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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Confusion regarding the convexity of a function

I want to know how come the function f(y)=1/y is convex?
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Minimize a complex quadratic subject to two convex quadratic constraints

I have the following the optimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~&||\mathbb{x}^H\mathbb{u}||_2^2-2*Real\{ \mathbb{x^Hu}\} \\\ ...
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Show that $Z=T(2S−I)+I−S$ is firmly nonexpansive

Suppose $T$ and $S$ are firmly nonexpansive mappings from $\mathbb R^n$ to $\mathbb R^n$. Let $I$ be identity operator. I want to show that $Z=T(2S−I)+I−S$ is firmly nonexpansive. Definition. We say ...
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51 views

Showing that $T+S$ is firmly nonexpansive

Show that $T+S$ is firmly nonexpansive considering that $T$ and $S$ are firmly nonexpansive mappings from $\mathbb R^n$ to $\mathbb R^n$. Definition: We say that $F$ is firmly nonexpansive if: ...
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Parameterized convex optimization

I'm trying to formulate a game so that at Nash equilibrium I achieve supply equales demand. Then I ran into this problem. For all $i,$ $v_{i}\left(x_{i}\right)$ is concave in $x_{i}$. The value ...
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117 views

proving compactness and convexity of a set

Suppose functions $f(x)$ and $g(x)$ are continuous with domain $X \subset \mathbb{R} $ which is nonempty, convex and compact, can we show that $$S \equiv (f(x), g(x)) $$ for all $x \in X$ is ...
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Strong Separation of Closures

Let $\bar D $ and $\bar E$ denote the closures of $D$ and $E$ respectively. If $ D\subset \mathbb R^n$, $E \subset \mathbb R^n$ and they are strongly separated. Show that $\bar D $ and $\bar E$ can ...
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Making sense out of “field”, “algebra”, “ring” and “semi-ring” in names of set systems

There are some set systems with algebraic titles, such as "field", "algebra", "ring" and "semi-ring" (and possibly other titles), in their names. Examples are a sigma field (aka sigma algebra, ...
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2answers
118 views

is product of norms convex?

Is a function of the form $f(x) = \|x\|_1\|x\|_2$ convex in x? I have tried plotting it in wolfram alpha and it appears convex, althought I ahve not been able to show it yet
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Are these sets convex polyhedrons?

I need some help with convex polyhedrons. First of all, I will write my definition of "convex polyhedrons", since Im not sure about translating this term into English (you can edit the name if you ...
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When does it make sense to define a generator of a set system?

In a set system, such as a topology, sigma algebra or convexity structure, a generator is defined to be a class of subsets such that the given set system is the coarsest such set system ...
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Is a closed set in a TVS over $\mathbb{R}$ convex?

From Theory of Convex Structures by M. L. J. Van De Vel, on a set $X$, a topology and a convexity structure are said to be compatible, if the convexity structure is generated by the closed sets. The ...
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51 views

Two different opinions on whether a topological vector space is a uniform space

Van de Vel's Theory on Convexity Structures says a TVS is uniform iff it is locally convex: 3.10.1. Proposition. Let $X$ be a topological vector space, equipped with the standard convexity and ...
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convex relaxations

Is there a notion of "best" convex relaxation of a particular function in a normed vector space? For example, the $\ell^0$ pseudo-norm can be relaxed into the $\ell^1$ norm, and that allows us to ...
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332 views

How to characterize the convex hull/closure operator

From Wikipedia Every subset $A$ of a vector space $S$ over the real numbers, or, more generally, some ordered field, is contained within a smallest convex set (called the convex hull of $A$), ...
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267 views

Anderson's Inequality for Gaussian measures

Let $C\subset \mathbb R^n$ be convex and symmetric about the origin. I am trying to prove that $\gamma(C) \geq \gamma(C+x)$ for any $x\in \mathbb R^n$, where $\gamma$ is the standard Gaussian measure. ...
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51 views

Does the following condition characterise convexity of a set?

Conjecture: A set $X \subseteq \mathbb{R}^n$ is convex if and only if the following holds. For any $x \in X$ and any vector $v \in \mathbb{R}^n$ such that $x+v \notin X$, it holds that for any scalar ...
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62 views

Find a plane that covers a convex function's graph

Let $f(x)$ be a convex function on $\Pi = [a,b] \times [c,d]$ such that there exists $y \in \mathbb{R}^2$ with property $$ y\cdot x\geq f(x), \;\;\; \forall x \in \Pi. $$ I want to find such $y$. ...
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Easy but hard question regarding concave functions!

I have a question about concave functions. Let $f:[0,T]\rightarrow R^+$ is a concave function. Let $S=\int_0 ^ T f(x)dx$ and $R=\frac S T$. Show that there is an interval $[a,b]$, where $0\leq a \leq ...
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Subgradient of convex minimization duality

$$\min(f_0(x))$$ $$\text{s.t. }f_i(x) \le y_i \forall i, i = 1 ,\ldots, m$$ $$f_i : \text{convex};\quad x : \text{variable}$$ It is also considered that $g(y)$ is the optimal value of the problem ...
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184 views

Is it possible to replace function by its concave envelope

Let $f(x) \in C[-1,2]$. Consider an optimization problem $$ J[\mu] = \int\limits_{-1}^{2}f(x) \, \mu(dx) \to \max\limits_{\mu - \text{Borel probability measure}} $$ with restriction $$ ...
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Convexity of cubic vector function?

First question in these forums so go easy on me. I have a function $f_i(x):\mathbb{R}^N\to\mathbb{R}$ which is defined by $$f_i(x) = \frac{(x^TAe_i)x^TAx}{(x^TA+b^T){\bf 1}}$$ where ...
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A consequence of convexity

Let $f:\mathbb{R}\to\mathbb{R}$ a convex decreasing function. Let $x_0 < x_1 < x_2$. Studying the behaviour of the difference quotient, it is clear that $$f(x_0)-f(x_2) \leq M (f(x_0)-f(x_1))$$ ...
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Boyd & Vandenberghe, question 2.31(d). Stuck on simple problem regarding interior of a dual cone.

In Boyd & Vandenberghe's "Convex Optimization", question 2.31(d) asks to prove that the interior of the dual cone $K^*$ is equal to (1) $\text{int } K^* = \{ z \mid z^\top x > 0 $ for all $ x ...
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221 views

How to show that a function is piecewise linear

Let z(t) = min $(c+t d)^T x$ s.t $Ax <= b$ Show that Z(t) is a concave, piecewise linear function of t. I'm really not sure how to even start proving this, I would really ...
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Show that maximisers are in corners

Let $\Pi$ be a rectangle $[a,b] \times [c,d]$ containing $0$: $a < 0 < b$, $c < 0 < d$, let $f(x)$ be a convex continuous function on $\Pi$. Define a functional $$ J(x_1,x_2,x_3) = p_1 ...
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Proof of an obvious fact about convex polygons.

Consider a closed simple polygon in the plane. It is intuitively obvious that the polygon is convex if and only if all the interior angles measure less than or equal to $\pi$ radians. I have never ...
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101 views

About sequence of convex function

Let $(f_n)_{n\in\mathbb{N}}$ be convex $C^2$ functions such that $$f_n(x)\xrightarrow[n\to\infty]{}f(x)\in\mathbb{R}\ \forall x\in\mathbb{R}.$$ Do you think it's true that for Lebesgue-almost every ...
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Find optimal measure

Let $\Omega$ be a convex compact set in $\mathbb{R}^n$, $f\colon \Omega \to \mathbb{R}$ be a convex function. Consider an optimization problem $$ \int\limits_{\Omega}f(x)\,\mu(dx) \to ...
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Limit of derivatives of convex functions

Let $(f_n)_ {n\in\mathbb{N}}$ be a sequence of convex differentiable functions on $\mathbb{R}$. Suppose that $f_n(x)\xrightarrow[n\to\infty]{}f(x)$ for all $x\in\mathbb{R}$. Let ...
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Does linearity decompose down convex sums?

I'm doing some convex optimisation where I'm minimising sum function $f(x) = \sum g_i(x)$, where the $g$'s are convex (and hence so is $f$) and the sum is finite. In doing so it turns out that $f$ is ...
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Proof of Non-Convexity

Am looking for a proof of non-convexity of the quotient of two matrix trace functions as given by $\frac{\operatorname{Tr}X^TAX}{\operatorname{Tr}X^TBX}$, when $TrX^TBX>0$ for two different ...
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about convexification

Let $f: \mathbb{R}^{n} \rightarrow \overline {\mathbb{R}}$. Called conjugate in the sense of Young-Fenchel of $f$, the following function: $$f^{*}(x^{*})=Sup\lbrace \langle x,x^{*}\rangle -f(x) : x ...
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$f$ is concave and convex in its arguments.

Suppose that X be a reflexive Banach space and function $f:X×X↦R$ which is concave in its first argument and convex in its second one. How to prove $f(x,x)=0$ for all $x∈X$?
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Convex functions: bounding the difference

Suppose you are given a convex function $f: R^d \rightarrow R$. Let us say you are given $x,x' \in R^d$ and there are $x_1, x_2, \ldots, x_n \in R^d$ such that $$\sum_{i=1}^n (x_i - x') = x - x'.$$ ...
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Why is the set of subgradients a convex set?

I'm struggling to understand an example we were given. The problem description is: Let $f$ be a convex function in $E^n$. Prove that the set of subgradients of $f$ in a given point form a ... convex ...
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How to prove this sign of the derivative?

Suppose that $u:[0,\delta]\rightarrow\mathbf{R}$, $u\in C^2((0,\delta))\cap C([0,\delta])$ such that $$u(0)=0,$$ $$u>0 \ \ in \ (0,\delta],$$$$u''>0$$ Then $u'>0$ in $(0,\delta)$.
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Minimal point of a intersection of N convex sets

I would like to prove that the minimal point of a intersection of $N$ convex sets in $\mathbb{R}^2$ is also the minimal point of the intersection of two of the aforementioned sets. Rephrasing the ...
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70 views

Convexity of a function

Suppose we have $F: R^n \longrightarrow R$ , $P: R^n \longrightarrow R^n$ and $G: R^n \longrightarrow R$ all nice- let's say given by polynomial and $P$ is invertible - such that $F(x) =G( P(x) )$. ...
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1answer
270 views

Existence of a direction of a convex set

Assume we are given a convex set $A$. A direction of this set is a unit vector $\bar x$ for which $\forall a \in A, \ \forall c>0, \ (a+c \bar x) \in A$. In other words it is a direction you can ...
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A convex combination coefficients

$x \in \mathbb{R}^{n}$ is a convex combination $C$ if there $p=p(x)\in \mathbb{N}$, $\lbrace \lambda_i\rbrace_{i=1}^{p} \subseteq [0,1]$ y $\lbrace x_i\rbrace_{i=1}^{p} \subseteq C$ such that $$ ...
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Regularity Conditions for Constrained Optimization

Question: Let $G$ be a convex mapping from $\Omega \subseteq X$ into a normed space $Z$, and assume that $P \subseteq Z$ be a positive cone with nonempty interior. Show that the following two ...
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Convexity of certain set

I need to show that $S = \{(x,y) \in \mathbb{R}^2 : y \geqslant x^2\}$ is a convex set, but I'm having a bit of trouble. Simply applying the definition of convexity has got me nowhere.
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Is this set compact?

Given two densities $f$ and $g$ the squared Hellinger distance between them is defined as follows $$ S(g,f)=H^2(g,f)=\frac{1}{2}\int_{\mathbb{R}}\left(\sqrt{g(y)}-\sqrt{f(y)}\right)^2 \mbox{d}y $$ I ...
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158 views

Is $S^\circ$ convex if $S$ convex?

Suppose $S\subset \mathbb{R^n}$ and $S^\circ$ denoted as the interior of $S$.Is $S^\circ$ convex if $S$ convex? $S$ is Convex mean $ \forall x,y\in S, kx+(1-k)y\in S, k\in [0,1]$ I know how to prove ...
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Matrices produced by bivariate convex functions

Suppose we have an $n \times n$ real-valued matrix $A = (A_{ij})$. When is it the case that there exists a bivariate "convex" function $f: \mathbb{R}^2 \to \mathbb{R}$, and a permutation $\pi$ on ...
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290 views

prove that m(a,y) is non decreasing in a and y and concave in a.

I am a university student specializing mathematics for economists. I am in a preparation for my final exam. My prof gave me some questions that might be on the exam. One of the question dragged me ...
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Concave lower bound on inner product of two distributions.

Given two discrete distributions $p, q$ which lie in an $m$-dimensional simplex, is it possible to provide a concave lower bound on the inner product between these distributions. That is we wish to ...
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996 views

Show this function is convex.

Could someone point me in the right direction for proving the following? Given that $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ is an affine map given by $f(x)=A\mathbf{x}+\mathbf{b}$, ...
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385 views

Gradient of log softmax in matrix form

Suppose $J(\mathbf{A})$ is defined as follows $$J=\text{tr}(\log \mathbf{P})$$ $$\mathbf{P}=\frac{e^\mathbf{A}}{\mathbf{1} \mathbf{1}' e^\mathbf{A}}$$ where division, exp and log are taken pointwise, ...