Questions tagged [convex-analysis]

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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A Slight Difference Between the Definition of Sub-gradient in Different Literatures

To start, I would like to state both definitions for sub-gradient and where they came from. Definition 1 Let $f$ be $\mathbb E \mapsto \mathbb {\bar R}$ with $x\in \text{dom}(f)$ then $v\in \mathbb E$...
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Strong convergence of subgradients of a convex Fréchet differentiable function on a normed space

This thread is meant to record a question that I feel interesting during my self-study. I'm very happy to receive your suggestion and comments. See: SE blog: Answer own Question and MSE meta: Answer ...
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Weak$^*$ convergence of subgradients of a convex continuous function on a normed space

This thread is meant to record a question that I feel interesting during my self-study. I'm very happy to receive your suggestion and comments. See: SE blog: Answer own Question and MSE meta: Answer ...
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Is this function strongly convex? approaching boundary case

The definition for strongly convex is: A differentiable function $f$ is strongly convex if $$ f(y) \geq f(x)+\nabla f(x)^{T}(y-x)+\frac{\mu}{2}\|y-x\|^{2} $$ for some $\mu>0$ and all $x, y$. And ...
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Assume $A$ is open convex and $f$ convex continuous. Then $f$ is Gâteaux differentiable at $a \in A$ if and only if $\partial f (a)$ is a singleton

This thread is meant to record a question that I feel interesting during my self-study. I'm very happy to receive your suggestion and comments. See: SE blog: Answer own Question and MSE meta: Answer ...
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Characterize directional derivative of a convex continuous function by subdifferential

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Characterize subdifferential of a convex function by directional derivative

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Assume $A$ is open convex and $f$ convex continuous. Then $f$ is $L$-lipschitz on $A$ if and only if $\partial f(A) \subset L B_{X^*}$

This thread is meant to record a question that I feel interesting during my self-study. I'm very happy to receive your suggestion and comments. See: SE blog: Answer own Question and MSE meta: Answer ...
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Random convex functions

I am collaborating with a couple of machine learning specialists on a project related to phase retrieval. The problem we are investigating has solutions which are convex 2D functions on a rectangular ...
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Showing that there always exists an optimal mixed strategy for two player zero sum games

Exercise Let there be a game of two players $A,B$ and $x',y'$ optimal strategies such that: \begin{align} f(x') = \max_{x \in A} \min_{y \in B} E(x,y) = \min_{y \in B} \max_{x \in A} E(x,y) = g(y'). \...
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Approximate a convex proper lower semi-continuous function by an increasing sequence of Lipchitz-continuous convex ones

I'm trying to prove this well-known approximation of proper l.s.c. convex function. Could you have a check on my attempt? Let $(X, | \cdot|)$ be a normed space and $f:X \to \mathbb R \cup \{+\infty\}$...
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must max of sum lie in convex subspace? [closed]

$f$ and $g$ are two concave functions defined on a convex subset $A$ of a vector space. Each of $f$ and $g$ achieves its maximum on a convex subset $B$ of $A$. Does it imply that the maximum of $f+g$ ...
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Example of convex function with empty subdifferential at some point

Let $X$ be a normed space and $C$ an open convex subset of $X$. Let $f: C \to \mathbb R$ be convex. If $f$ is continuous at some point in $C$, then $f$ has non-empty subdifferential at all points in $...
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Show that the set $\{(x_1,x_2) : 4 \sqrt{x_1} + x_2 \geq 20\}$ is a convex set.

Suppose $f : \mathbb{R}^2 \to \mathbb{R}$ is given by $f(x_1,x_2) = 4 \sqrt{x_1} + x_2$. Show that the set $\{(x_1,x_2) : f(x_1,x_2) \geq 20\}$ is a convex set. We need to show that for any $x = (a,b)...
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The relation between convex conjugate and subdifferential

I'm reading about convex conjugate and its relation to subdifferential. In order to characterise subgradients we will use the convex conjugate defined below. This is essentially a special case of the ...
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An example of convex function that is not lower semi-continuous [duplicate]

Let $f:\mathbb R^n \to \mathbb R \cup \{+\infty\}$ be convex. Let $D := \{x\in \mathbb R^n \mid f(x) \in \mathbb R\}$ be the domain of $f$. Then $f$ is continuous on $\operatorname{int} D$. Could you ...
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Krein Milman theorem, when the closure is not needed

Krein-Milman theorem states that If ${K}$ is a non-empty compact convex subset of a locally convex space $ {X}$, then ${\text{ext}\;K\neq\emptyset}$ and ${K=\overline{\text{co}}(\text{ext}\;K)}.$ Note ...
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Continuous convex function on a normed vector space has non-empty subdifferential at every point

I'm trying to prove this well-known result. Could you have a check on my attempt? Theorem: Let $X$ be a normed vector space and $A$ an open convex subset of $X$. Let $f:A \to \mathbb R$ be convex ...
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Quasiconcavity/ Quasiconvexity of $f: \mathbb R \to \mathbb R\ldots$ [closed]

The function $f: \mathbb{R} \to \mathbb{R}$ given by $$f(x)=\begin{cases}\frac{x}{|x|}, &\text{if $x\neq 0$}\\\\1, &\text{if $x=0$}\end{cases}$$ is Concave Convex Neither Concave nor Convex ...
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Expected utility function is continuous over discrete probability distributions

I'm having trouble proving that the follwing function is continuous: Let $A$ be a non-empty set (not necessarily finite) and $$ X = \left\{ x : A \to [0, 1] \ \middle| \ \text{supp}(x) \ \text{is ...
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Optimization, Convex Analysis, Geometry

Let A,B,C three points on the plane. Where is the point D that minimizes the sum of distances $\left\|d-a \right\|+\left\|d-b \right\|+\left\|d-c \right\|$ where $a=(a_{1},a_{2}), b=(b_{1},b_{2}), c=(...
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3 answers
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Smooth convex set intersects any vector non-coplanar with supporting hyperplane?

Suppose I have a convex, compact, and $n$-dimensional set $K \subseteq \mathbb{R}^n$, and the origin $o$ is a smooth point on the boundary of $K$. Let $b_1, ..., b_{n-1}$ be a basis for the unique ...
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1 answer
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An extension on Jensen inequality

Suppose $f(x)$ is convex in $[0,1]$, then define $$ x_{n}=\frac{1}{2 n}\left[f(0)+2 f\left(\frac{1}{n}\right)+2 f\left(\frac{2}{n}\right)+\cdots+2 f\left(\frac{n-1}{n}\right)+f(1)\right] $$ I want to ...
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Vertices of a Polyhedron

I ask your expertise for the following point: let P be a polyhedron unbounded and without lines. According to a standard decomposition, P can be written as: $$P= M + K$$ (+ stands for the Minkowsky ...
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Properties of Banach.-Mazur distance [closed]

I’m trying to prove the properties of symmetry,reflexivity like and multiplicative triangular inequality for Banach-Mazur distance.. Can you please help me?
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Substitute extremum back into functional (envelope theorem for functional)

Say I have a functional of the form, $$ F(x, f(x); c) = \max_{f(x),c} \left\{ f(x)c - 1/2 c^2 \right\} $$ with $c$ a parameter. Can I take first-order conditions with respect to c and substitute $c^* ...
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3 votes
1 answer
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Convex function using intuition alone

Suppose $K$ is convex and $F(x) = 1$ for $x$ in $K$ and $F(x) = 0$ for $x$ not in $K$. Is $F$ a convex function ? What if the $0$ and $1$ are reversed ? I think in both cases, the function is not ...
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invariance of convex sets

Let $T: S_X \rightarrow S_Y$ be an onto isometry between the unit spheres of two (real) Banach spaces, and $C \subseteq S_X$ a maximal convex subset of $S_X$. Is it true that $T(C)$ is also a convex ...
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How do you compute this orthogonal projection

Suppose that $K \in \mathbb{R}^{n \times n}$ is a symmetric positive semidefinite matrix and $\mathbb{R}^n_+$ is the positive quadrant in $\mathbb{R}^n$ (including the boundaries). Let $X = \{x \in \...
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3 votes
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$V \subseteq T_1 U$, and $V \subseteq T_2 U$, is $V \subseteq [\lambda T_1+ (1-\lambda) T_2] U$?

$U, V$ are two convex subsets of vector spaces. $T_1, T_2$ are linear operators. $0<\lambda<1$. $V \subseteq T_1 U \implies \forall v, \exists u_1$ such that $v = T_1 u_1$. $V \subseteq T_2 U \...
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Implications of increasing componentwise convex functions and composition of such functions

in my textbook I have the following implications. Increasing directionally convex functions and increasing convex functions are both subsets of increasing componentwise convex functions. Why is it an ...
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Convex function with $L^p$-Norm values

a question is as follows. Let $f$ be some measurable (to some general measure) function on $[0,1]$. We define the map $\phi$ this way: $\phi: a\to \|f\|_{1/a}$ for all such $a>0$, that $f\in L^{1/a}...
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Proof about concave decreasing function with some constraints

Hi it's a follow up of How to prove that the functional equation $f(x)+f(y)=f(\frac{xf'(x)+yf'(y)}{f'(x)+f'(y)})+f(\frac{x+y}2)$ is verified only by some basic functions? : Conjecture :...
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-3 votes
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show the convexity of composed function

need to show that $g \circ L$ is always (strictly) convex : \begin{alignat*}{3} &L&&(A&&) && = X^TAX \geq0 \qquad &&\text{where }X \in \mathbb{R}^n, \quad A \&...
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-3 votes
2 answers
62 views

convex function with global minimum but no local minimum [closed]

I am trying to find a convex set and function that satisfy having a global minimum but not a local minimum. I have been told that this is achievable but I am having a hard time conceptualizing such a ...
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is the mix of convex and linear functions always convex function?

I want to prove that the following composed function $g \circ L$ is always (strictly) convex : \begin{alignat*}{3} &g&&(t&&) && =-\log(1-e^{-t}) \qquad && \text{...
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2 votes
1 answer
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Inequality about $x^p$ for $p\geq 2$ a real number [closed]

I am stuck with the following inequality $$g^2(n+2)+2g(n+2)g(n)-g^2(n+1)-2g(n+1)g(n+2)\geq 0,$$ for all integer $n\geq 1$. Here, $g(x)=x^p$ where $p\geq 2$ is a real number. I need help.
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1 vote
2 answers
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Self study for exam in convex analysis regarding optimality conditions

I am working for my oral exam. Here is a question that you might want to help me with. "Define a constrained minimization problem for a smooth (convex) function with smooth constraints. Provide a ...
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1 vote
1 answer
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Closure and interior of convex set is convex?

For a metric space $(X,d)$ and points $x,y \in X$ we define the metric segment between them as the following set: $\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$ We then say ...
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1 answer
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The set of all convex combinations of $n+1$ vectors from $A$ is the convex hull of $A$ if $A \subseteq \mathbb{R}^n$

Let $A\subseteq \mathbb{R}^n$. If we mark: $$A_k=\{\sum^k_{i=1}\lambda_i a_i: \sum^k_{i=1} \lambda_i=1,a_i\in A, \lambda_i \geq 0 \}$$ Then we want to show that: $$A_{n+1}=C(A)$$ Where $C(A)$ is the ...
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7 votes
2 answers
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Is there a closed, convex subset $C$ of $\mathbb{R}^3$, such that any proper, closed, convex shape in the plane is a section of $C$?

To be precise, by a section of $C$ I mean the intersection of $C$ with a hyperplane. I want every proper, closed, convex subset of the plane, up to translation and rotation, to be appear as sectiond. ...
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Is the composition of two optimal transport maps still optimal (under some assumptions)?

Consider three absolutely continuous probability measures $\mu$, $u$, and $\nu$ on $\mathbb R^d$ ($d \geq 1$), all of which have finite second moments. A transport map from $\mu$ to $\nu$ is called ...
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Maximize sharpe ratio. Is $x_0 x_1 \ge x_2$ convex?

I'm dealing with the portfolio optimization case study of mosek and want to add limits on the total number of assets to be re-weighted and the turnover of each asset. The objective is to maximize ...
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Attempting to prove $f(x,y)=\max\{x,y\}$ is convex using the Hessian matrix

I'm trying to prove that the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$, which is defined by $f(x,y)=\max\{x,y\}$, is convex. My lecturer proved this by showing (by definition) that $f(x\cdot t +(...
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Preparing for oral exam in convex analysis help to understand question

I am preparing for my oral exam in convex analysis and I just wonder what my teacher exactly means with the following question? "Elaborate on the precise meaning of separation between a (closed) ...
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1 answer
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Is this claim about Minimax proof correct?

Context I am reading about Convex Analysis and more specifically the famous Minimax Theorem. It goes on about what to do if the function is convex, strictly convex, concave and strictly concave. All ...
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1 answer
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Let $A$ be a convex. Is it true that $\operatorname{aint} (A) \neq \emptyset \implies \operatorname{aint} (A) = \operatorname{int} (A)$?

I have recently come across this result. Theorem: Let $A$ be a convex subset of a t.v.s. $X$. If $\operatorname{int} (A) \neq \emptyset$, then $\operatorname{aint} (A) = \operatorname{int} (A)$. ...
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0 votes
2 answers
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How should we optimize this optimization problem

The optimization problem is given by \begin{equation} \begin{aligned} F(\boldsymbol{\tau})&=\sum\limits_{m=1}^{M} \frac{\left( \tau_{m-1} e^{-\frac{\tau_{m-1}}{\lambda}} -\tau_{m} e^{-\frac{\tau_{...
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1 vote
1 answer
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Prove that a special function $\ell(\bar{x}+t \mathbf{a})$ formed by concave function is decreasing of $t$ when $t>0$. [closed]

Let $x_{1}, x_{2}, \ldots, x_{n}$ be $d$-dimensional vectors of real numbers with $n$ sufficiently large but the exact value is not of importance. A function of $\mu$ is defined to be $$ \ell(\mu)=\...
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5 votes
1 answer
101 views

A Clarkson-type inequality $|a+b+c|^q + \sum\limits_{\mathrm{cyc}} |a + b - c|^q \ge \sum\limits_{\mathrm{cyc}} (|a + b|^q + |a - b|^q)$

Show that, for $a,b,c \in \mathbb{R}$ and $q \geq 2$: $$ |a+b|^{q} + |a-b|^{q} + |a+c|^q + |a-c|^q + |b+c|^q + |b-c|^q \\\leq \\|a+b+c|^q + |a+b-c|^q + |a-b+c|^q + |a-b-c|^q $$ When $c=0$, this ...
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