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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Problem with showing convexity of a function

I want to show that $f_n(\zeta) = \frac{1}{n} \log \sum_{w \in W_n} e^{\zeta K_n(w)} P_n (w)$ , with $\zeta \in \mathbb{R}$ is convex. I will not explain what $W_n, K_n$ and $P_n$ are, because this is ...
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Optimizing a convex combination

Assume that functions $f_i:X\to\mathbb{R}$ are fixed among weights $\alpha_1,...,\alpha_n$ with $\sum_i \alpha_i=1$, and assume that each function $f_i$ is maximized at a point $x_i\in X$, i.e. ...
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Closed convex set as the intersection of (tangent) half spaces

Theorem 18.8 in the book by Rockafellar establishes that any $n$-dimensional closed convex set $C$ in $R^n$ can be expressed as the intersection of the closed half spaces tangent to $C$. See here for ...
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Hahn-Banach separation theorem with a countable subset of functionals

For a separable Banach space $X$, the unit sphere of $X^*$ always contains a countable set $D$ such that $$ \left\Vert x \right\Vert = \sup_{f \in D} \left\vert f(x) \right\vert \qquad \mbox{ for ...
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The distance is attained by a unique point

Theorem: Let $K$ be a convex and closed subset of a Hilbert space $X$ and $x \in X$. Then there is a unique $y_x \in K$ such that $$\|x-y_x\|=d(x,K):=\inf \{\|x-y\|: y \in K \}$$ Remarks: if $K$ ...
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Equidistant sets and extremal points

Let $\lVert\cdot \rVert$ be any norm on $\mathbb{R}^n$ and let $x\in \mathbb{R}^n$ be a point. Let $\textrm{ext}(B_{\lVert x\rVert})$ denote the set of all extremal points of $B_{\lVert x ...
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Convex conjugate of l1 norm function with weight

The conjugate of $f(x)=\|y\|_1$ is, by definition, $$ f^*(z) = \sup_y \{y^Tz - \|y\|_1\} $$ Also we can write $$f(y)=\|y\|_1 = \max_{\|p\|_\infty\leq 1} y^Tp $$ By using this, we can get the ...
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65 views

Minkowski functional of a convex set is a convex function.

Let $X$ be a real vector space, and $K$ be a convex set with $2$ properties: $0\in K$ and $\forall x\in X, \exists t >0$ s.t. $x/t\in K$. Define the Minkowski functional of the set $K$ to be ...
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Problems with Proof of Jensen's Inequality (Durrett's “Probability Theory and Examples”)

I have some questions concerning the proof of the Jensen's Inequality I found in Durrett's "Probability Theory and Examples" [pp.23-24]. In the following there is the proof, with the questions I have ...
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Examples of $C^1$ differentiable convex functions.

Could you please provide examples of convex functions that are differentiable, but their derivatives are not differentiable.
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Let $f:X\rightarrow ]-\infty,+\infty]$, let $b\in x$ and let $c\in X$. Find Fenchel conjugates of the functions:

Let $f:X\rightarrow ]-\infty,+\infty]$, let $b\in x$ and let $c\in X$. Find Fenchel conjugates of the functions: $(i)\;\;\;\; f(x)+\langle c,x\rangle,$ $(ii)\;\;\;\; f(x-c).$ For (i) I'm thinking ...
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36 views

Is the function determinant $A \rightarrow \det(A)$ a non-convex fuction?

Is the function $$ \det: A\in \mathbb{M}^{n \times n}(\mathbb{R}) \rightarrow \det (A)$$ a convex function? I think the answer is no, but I cannot prove it directly using the definition of convex ...
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notions of geometry needed for optimization

Could you tell me the notions of geometry (not topology) that are needed before starting courses on convex analysis and optimization ? Thank you.
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Follow up question for: Convexity of the product of two functions in higher dimensions

The question referred to in the title ( Convexity of the product of two functions in higher dimensions) is already answered. However I have a question regarding the answer and I am not able to post it ...
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Proving Convexity for a function $f(x) = \frac1{g(x)}$?

So I have a function $g$ that maps from some subspace, $S$, of $\mathbb{R}^n$ to $\mathbb{R}$. $g$ is concave such that $g(x) > 0$ for all $x$ in this subspace, $S$, of $\mathbb{R}^n$. $f(x)$ is ...
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Suppose that $X=\mathbb{R}^n$ and that $K=\mathbb{R}_{+}^{n}$.

Suppose that $X=\mathbb{R}^n$ and that $K=\mathbb{R}_{+}^{n}$. using the definition of the Fenchel conjugate verify that $\iota_{K}^{*}=\iota_{-k}$ where $\iota_{K}$ is the indicator function. my ...
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Formulate the dual problem for primal problem with absolute value constraint

Let $y \in R$, the goal is to find the dual problem to: $$\min y\\ s.t. |y| \leq 0$$ The lagrangian of the problem is: $$L(y, \lambda) = y + \lambda|y|$$ The dual function is: $$g(\lambda) = ...
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Non-binding constraints with positive shadow prices (matlab)

The output of fmincon indicates positive shadow price for linear constraints, although the corresponding constraints are not binding. What could be wrong mathematically? I've checked the code but ...
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Prove that if g,h convex functions and g is positive ,then (goh) is a convex function ,too.

So, I am stuck and I can't think of an answer to the question above. Any help? Note : g is convex and positive and h is just convex.** We want to prove that (goh) is convex** Note2 : We do not know ...
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51 views

Geometric interpretation of monotone operators on a Hilbert space

Recall that a monotone operator is defined by the relationship as follows: $$\langle y - x, F(y) - F(x)\rangle \geq 0, \quad \forall x,y \in X$$ ($X$ is a Hilbert space) What is a good geometric ...
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An inequality using convex functions: If $\frac{a^2}{1+a^2}+\frac{b^2}{1+b^2}+\frac{c^2}{1+c^2} = 1$ then $abc \le \frac1{2\sqrt 2}$

I see this in a Chinese convex analysis book. Suppose that $a$,$b$,$c$ are positive real numbers satisfying \begin{equation} \frac{a^2}{1+a^2}+\frac{b^2}{1+b^2}+\frac{c^2}{1+c^2} = 1. ...
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Convex hull of finite union of closed sets

Suppose $C_1,\cdots,C_n$ are closed, compact, convex subsets of a locally convex topological vector space, then is $\text{Conv}(C_1 \cup \cdots \cup C_n) = \overline{\text{Conv}(C_1 \cup \cdots \cup ...
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Convexity of the log barrier function

Let's consider the following convex optimization problem of minimizing the log barrier function: $$\min_{\textbf{x}\in \Re^n}f(\textbf{x})=\min_{\textbf{x}\in ...
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Dimension of a set of points and directions

Let $S_0$ be a set of points in $R^n$, and $S_1$ be a set of directions in $R^n$, the dimension of the set $S=S_0\cup S_1$ is defined as the dimension of the set $$ \text{aff}\big( \text{conv}(S_0) ...
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Convergence of sequence to minima of convex function

I have a cost function $f$ that is continuous but it might not be smooth. We know that the level set $\{x:f(x)\le f(x^0)\}$ is bounded, and I have an optimization method that generates a sequence ...
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1answer
30 views

Proof about vertices of a convex hull

how to prove that, given a set defined as $S_{k}$ = {y: y = Ax, $\|x\|_{\infty}\leq$ 1} its convex hull conv($S_{k}$) has its vertices defined by those vectors $x$ such that $\|x\|_{\infty}$ = 1. ...
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Convex subsets and Linear functionals

Let $E$ be a convex subset of a normed space $X$ and $x\in E$. Then $x\in \overline{E}$ if and only if $\Re f(x)\geq 1$ for every $f\in X'$ such that $\Re f\geq 1$ on $E$ and $\Re f(x)\leq 1$ for ...
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Unbounded convex conjugate

This is my first time posting a thread. I apologize if I somehow do not comply with the rules (please remind me if it happens, so next time I can do it correctly:) Today I was having an optimization ...
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Line Segment is an edge of the Convex hull

I have to prove the following: The line segment xz is an edge of the convex hull CH(A) iff all other points of A lie in one of the closed half-planes induced by the supporting line l(x, z) of xz, ...
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a property for points in convex hull

Let $A\subset\mathbb{R}^2$ and $b=(b_1,b_2)$ is in the convex hull of $A$. Prove that for any $x=(x_1,x_2)\in\mathbb{R}^2$, there exists $a=(a_1,a_2)\in A$ such that $a_1x_1+a_2x_2\le b_1x_1+b_2x_2$. ...
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Intuitive interpretation of proof that projections are non-expansive

Is there an intuitive (e.g. graphical) interpretation of the proof that projections on closed convex sets are non-expansive? Most proofs, e.g. the one given here, are presented as a sequence of ...
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Convexity criterion for piecewise regular planar curves

A theorem of classical differential geometry states, that a simple, closed and regular $\mathcal{C}^2$-curve $\gamma:[0;1]\to\mathbb{R}^2$ is convex, iff its signed curvature doesn't change sign. ...
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Is $x^TAy$ convex or concave, for $x,y$ not identically equivalent?

It is well known if we had something like: $f(x) = x^TQx$ A quadratic form, is positive semidefinite of $Q$ is positive semidefinite How is the structure of $f(x,y) = x^TQy$ analyzed? i.e. what ...
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Convex hull in projective space

Let $S \subseteq \mathbb R\mathbb P^n$ be a closed connected set that does not intersect every hyperplane. If I choose any affine chart containing $S$, I can consider its convex hull, and it seems ...
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Uniquness of convex combination

Let $a$ and $b$ be real numbers such that $0<a<b$ and let $x$ be such that $a<x<b$. How can I determine $a_x$ and $b_x$ such that $$ x=a_x\cdot a+b_x\cdot b~~,~~a_x\geq 0,~~b_x\geq ...
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Relationship between affine functions and affine sets?

A function $f: \mathbf{R}^n \to \mathbf{R}^m$ is affine if it is a sum of a linear function and a constant ($f(x) = Ax + b$). A set $S \subseteq \mathbf{R}^n$ is affine if for any $x_1,x_2 \in S$ and ...
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Concavity of continuous positive functions with extra properties

Suppose $f$ is a continuous positive function and it is not constant. Moreover satisfies the following $$f(\lambda x+(1-\lambda)x')\geq \lambda^4f(x)+(1-\lambda)^4f(x')\,\,\,\, $$ ...
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Proof that a set is non-star-shaped.

Note: A subset $S\subseteq \mathbb{R}^2$ is called star-shaped if there exists an element $x\in S$ such that for every $y\in S$, the line segment connecting $x$ to $y$ is contained in $S$. We want to ...
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Concerning the optimality condition of convex functions

So I am studying convex optimization and I came across this theorem regarding the minimizer of a function in a space $\mathcal{X}$. In particular, the theorem states that if $f:\mathcal{X} \rightarrow ...
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What is the tangent cone of the nonnegative orthant?

Definition: $T_c(x)$ is the tangent cone of $C$ at point $x \in C$ $T_c(x) = closure\{z \in \mathbb{R}^n, z = k(y - x), \forall \thinspace y \in C, k \geq 0 \}$ And $T_c(x)$ is a closed convex ...
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The distance between two convex functions

I have a sequence $(u_t)\subset L^2(\Omega)$ so that the function $f(t)= \|u_t-u_0\|_{L^2}$ is continuous and strictly increasing and $f(0)=0$. Now, given $u'=u_0-v$ where $\int v =0$ and $\int v^2=c$ ...
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Find the Fenchel conjugate $f^{*}$ of the function $f:\mathbb{R}\rightarrow ]-\infty,+\infty]$ given by

Find the Fenchel conjugate $f^{*}$ of the function $f:\mathbb{R}\rightarrow ]-\infty,+\infty]$ given by \begin{equation*} f(x) = \begin{cases} +\infty & \quad \text{if } x\leq0\\ ...
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Fenchel-Rockafellar duality problem: Show that weak duality holds, i.e., p≥−d .

I am looking for help with motivation for the Fenchel-Rockafellar duality problem. Specifically the following: Let $\;f : X \rightarrow ]-\infty, +\infty]$ be convex and lower semicontinuous, let $A ...
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How to prove that the ball is convex.

I want to conclude that $B_r(x_0)$ is convex from the fact that $B_1(0)$ is convex. So I was trying to use the answer provided here Proving that closed (and open) balls are convex But the thing is ...
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Show that the dual of the dual is the primal for a min problem

I have LP of the form min $c^tx$ such that $Ax\leq b$, there is no restricition on $x$. I need to show the dual of the dual of this LP is the original LP. To get the dual of my LP, I need to put it ...
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Is the inverse image of a set under a convex function convex?

So all I have is that $\mathscr{H}$ is a Hilbert space and that $f:\mathscr{H}\to\mathbb{R}$ is a convex function. i.e. for all $x,y\in\mathscr{H}$ and $\alpha\in[0,1]$, $f(\alpha x ...
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Convex function on $\mathbb{R}^n$ composed with $n$ subharmonic functions

Currently, I am working on the following question (which appears in Hörmander's Notions of Convexity, Exercise 3.2.6) Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is a convex function which is increasing ...
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Subtle difference between convex and strictly convex, why?

A function is convex if $f(\theta x + (1-\theta) y) \leq \theta f(x) + (1-\theta) f(y)$, $\theta \in [0,1]$ A function is strictly convex if $f(\theta x + (1-\theta) y) < \theta f(x) + ...
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Segment ordered density conjecture revisited

I have a set $S\subset\mathbb {R}^2$ with the following property (P) $\forall x,y\in S$, $\forall\mathscr{C}$ a convex set that contains $x$ in its interior, $bd\mathscr{C}\cap [x,y]\subset ...
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the boundary normal vectors

Suppose $X, Y$ are convex compact sets in $\mathbb R^n$ and $z \in \mathbb R^n$. Would it be the case that $$\max_{x\in X} [ \langle x, z \rangle-\max_{y\in Y} \langle x ,y \rangle ]=\max_{x\in ...