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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How to determine whether a function of many variables is convex or non-convex?

A function $f$ is convex if $$f(\theta x + (1 − \theta)y) \leq \theta f(x) + (1 − \theta)f(y)$$ for all $x, y \in \mathcal{D}(f)$, the domain of $f$, and $\theta \in [0, 1]$. How do I determine ...
4
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1answer
55 views

Is the function $f(x) = |x|$ convex?

I am asking because some Internet pages contradict. In Wikipedia, the definition I found of a convex function is: "Let $X$ be a convex set in a real vector space and let $f: X \to R$ be a function. ...
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10 views

Does expectation operation preserve convexity? [on hold]

We have f(x) is a convex function. Is $E[f(x)] $ is convex? If yes, how can we prove it? Thanks, Tan-
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1answer
8 views

Minimum of biconjugate of a nonconvex $f(x)$ is the minimum of $f(x)$ also?

In Fazel (2002) Matrix rank minimization with applications, Ch. 5.1.4-5.1.5, the author finds an analytic expression for the convex biconjugate of their nonconvex function; however, they state that ...
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282 views

Dual norm of the matrix $L^1$ norm is infinity norm (and vice versa)

Recall that for a given norm $\|\cdot\|$ on $\mathbb{R}^n$, the dual norm is defined as a function $\|\cdot\|_*: \mathbb{R}^n \rightarrow \mathbb{R}$ with: $\|y\|_* = \max \limits_x \{x^Ty: \|x\|\le1 ...
2
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1answer
35 views

How to solve the convex combination problem of matrix?

Let $A \succeq B$ denote matrix $A-B$ is positive semidefinite, and here is the definition of redundant(all the matrix dimensions are $N\times N$ ): Given a set of matrix $\{B_i\}_{i=1}^{l}$, if ...
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24 views

polyhedral function

A function $f: \mathbb{R}^n \mapsto (-\infty,\infty]$ is polyhedral if its epigraph is a polyhedral, i.e. $$\text{epi}f=\{(x,t)\in \mathbb{R}^{n+1} | \ \ C\left( \begin{matrix} x\\ t \end{matrix} ...
2
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1answer
29 views

Relative Interior of a Convex Hull

Given pts $y_0,...,y_k \in \mathbb{R}^n$, their convex hull is Co($y_0,...,y_k$):={$\sum_{i=0}^k a_i y_i$ : each $a_i \geq 0$, $\sum_{i=0}^k a_i =1$}. Their affine hull is ...
3
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1answer
32 views

Are the stationary points of a strongly convex function unique in each dimension?

Consider a strongly convex function $~f: \mathbb{R}^n \rightarrow \mathbb{R^+}~$ with a unique minimum at the point $x^* \in \mathbb{R}^n$. I am wondering: if I have another point $y \in ...
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46 views

Is the ratio trace problem convex?

I have a ratio trace problem described as follows: $\arg\max_{w} \text{trace}((w^tAw)\cdot \text{inv}(w^tBw))$, where A and B are full rank matrices. This problem can be solved via generalized ...
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88 views

Convex Optimization Closed Form Solution

I'm currently studying for my exame in convex optimization. It covers problem formulation, first and second order conditions of optimality, unconstrained optimization, constrained optimization with ...
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1answer
31 views

Is the function (sum-of-squares) / sum convex on nonnegative input?

Let $$f \colon \mathbb{R}_{> 0}^n \to \mathbb R$$ be defined by $$f(x_1,\dotsc,x_n) = \begin{cases} 0 &\text{if }x_1 = \dotsb = x_n = 0\text{,}\\ \frac{\sum_i x_i^2}{\sum_i x_i} ...
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0answers
14 views

Intersecting Simplices with Normballs

Let $e$ be the vector of all ones 's consider the standard simplex $$\Delta_m:=\{x\in\mathbb{R}^m_+: \langle x, e\rangle=1\}.$$ Then the truncated simplex $\Delta_m^d$ is given as ...
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Are convex functions enough to determine a measure?

Suppose we are talking about $\mathbb{R}^n$. We know that if $\mu$, $\nu$ are two finite Borel measures such that $$\int_{\mathbb{R}^n}f(x) \, d\mu(x)=\int_{\mathbb{R}^n}f(x) \, d\nu(x),$$ for all ...
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1answer
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Convex sets in $\mathbb R^n$: Do they have a particular form ? Does the gradient of a linear convex function $f$ exist on such a set?

Convex sets in $\mathbb R^n$: Do they have a particular form ? Does the gradient of a linear convex function $f$ exist on such a set ? Suppose that $S$ is a convex set in $\mathbb R$. Then $S$ is ...
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227 views

Subsets of R2 that are convex, closed, and have non-empty interiors?

Can someone give me some guidance with this problem? Thanks. Suppose that $A, B \subset \mathbb{R}$ are convex, closed, and have non-empty interiors. Prove that $A, B$ are the closure of their ...
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2answers
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$A$ is a convex subset with non-empty interior and $D$ is dense in $\mathbb R^n$ ; then $\mathbb R^n$ , $U\cap D \cap A \ne \phi$? [on hold]

Let $A$ be a convex subset , with non-empty interior , of $\mathbb R^n$ and $D$ be a dense subset of $\mathbb R^n$ ; then is it true that for every open subset $U$ of $\mathbb R^n$ , $U\cap D \cap A$ ...
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3answers
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$A$ be a convex subset and $D$ be dense in a real NLS ; then for every non-empty open subset $U$ of the NLS , $U\cap D\cap V \ne \phi$?

Let $A$ be a convex subset of a real normed linear space $V$ and $D$ be a dense subset of $V$ ; then is it true that for every non-empty open subset $U$ of $V$ , $U\cap D\cap A $ is non-empty ?
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44 views

Is the projection function convex?

Define the following function $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ to be the projection function onto a convex and a closed set C $f(x)=\arg\min_{y\in C} ||x-y||_2^2 $ Denote $f_i(x)$ ...
0
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1answer
102 views

Firm non-expansiveness in the context of proximal operators

$\newcommand{\prox}{\operatorname{prox}}$ Probably the most remarkable property of the proximal operator is the fixed point property: The point $x^*$ minimizes $f$ if and only if $x^* = \prox_f(x^*) ...
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weighted geometric mean is concave

Ask for a hint to show following concave: $h(y) = y_1^{\theta1}...y_m^{\theta m}$ with $\theta_1+...+\theta_m=1$ and $\theta_i \geq 0$ If I do not want to use Hessians to show, any better way to ...
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1answer
28 views

Is this function composition convex?

Say we have two functions $f:R^n\rightarrow R$ , $g:R^m\rightarrow R^n$. Given that $f$ is convex, under what conditions on $f$ and $g$ we will be able to say that the composition function ...
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1answer
9 views

Interior of polar cone and self-concordant function

Suppose $f$ is a self-concordant(see 9.6.2) barrier of a proper cone $K$ ( solid,convex, closed and pointed) in $\mathbb{R}^n$. It looks like the value of all $\nabla f$ is just the interior of the ...
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2answers
46 views

Prove that the sum of convex functions is again convex.

I must to prove that the sum of convex functions is again convex. I know the definition of convex function: $f(tx_1+(1-t)x_2)\leq f(x_1)+(1-t)f(x_2)$ - this the first convex function, then I have the ...
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3answers
271 views

Convexity vs convexity on every line

I was reading this lecture on convex functions and I came across this $f\colon \Bbb R^n\to \Bbb R$ is convex if and only if the function $g\colon \Bbb R\to \Bbb R$, $g(t) = f(x+tv)$, ...
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1answer
25 views

Prove if C is midpoint convex and closed then its a convex set [duplicate]

Midpoint convexity. A set C is midpoint convex if whenever two points a,b are in C, the average or midpoint (a + b)/2 is in C.Prove that if C is closed and midpoint convex, then C is convex. ...
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Does this relation imply convexity?

I'm trying to figure out wheter the following condition inplies convexity or not. Let $\cal{X}$ be an inner product space with inner product $\langle \cdot, \cdot \rangle$ and a norm $\|\cdot\|$ (not ...
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1answer
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Can I write $\mathbb{S}_+^3$ as a norm cone?

Let $\mathbb{S}^3_+$ be the set of $3\times 3$ symmetric semi-definite positive matrix. I wonder whether I can write $\mathbb{S}^3_+$ as a norm cone, i.e., $$\exists A\in \mathbb{R}^{m\times 9}, C, ...
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0answers
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Unique solution of LP

Hi I am working on the following question: If $c \in int(N_P(x))$, then $x$ is a unique solution. I have proven that this is true if $x$ is a vertex. Well I am wondering if the following is a ...
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2answers
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Convex and conic hull, geometric interpretation

$$\operatorname{conv}\,X=\left\{\sum_{i=1}^N \lambda_i x_i \,\Bigg\vert\, N\in\Bbb N,\, x_i\in X,\, \sum_{i=1}^N \lambda_i = 1,\lambda_i \geq 0\right\}$$ $$\operatorname{cone}\,X=\left\{\sum_{i=1}^N ...
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1answer
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Can I write $\mathbb{R}^n_+$ as a norm cone?

Let $\mathbb{R}^n_+=\{x=(x_1,\dots,x_n):x_i\geq 0,\forall i \},n\geq 2$. I wonder whether I can write $\mathbb{R}^n_+$ as a norm cone, i.e., $$\exists A, c, \|\cdot\|, s.t. x \in \mathbb{R}^n_+ \iff ...
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35 views

Question about the proof of Caratheodory's theorem

In the proof available here, I do not understand why $\alpha>0$. How can we know for sure that $\lambda_i>0$?
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22 views

What is the right isomorphism for convex set in $\mathbb{R}^n$

Like we have linear transformation for vector space, I wonder what kind of 'transformation' or 'homomorphism' or 'isomorphism'( when the map is bijective) to look at for convex set in $\mathbb{R}^n$. ...
0
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1answer
43 views

What is a proximity operator? why do we need it?

I am going to deal with convex optimization problems and I am not a math student so I may have some problems in understanding some topics. As you know, many of the optimization problems lead to a cost ...
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2answers
158 views

On the decomposition of stochastic matrices as convex combinations of zero-one matrices

Let "stochastic" matrix be the matrix whose rows sum to one and deterministic matrix be a stochastic matrix whose all rows consist of a one and zero. For example $\left [ \begin{array}{ccc} 1 & ...
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1answer
40 views

About the convexity of $\sin x$ for $\pi\leq x\leq 2\pi$ [closed]

To prove the convexity of $\sin x$ over $[\pi,2\pi]$ through the second derivative is easy, but I would be interested in a (possibly) simple proof of convexity that avoids derivatives. Can you provide ...
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Proof Attempt of Brouwer (via Separating Hyperplane Theorem)

In part motivated by the discussion here, I have been playing with trying to prove Brouwer's theorem appealing as minimally as possible to topology. In the 1-dimensional case I believe one can ...
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4answers
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Direct proof for convexity of $e^x$ [closed]

Is there any direct proof without using second derivative for convexity of $e^x$?
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Strong convexity of quadratic function

Assume that $Q$ is a positive definite matrix, is it true to say that the function $f(v)=v^TQv$ is strongly convex with respect to the norm $||u||=\sqrt{u^TQu}$? Thanks
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Convexity increases the “cost” of long steps

Let $V(n)$ be a non-decreasing, convex function on $\mathbb{N}$ such that $V(0)=0$, $V(1)=1$. Let $(r_i)_{i=1}^{N}$ and $(r^{\prime}_i)_{i=1}^{N^{\prime}}$, $N^{\prime} > N$, be two sequences of ...
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(Just for clarification) - Is a convex, piecewise continuous function f on an closed interval continuous?

Lets say f is defined on an Interval $I = [a,b] $. Since f is convex, one immediately knows that f is continuous on $I^°$ , however left are the points $a$ and $b$ The piecewise continuity of f ...
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Notion of outer normal cone and supporting cone if $x \in$ relint($C$)

In my lecture we defined the outer normal cone $ N_c(x^*)= \{ c\ \in \mathbb{R^n} : \max\limits_{x \in C} \ \ c^Tx = c^Tx^* \}$ and the supporting cone $S_C(x^*)= \bigcap\limits_{c \in N_c(x^*)} ...
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How to prove that $e^x$ is convex? [closed]

I need a help with proving convexity of $e^x$ function. We know $e^x$ is a convex function. $$f(c(x')+(1-c)(x'')) ≤ c\cdot f(x')+(1-c)\cdot f(x''), \quad 0 ≤ c ≤ 1$$ Thanks.
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minimal distance betwen a point and and the halfspace containing a convex set

Let $L^2(I)$ be the usual $L_2$ space with $L_2$ norm and $S$ a convex and compact subset of $L^2(I)$. Suppose $g^*\notin S$ and $$\min_{f\in S} \|f-g^*\|$$ has the unique solution $f^*\in S$. ...
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Convex analysis of a vectorial function

In the context of equilibrium equations on structural concrete study, I deal with an system (3 non linear equations) that relates an entry of variables $(\epsilon_0,\kappa_x,\kappa_y)$ (which ...
3
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1answer
188 views

Proof that a set is convex

$$k\in \mathbb{R}_{+}$$ $$M=\left \{ (x_1,x_2)\in \mathbb{R}_{++}^2 \mid x_1 x_2\geq k\right \}$$ Prove that the set $M$ is convex. A hint is given (quoted from the text): We could choose to ...
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3answers
46 views

Proving that a certain set is convex

I'm trying to prove that the set $W = \{x \in \mathbb{R}^2 : 2x_{1}^{2} + 3x_{2}^{2} \leq 4\}$ is convex. I've been trying to do this using the definition of W being convex when for all $x, y \in W $ ...
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1answer
40 views

Convexity of a certain set

Would someone please help me? I know that the set $$\{(x,y)\mid \cos(x+y)\geq \frac{\sqrt 2}{2}\}$$ is convex, but I am seeking for a simple proof?
5
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1answer
37 views

Function defined by integrals convex?

Let $g$ be a positive integrable function in $[0,\infty)$, and $G$ its integral, that is $G(t) = \int_0^t g(u) \, du$. Is the function f, defined as $$ f(t) = \int_0^\infty g(u) e^{-(G(u+t) - G(u))} ...
11
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1answer
128 views

Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ ...