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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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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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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19 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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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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23 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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23 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
41 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
20 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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1answer
7 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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27 views

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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24 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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38 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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1answer
55 views

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
42 views

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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49 views

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} ...
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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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1answer
15 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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54 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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29 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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1answer
65 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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46 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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22 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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29 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
35 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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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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52 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 ...
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1answer
35 views

How to pivot to an adjacent vertex in simplex method

In the simplex method, we need to move from one vertex of the polyhedron to an adjacent one. Suppose the polyhedron is $P=\{x\in\mathbb{R}^n\mid Ax=b,x\geq0\}$ with rank$A=m<n$. For a ...
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29 views

What is the convex hull of $ \{t \to e^{-\lambda t} : \lambda >0\}? $

What is the convex hull of $$ \{t \to e^{-\lambda t} : \lambda >0\}? $$ (Interpreted as the set of all functions on the above form.) Reference or argument is great.
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16 views

Is it covex function?$J_{new}(u)=\int_{\Omega} \sum_{i=1}^{N} \lambda_if(x)u_i(x)dx$

I have a function such as $$J(u)=\int_{\Omega} \sum_{i=1}^{N} f(x)u_i(x)dx$$ where $f(x):\Omega \to R$, $0 \le u_i(x) \le 1,\sum_i u_i(x)=1$ Given that $J(u)$ is a convex function w.r.t $u$. Now I ...
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1answer
56 views

Minimizing the function with a log determinant and trace function?

I am trying to minimize the following argument, which is unbounded in case one of the eigenvalues of $A$ is equal to zero. $\arg min_{S} \log|S^H A S| - tr\{ \Sigma^{-1}S^HAS\}$ Let $A > 0$, ...
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37 views

A question on convex hull

Let $a_1, a_2,\ldots, a_n$ be $n$ points in the $d$-dimensional Euclidean space. Suppose that $x$ is a point which does not belong to the convex hull of $a_1, a_2,\ldots, a_n$. My question is, does ...
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Homeomorphic images of convex sets

Let $K$ be a convex set of a topological vector space $X$. Is there anything we can say about the image $f(K)$ under a continous or homeomorphic map $f \colon X \to X$ ?
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How to give a rigorous proof of a fact about convex polygon?

I claim that there exists universal constants $0<\delta_1(m), \delta_2(m)<1$ such that for any convex polygon $P$ in $\mathbb{R}^n$ with $m$ faces, \begin{equation} \frac{\mathcal{H}^{n-1}(\{x ...
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1answer
40 views

Matrix norm in the objective of an optimization problem

I am stuck with the following optimization problem from research. The optimization problem have the following objective function: $\|Q-H\|_\infty$. Here $Q$ is a PSD matrix and $H$ is a symmetric ...
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15 views

how to show a concave function on discrete domain increases in x?

Define $g(x)=\frac{f(x)/x}{f(x)-f(x-1)}$ where x$\in$ $\mathbb{Z}$. Known that $f(x)$ has the concave extension in every consecutive $x$, i.e: $f(x+1)+f(x-1)-2f(x)<0$ holds $\forall x$. My question ...
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Question about sub differentiability of convex function

I am reading this book to study sub differentiability. On page 20 it says, as $V$ is a normed linear space for simplification, "a continuous affine function $l(v)$: $V\to \mathbb R$ everywhere less ...
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3answers
44 views

How to compute the projection of a polyhedron

Suppose that we have a polyhedron in $(x,y)$: $P=\{ (x,y) \mid A_1 x +A_2 y \leq b \}$ How can I find the polyhedron $P_x=\{ x \mid (x,y)\in P \}$? In other words, I would like to write $P_x=\{x ...
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66 views

In linear programming, how to check whether a convex polyhedron is contained in another

Suppose we have two convex polyhedra $P_1=\{x\in \mathbb{R}^n \mid A_1 x \geq b_1 \}$ and $P_2=\{x\in \mathbb{R}^n \mid A_2 x \geq b_2 \}$ Is there a way to check whether $P_1 \subseteq P_2$? I was ...
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1answer
38 views

A question about Caratheodory's Theorem of Convex Sets

As I understand it, Caratheodory's Theorem of Convex sets essentially states If $Q$ is a set in a vector space of dimension $n$ and x lies in the convex hull of $Q$, then x can be written as a ...
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1answer
20 views

Subgradient of the optimal value of a linear program with respect to its parameters

Consider the linear program $f(c)=\min\{c'x\mid x\in\mathbb{R}^n,Ax=b,x\geq0\}$. Are there any results on what is $\partial f/\partial b$?
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24 views

Question on relation between convex sets

The book I am referring regarding bell inequalites has the following two things it mentions about two convex sets which I am unable to understand. There are two convex sets $C$ ($C$ is a polytope ) ...
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31 views

Convex envelopes of bivariate functions

In order to convexify my nonlinear non-convex program I need convex envelopes for the function $(x/y)^2$, both x,y are positive. I am only aware of the convex envelopes of the type $xy$ from here ...
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38 views

Are differentiable and strictly decreasing functions always concave?

If a demand function is continuously differentiable and strictly decreasing in price, does that mean it will be always concave?
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1answer
57 views

Is the expectation of log-concave function still log-concave?

I know the expectation preserves the concavity (or convexity), but I was wondering is it still true that the expectation of log-concave function still log-concave; to be more precise, Let ...
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1answer
39 views

The Legendre-Fenchel transform of $BV$ semi-norm

I am reading a numerical paper in which it calls some "easy" facts from convex analysis but I can't justify it... Let $\Omega\subset \mathbb R^2$ be open. Define for a function $u\in L^1(\Omega)$ and ...
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41 views

Prove the concavity of the transformation from a concave function to another

Let's say we have $f_1$ and $f_2$, both strictly increasing and strictly concave on $[0,+\infty)$. $f_1(0)=f_2(0)=0$ and the difference $f_1-f_2$ is strictly positive and strictly increasing. That is, ...
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Convex signal reconstruction for convex generator function?

Let $f : \mathbb{R} \mapsto \mathbb{R}$ be a convex (not affine) function and suppose that $y = f(x)$. We want to reconstruct the input $x$ for a given $y$, and the standard approach would be to ...
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73 views

proving that $\text{aff}C-\text{aff}C\subset\text{aff}\,(C-C)$

In proof of Theorem 6.4.1 of Auslender's book about asymptotic cones, the author assumes that $\text{rge}\,A\subset\text{aff}\,C$ and for $\epsilon>0$ claims that ...