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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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46 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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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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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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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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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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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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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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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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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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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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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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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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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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Is convex hull linear subspace of linear hull?

We have some convex and compact supset $G$ of banach space $B$ and finitely many points ${x_1,...,x_N}$ . The question is : does the convex hull $C$ of ${x_1,...,x_N}$ a linear subspace of space ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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 functionals, min-max of linear functionals, etc

I was reading in this book on the well-known Courant-Fischer min-max Theorem, which states that if $\lambda_1(A)\geq \lambda_2(A)\geq \lambda_n(A)$ are the $n$ eigenvalues of a $n\times n$ Hermitian ...
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An affine set $C$ contains every affine combinations of its points

Show that an affine set $C$ contains every affine combinations of its points. Proof by induction: From the definition of an affine set, we know that $\forall x_1,x_2\in C \text{ and } \theta_i\in ...