ...
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27 views

$\sup\{t_{1}f_{n}(x_{1})+t_{2}f_{n}(x_{2})\mid n\geq n_{0}\}=\sup\{t_{1}f_{n}(x_{1})\mid n\geq n_{0}\}+\sup\{t_{2}f_{n}(x_{2})\mid n\geq n_{0}\} $

I was thinking, if this is correct: Let $f_n$ is a series of convex, limited function $I \rightarrow \mathbb{R}$ $t_1, t_2 \in \mathbb{R} \ \ \ \ \ t_1 + t_2 = 1$ Is that a true statement : ...
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Some questions on Minkowski's functional

I'm reading Wikipedia's article on Minkowski's functional. They state that if the set K used in defining Minkowski's functional pK is convex then pK is sub-additive. They argue as follows: Suppose ...
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Need $f:[0,\infty)\to[0,\infty)$ such that $f$ is not convex but $f(x^p)$ is for $p>1$.

To be more specific I need to find an $f:[0,+\infty)\to[0,+\infty)$ which satisfies the following: (somewhat trivial stuff) The function $f$ is continuous, nondecreasing, there exists $k>0$ such ...
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71 views

Is the correlation function convex or not?

Suppose the function for statistical correlation is a non linear constraint in a non linear programming model: $$ \frac{\sum_{t=1}^T (p_t - \bar{p})(R_t - \bar{R})}{\sqrt{\sum_{t=1}^T (p_t - ...
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1answer
68 views

Is there a notion of 'pre-convex' sets?

A set $A$ in $\mathbb{R}^n$ is convex if for any two points $p,q \in A$ and real $\lambda \in [0,1]$, the point $\lambda p + (1-\lambda)q$ is also in $A$. There are many beautiful theorems about ...
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62 views

Convex hull area of projected points are convex respect to rotations?

Let $A$ be a finite list of points in $\mathbb{R}^3$ and $c$ the centroid of $A$. Let $P$ be an orthographic projection onto a plane in $\mathbb{R}^2$ and $h$ be the convex hull of $P(A)$. Let further ...
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216 views

Is this sum of convex and concave functions a convex function?

Is this a convex function in $X$, where all the entries are real and $Y,\beta$ are constants where $X,Y$ are rectangular matrices and $\beta$ is a constant vector and $A,B$ are constant p.s.d ...
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1answer
147 views

Strictly Convex Function and Well-Separated Minimum

Suppose $\Theta \subset \mathbb{R}^d$ is a convex set, and $f:\Theta \rightarrow \mathbb{R}$ is a strictly convex function that has a minimum at $\theta_0\in\Theta$. Is it true then that $\forall ...
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1answer
55 views

Distributed Robust Optimization

Consider the following constrained optimization problem $\mathcal{P}$. $$ \min_{x \in X \subseteq \mathbb{R}^n} f(x) \ \text{sub. to: } g(x,y) \leq 0 \ \forall y \in Y \subseteq \mathbb{R}^m $$ ...
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1answer
76 views

Distance between a point to a $2d$ ellipse in $3d$ ambient space

Suppose we are working in the 3D Euclidean space. We are given an arbitrary point $p$ and a 2d ellipse: $$E=\{x:x^TQx\leq1,x^Tq=0\},$$ where $Q$ is a positive definite matrix and $q$ is an ...
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1answer
124 views

Is this matrix function convex or non-convex?

Given, $g(Z)=Tr(Z^Tf(Z)Z)$ , where $f(Z)=h(Z)-ZZ^T$ is a p.s.d matrix formed using entries in $Z$, where again $h(Z)$ is a diagonal matrix with its $i$'th diagonal entry being ...
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1answer
45 views

Achieving equality in the definition of the support function

Suppose that $B$ is a convex body (compact closed with nonempty interior), and let $$h_B(u) = \sup_{x \in B} \langle u,x\rangle$$ be its support function. Is there a nice description of the set $E ...