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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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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387 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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562 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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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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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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27 views

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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173 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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185 views

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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70 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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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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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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213 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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137 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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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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71 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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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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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 ...
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128 views

Convex and concave functions

This maybe a silly question... So mercy me. Let $m,v:[0,S]\to \mathbb{R}$ be two Lebesgue integrable, monotone functions, say $m$ decreasing and $v$ increasing and set: $$M(s):=\int_0^s m(\sigma)\ ...
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Relationship between a convex function and a convex set

Here is an assertion I have read from these lecture notes: Let $f(x)$ be a convex function, then the set $I_\beta= \{f(x)\leq \beta\}$ is convex for every $\beta$ This is not hard to prove. we ...
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Is the convex hull of closed set in $R^{n}$ is closed?

Is convex hull of closed set in $R^{n}$ closed?
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Quasi-Convexity

Can I get the conclusion that the function of matrix $P$ and $Q$ \begin{equation} \mathrm{tr}\left( PQ\right) \end{equation} is a quasi-concave function for $P>0$, and $Q>0$? It is true for ...
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203 views

Projection operator property

Let $\pi_M(a)$(projection operator) be the closest point of $M$ from the point $a$ . How one can prove if $M$ is convex set of $\mathbb R^n$ then projection operator has this property? ...
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When finding root, does Newton's method fail if the function is non-differentiable?

According to wikipedia's description, the Newton's method finding a root presumes a differentiable function. Then, will it fail when encountering non-differentiable function? For example, can it find ...
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Is it problematic when using Newton Descent with discontinuous Hessian?

Is there any side effect when applying Newton Descent to a convex function whose Hessian is discontinuous?
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197 views

Fenchel conjugate of non smooth function

Is it valid to derive Fenchel conjugate for a non-smooth function? Checking its definition $f^*(y) = sup_{x \in \mathsf{dom}f} (y^Tx - f(x))$, I think this would be OK, but I'm not sure about that. ...
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maximise the function

$f(x,y,z)=Ae^{-(x+y)}+Be^{-(x+y+z)}-C$, In above function A,B and C are constants. $x,y$ and $z$ are dependent on each other too (As an example when x is changed y and z are changed too..like wise). ...
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Why is the negative entropy Lipschitz with respect to the $1$-norm (Over)?

Let $\left\|x \right\| = \sum_{i=1}^{i=n}\left|x^i\right|$ and $d\left(x\right)=\sum_{i=1}^{i=n}x^i\ln x^i$ where $x\in R^n $ and $ \sum_{i=1}^{i=n}x^i=1$ How to prove: For all $x, x'$, $$\left| ...
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Proving that $|x|^p,p \geq 1$ is convex

I want to show that $|x|^p,p \geq 1$ is convex, for this i have to prove the inequality $|(1-\lambda )x+\lambda y)|^p \leq (1-\lambda)|x|^p+\lambda |y|^p $ for $\lambda \in (0,1)$ Can anyone prove ...
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Convex optimization and linear programming please help! :)

How would I write the following as a standard form LP? Minimizing $\sum_{i=1}^n x_i + c\max(a_i-x_i)$ for $a_i \ge 0$ and what is the optimal value for when $c=n$ How to express minimize $\frac{1}{2} ...
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Some convex optimization questions

Is minimizing number of $\{{i : x_i \ne 0}\}$ subject to $Ax=b$ a convex problem? Why is it computationally hard? What is polar cone of $\{x \in \mathbb{R}^2:0\le x_1 \le x_2\}$? Are ...
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If both $f$ and $-f$ are convex functions, then $f$ is affine

Prove that if both $f$ and $-f$ are convex functions, then $f$ is affine My attempt If $f$ is convex, $f(\lambda y +(1-\lambda)x) \le \lambda f(y) + (1-\lambda)f(x)$ If $-f$ is convex, ...
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Projection: two closed convex sets

I am really struggling with this problem: $C$ and $D$ are closed, convex subsets of ${R}^n$ with non-empty intersection, i.e. $C \cap D \neq \emptyset $ . Is it true that projection $p_{C\cap ...
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Definition of direct products of two cones or of two convex subsets?

When reading a comment after this reply, I was wondering what the definitions of direct product of two cones? More generally, what is the direct product of two convex subsets? This case is what I ...
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Log-concave functions whose sums are still log-concave: possible to find a subset?

Rationale: I am puzzled by a problem of log-concavity, which arises in population dynamics where the curvature of the logarithm of sums is a quantity of interest. It is well-known that sums of ...
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How to understand convex duality intuitively

Is there an intuitive way to understand the convex duality? If the primal problem is minimization, the dual is maximization over another set of variables - but I would love to have a geometric ...
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cosh x inequality

While reading an article on Hoeffding's Inequality, I came across a curious inequality. Namely $$\cosh x \leq e^{x^2/2} \quad \forall x \in \mathbb{R}$$ I tried many ways to prove it and finally, ...
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How can I reformulate my problem to make it convex?

I would like to find the symmetric positive definite matrix $S\in \mathcal{M}_{m,m}$ that minimizes the function $f(S)=\mathrm{trace}(S)+m^2\mathrm{trace}(S^{-2})$ which has been proven to be convex ...
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A question regarding the convex envelope of a function

I know that by definition, the convex envelope of a function $f$ ($f$ not necessarily convex), denoted $\operatorname{conv}f$, is the largest convex function majorized by $f$. That is, it is a convex ...
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258 views

lsc function on compact set it attains its maximum minimum?

Is this true if so how to show it? if not true can you give a counter example: A lower semicontinuous function f on a compact set K attaings its minimum on K. A lower semicontinuous function f on a ...
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Continous map assuming positive value in the closure of a convex set

Let $\phi:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous map w.r.t. Euclidean norm. Let $C\subseteq\mathbb{R}^d$ be a convex subset. Assume that there exists $y\in\overline{C}$ such that ...
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Proof that the set of doubly-stochastic matrices forms a convex polytope?

Does the set of all doubly-stochastic matrices form a convex polytope? In general, I wonder how the proofs of convexity and geometry can be established for sets of matrices of this kind? Anything to ...
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145 views

Convexity of $\mathrm{trace}(S) + m^2\mathrm{trace}(S^{-2})$

I have the following function $f(S)=\mathrm{trace}(S)+m^2\mathrm{trace}(S^{-2})$ where $S\in \mathcal{M}_{m,m}$ symmetric positive definite matrix. I'm trying to prove the convexity of this function ...
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$\int_0^1 u(t)\phi''(t)dt \geq 0,\ \forall \phi\in C_0^1((0,1)), \ \phi\geq 0$. Is $u$ convex?

Suppose that $u\in C([0,1])\cap C^1((0,1))$ satisfies for all $\phi\in C_0^2((0,1))$, $\phi\geq 0$ $$\int_0^1 u(t)\phi''(t)dt \geq 0$$ Can we conclude that $u$ is convex? Note: $C_0^2((0,1))$ is ...