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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9
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2answers
108 views

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 ...
0
votes
0answers
9 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 ...
0
votes
1answer
30 views

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 ...
0
votes
0answers
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 ...
2
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2answers
79 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 ...
-3
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2answers
26 views

$A$ 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 $\mathbb R^n$ , $U\cap D \cap A$ ...
0
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3answers
12 views

$A$ 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 ?
0
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2answers
41 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
votes
1answer
101 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^*) ...
0
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0answers
114 views

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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 $g\colon \Bbb R\to \Bbb R$, $g(t) = f(x+tv)$, ...
0
votes
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. ...
2
votes
2answers
51 views

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 ...
1
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1answer
18 views

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, ...
1
vote
0answers
34 views

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 ...
1
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2answers
27 views

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 ...
1
vote
1answer
22 views

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 ...
0
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2answers
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 $\lambda_i>0$?
0
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0answers
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
votes
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 ...
1
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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 & ...
0
votes
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 ...
1
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0answers
47 views

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 ...
-1
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4answers
82 views

Direct proof for convexity of $e^x$ [closed]

Is there any direct proof without using second derivative for convexity of $e^x$?
0
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0answers
26 views

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
3
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0answers
28 views

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 ...
0
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1answer
20 views

(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 ...
0
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0answers
17 views

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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3answers
1k views

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.
2
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0answers
37 views

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 $f^*\in S$. ...
1
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0answers
30 views

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
votes
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 ...
0
votes
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 $ ...
1
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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
votes
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
votes
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,\ ...
0
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1answer
467 views

Dual cone of a L1 norm cone?

I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...
6
votes
3answers
2k views

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 ...
3
votes
0answers
23 views

How to sample from a convex hull?

Let us consider a couple of points $x^{(i)}\in \mathbb{R}^m$ where $i=1,\dots,n$. Convex hull is defined as $$ C = \left\{\sum_{i=1}^{m} \alpha_i x^{(i)} \mathrel{\Bigg|} (\forall i: \alpha_i\ge ...
0
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0answers
20 views

Indicator function to zero-set of a function

Given the indicator function $I_{C}: \mathbb{R} \rightarrow \mathbb{R}$ to a convex set $C \subset \mathbb{R}$ and a function $g(x): \mathbb{R}^n \rightarrow \mathbb{R}$ $$ I_{C}(g(x)) = ...
0
votes
1answer
30 views

Is the constraint $A^2 = B^2$ convex

I am trying to use a continuous constraint to replace a discrete equation $A = |B|$ in my model. Since the linear programming method for absolute value is inapplicable in my model, I come up with ...
0
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1answer
37 views

On convexity of $\frac{1}{x}$

I would like to prove convexity of $\frac{1}{x}$. It can be proved by using second derivative but I want without using second derivative. Can someone help me?
3
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0answers
37 views

Is minmax equivalent to maxmin?

More precisely, problem $1$ is as follows: \begin{eqnarray} &\max_{1{\le}i{\le}N}\min_{[\gamma_m^i]}\left[\lambda_i - \sum_{m=1}^M\phi_m\gamma_m^iF_m^i\right] \\ &\mbox{subject to} ...
0
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1answer
17 views

geometric representations in convex analysis

Do you have any advices that help having geometric representations in convex analysis ? (for instance examples you always keep in mind when you are working, websites with simulations, graphs , ...) ...
0
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1answer
17 views

Convexity proven as max of linear functions

i am studying convexity, and stumbled upon the statement and example below. Am i right to understand that the function in the example is convex because maximizing the equation on the right hand size ...
1
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1answer
30 views

Prove the solution $u_i$ of equation $\lambda r_i+\frac{1}{\theta}(u_i-v_i)+\beta \left (\sum_{i=1}^Nu_i-1\right )$

I have an cost function such as $$E(U)=\lambda \sum_{i=1}^{N} \int_{\Omega}r_iu_idx+\frac{1}{2\theta}\sum_{i=1}^{N} \int_{\Omega}(v_i-u_i)^2dx+ \frac{\beta}{2} \int_{\Omega}\left ( ...
0
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0answers
22 views

Find convex efficient columns in Matrix

Consider a path-incidence matrix $A$ of a graph, where vertices are e.g. machines, paths are alternative production paths for a given product and entries $a_{ij}$ denote the workcontent for machine ...