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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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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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 ...
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4answers
88 views

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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27 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
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21 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 ...
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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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39 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 $f^*\in S$. ...
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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
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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
41 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?
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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))} ...
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1answer
471 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 ...
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3answers
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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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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 ...
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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)) = ...
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1answer
31 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 ...
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1answer
38 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?
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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} ...
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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 , ...) ...
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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 ...
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1answer
31 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 ( ...
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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 ...
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22 views

Transformation between two optimization problems.

Problem $1$ is as follows: \begin{equation} \max_{1{\le}i{\le}N}\min_{\{v_i\}_{i=1}^N\in\textbf{V}_{\gamma}}\left[\lambda_i - v_i\right] \end{equation} Problem $2$ is as follows: \begin{eqnarray} ...
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1answer
35 views

Rectilinear convex hull

I am working on an algorithm, which takes as input as set points contained inside the Rectilinear Convex Hull of some fixed points in 2-dimension. I tried to find an implementation but met with little ...
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1answer
27 views

Cones: Face of intersection is intersection of faces?

I'm writing on behalf of a group project where we are currently looking at basic geometry; in particular we are interested in polyhedral fans. We wish to prove that (abusing terminology somewhat) the ...
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1answer
19 views

Convexity of a perspective of affine function

I was reading the well-known convex optimization PDF lesson by Boyd and Vandenberghe (more specifically chapter 3), and ran into a problem which I haven't been to solve. On slide 3-20, the ...
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5 views

concavity of functions of many variables

I have a function in many variables, the function is concave and non-increasing in each one of the variables, is the entire function concave?
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16 views

Partitioning in convex problem (variables in two subsets)

Consider the following problem from textbook Convex Optimization Algorithm p.10: \begin{equation} \begin{aligned} &{\text{min}} & & F(x)+G(y)\\ ...
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A proposition of relative interior point

One proposition from Convex Optimization Algorithm p.473: $X$ is a nonempty convex subset of $\mathbb{R}^n$ $f:X \rightarrow \mathbb{R}$ is a concave ...
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2answers
23 views

Class of differentiable functions and Lipschitz continuity

I am reading lectures notes by Dr. Yuvi Nesterov's "Introductory Lectures on Convex Programming ". On page 25, Lemma 1.2.2, to prove $f''(x) \leq L$, (where $f(x) \in C_L^{2,1}(R^n)$, $L$ is Lipschitz ...
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2answers
37 views

Is $L_2$-norm strictly convex?

I am new to convex analysis, and just wondering whether there is a simple check to see whether $L_2$-norm is strictly convex. How to mathematically prove/disprove this? $L_2$-norm: $\| x\|_2 = ...
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Why are convex metric spaces defined this way?

If my understanding is correct, a metric space $(X, d)$ is called convex if for all $x \in X$, and $y \in X - \{x\}$ there exists some $z \in X -\{x,y\}$ such that: $$d(x,z) + d(y,z) = d(x,y)$$ I can ...
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1answer
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How to resolve total variation $F(u)=f(u)+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$

Given $u(x)=[u_1(x)..u_N(x)]$, $0 \le u_i(x) \le 1, \sum_i^N u_i(x)=1$ and the cost function is: $$F(u)=f(u(x))+\lambda\sum_i^N\int_{\Omega}|\nabla u_i(x)|dx$$ where $u_i(x)$ is a value that indicate ...
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1answer
15 views

Distance from a compact convex "monotonicity''

If $C$ is a convex compact set in $\mathbb{R}^n$, we know that we can define the projection on $C$, $p : \mathbb{R}^3 \setminus C \to C $, such that : \begin{equation} \text{d}(x, p(x)) = \min_{y \in ...
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What modification is this of the notion of Renyi divergence?

Given two probability distributions $P$ and $Q$ over the same outcome and event space (assume finite if needed) one defines their Renyi divergence as $D_\alpha (P \vert \vert Q) = \frac{1}{\alpha -1} ...
3
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1answer
51 views

Uniform Sampling on Intersection of Simplices

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints $$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq ...
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13 views

Cumulant-Legendre

I have a short question: So suppose $b=\text{ess sup} X<\infty$, where $X$ is a random variable on $\mathbb{R}$. Now take $\Lambda (u)=\ln \mathbb{E}[e^{uX}]$, the cumulant, and ...
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1answer
50 views

How to find the domain of the support function

First the definition of the support function of set $S$: Let $S$ be a nonempty convex set. The support function $h$ of $S$ is the real-valued function defined by $$h(x)=\sup_{s\in S}\langle ...
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2answers
24 views

Deriving convexity from Taylor series expansion

Why is the function $f(x) = \sum^\infty_{k=1} (3x)^{2k}$ convex? What is the condition on the coefficients to deduce that $f$ convex?
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59 views

Arc length function of a helix/spiral is convex?

Given the arc-length of a parametric curve, $\int_a^b\|\gamma'(t)\|$ if the parametric curve was non-convex, can the arc length be a convex function?If the parametric curve was convex, will the arc ...
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43 views

How to computer the proximal operator of a indicator function?

For $G_{1}(x) = I_{c}(x), c =\{x|Ax=b\}$, the Proximal operator is: $Prox_{\gamma G_{1}}(x) = Proj_{c}(x) = x + A^{T}(AA^{T})^{-1}(y-Ax)$ I hope to know, how to derivative this result. And, for ...
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1answer
21 views

Integrable convex function vanishes at infinity

Why does a function that is Riemann-integrable in $[0, \infty)$ and that is convex vanishes at infinity?
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28 views

Why do we study multi-valued(set valued) mappings?

I am working on a problem that has to do with multi-valued mappings. Precisely, Iterative methods for Fixed points for multivalued mappings. However, I have no clear motivation for studying such ...
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1answer
33 views

Is this function convex or non-convex? How do you decide?

The problem is: find $$\min⁡ \mathrm{P}\left[{\log(1+p||H^H \mathbf{w}||^2)\over 1+p||G^H \mathbf{w}||^2}<R\right]$$ constraint to: $||\mathbf{w}||^2=1$ where $H$ and $G$ are matrices of ...
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1answer
667 views

convex hull function in matlab

Is there any way to compute the convex hull of a finite set of points in Matlab and gives the half-space representation as its result? I usually use a toolbox called MPT developed at ETHZ Zurich, but ...
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Why is the affine hull of the unit circle $\mathbb{R}^2$?

My question is addressed in Why is the affine hull of the unit circle $\mathbb R^2$? However, I am still confused. I thought that the affine of C in this case would be the interior of the circle. I ...
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1answer
29 views

Dot product - geometrical interpretation in convex analysis

I am studying a theorem on the characterization of solutions in nondifferentiable convex problems. Say that $\emptyset \neq C \subset \mathbb{R}^n$ is convex and $f: \mathbb{R}^n \to ...
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46 views

Concave optimization and corner solution

I have a optimization problem as follows: Assumptions: $f$ is an increasing and convex function on $R^+$ such that: $f(x): R^+\rightarrow R^+, \quad f(0)=0, \quad f'(x)\ge1,\quad f''(x)\ge 0 ...