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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Prove: dFunction with certain determinant properties is injective

Let $ G \subset \mathbb{R}^n $ be a convex set, $ f : G \to \mathbb{R}^n $ continuously differentiable and $$ \text{det} \begin{pmatrix} \partial_1 f_1 (c_1) & \cdots & \partial_n f_1 (...
6
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1answer
22 views

Strongest topology that makes vector space locally convex

Here is an exercise from Barvinok's "A Course in Convexity" (ex. III.3.3.3, p.119): Prove that the strongest topology that makes a vector space $V$ a locally convex topological vector space is the ...
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1answer
33 views

Why it is a subset of a convex set?

Let $C$ be a convex subset of $\mathbb{R}^n$. (and $0$ is supposed in $C$.) I've read in the "Convex Optimization Theory" book of Bertsekas (page 25) that this set: $$X = \left\{\sum_{i=1}^m \alpha_i ...
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1answer
40 views

Problem calculating the Dual of a convex problem

The problem is $$ (P) \hspace{1cm} \begin{array}{ll}\min & e^{-x} \\ \text{s.t.} & \frac{x^2}{y} \leq 0 \end{array} $$ over the domain $\mathcal{D}= \{(x,y)| y>0\}$, which I tried to ...
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1answer
26 views

Is $ G(x) = \nabla F(x)^* $ continuous when is $F$ continuously differentiable in Frechet's sense?

Let $F: X \to Y$ be a continuously differentiable function between banach spaces $X$ and $Y$(we also can assume $X$ is reflexive space) Let $x_0 \in X$, this tells us the derivative $\nabla F(x_0) \in ...
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0answers
36 views

Does $x_i \to x$ and $y_i^* \overset{w^*}{\rightarrow} y^*$ imply $ \nabla F(x_i)^* y_{i}^* \overset{w^*}{\rightarrow} \nabla F(x_0)^* y^* $

Let $F: X \to Y$ be a continuously differentiable function between banach spaces $X$ and $Y$. (we also can assume $X$ is reflexive space) Let $x_0 \in X$, this tells us the derivative $\nabla F(x_0) \...
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1answer
29 views

Why does strong convexity imply this inequality?

From Convex Optimization by Boyd and Vandenberghe: Let $f: \Bbb R^n \rightarrow \Bbb R$ be continuous and twice differentiable. Assume $$\|\nabla f(x)\|_2 \le \eta \le 3(1-2 \alpha)m^2/L$$ ...
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1answer
36 views

Star-Convex Set Centers Form Convex Set

I have the following proof that all centers of a star-convex set $U\subset\mathbb{R}^n$, $Z(U)$ form a convex set: Suppose $Z(U)$ is not convex, then there are elements $a,b\in Z(U)$ such that the ...
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1answer
35 views

Gradient always points away from the minima of convex functions?

I'm reading a paper by Léon Bottou, "Online Learning and Stochastic Approximations". He studies online learning with cost functions $C(w)$ satisfying two conditions: $C(w)$ has a single minimum $w^*$....
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32 views

Prove the convexity of this function [on hold]

Prove the convexity of this function without verifying the definiteness of its Hessian matrix $$ f(x_1,x_2) = x_1 *\log_2(1+x_2/x_1) $$ with $$ x_1,x_2 \geq 0 ;\: x_1,x_2 \in \Bbb R $$
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0answers
17 views

Geometric interpretation of convex inequality

Suppose we have a strongly convex function f(x), that is bounded below, so Hessian H satisfies H>= mI for some constant m. Then we can write an inequality on its Taylor expansion as (1) $f(y) >= f(...
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1answer
32 views

$S$ is convex and compact (proof)

Assume the set: $$ S = \left\{ (x,y,z)^T \in \mathbb{R}^3 : y = z, \, x^2 +2y^2 \leq 1 \right\} $$ How do we prove it's convex and compact? My attempt: Convexity: Let $a \in [0,1]$ and assume $ax + ...
3
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1answer
62 views

If $\int_0^1 f(x+\theta(y-x))d\theta \leq \frac{f(x)+f(y)}{2}$ then f is convex

Suppose that we have $f: \mathbb{R}\rightarrow\mathbb{R}$ is Riemann integrable and $$ \int_0^1 f(x+\theta(y-x))d\theta \leq \frac{f(x)+f(y)}{2} \ \ \ (\ast) $$ for all $x,y\in\mathbb{R}$. Is this ...
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1answer
22 views

why can we know when constraint set is non-empty,then duality gap must be zero

The primal problem $max \sum\limits_{i=1}^{M}\sum\limits_{k=1}^{K} \tau log_2(1+\frac{|g_{k,i}|^2P_{k,i}}{\sigma^2_n})$,${P_{k,i} \ge 0,\forall k,i}$ subjected to 1.$\tau P_{k,1} \le w_{k,1}^{*(l-...
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1answer
26 views

If $C$ is convex then $\cup_{y\in C} B(y,r)$ is convex.

I am studying for an upcoming exam on convex optimization and one of the practice exercises that I am working through is the following; Let $C\subseteq \mathbb{R}^n$ be a convex set. Is the set ...
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0answers
11 views

Lower bound for Strongly convex and Lipshcitz gradient function

I was reading this paper but could not understand one of it's condition. It says a function $f(x)$ is twice differentiable and strongly convex with parameter $m$ and Lipschitz continuous gradient with ...
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1answer
30 views

The closure of the interior of the boundary is a subset of the closure of the intersection between a set and the interior of the boundary

The statement is the following: Let X be a topological space and let A be a subset of X. Then $ cl(int(\partial A)) \subseteq cl(A \cap int (\partial A) ) $ Notation: $cl$ means closure, $\...
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0answers
50 views

Convexify a bilinear problem

I have an optimization problem with the following constraint: $$|f(x)| - y \: \Re \{ g(x) \} < 0$$ where $x,y$ are optimization variables (with $x \in \mathbb{R}^n$, $\{y \in \mathbb{R}: y > 0\...
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2answers
66 views

Existence and uniqueness of projections on closed convex sets

Let $X$ be a normed vector space. I'm interested in determining what are the minimal assumptions on $X$ that guarantee the existence and uniqueness of projections on closed convex sets and in ...
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0answers
44 views

If the interior of the boundary of a set is nonempty, then the interior of that set is empty

I was reading the post " A set which the interior of its boundary is not empty ", and I conjectured the following: Let $ (X, \tau ) $ be a topological space, and let $A \subseteq X $. If $int(\...
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0answers
15 views

convex hull for tangent function

I would like to ask if I can obtain a convex hull for the function tan(S2-S1) where S1 and S2 are unknown angles while knowing the maximum/minimum limits on (S2-S1).
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1answer
45 views

Convexity in single variable

I am trying to solve the below non-convex minimization problem $$ \underset{X_1,x_2,X_3,\Lambda}{\mathrm{argmin}} \|A-X_1X_2\Lambda X_3 \|_F^2 $$ where $A \in \mathbb{R}^{n \times n}$, $X_1 \in \...
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1answer
14 views

Function whose graph is a cone

I hope your day is going well. Let $u : \Omega \rightarrow \mathbb{R}$ a convex function on a open convex bounded set. I read "Let $v$ be the convex function whose graph is the cone with vertex $(x_{...
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0answers
16 views

Is the energy associated to a Tonelli Lagragian convex and superlinear?

A Tonelli Lagrangian on a compact Riemannian manifold $(Q,g)$ is a smooth function $L:TQ\to \mathbb{R}$ such that $L$ is convex, i.e. $$\frac{\partial^2 L}{\partial v^i \partial v^j}(x,v)w^iw^j>0\...
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2answers
171 views
+100

Image of $T:E \rightarrow \alpha E + (1-\alpha) E$ where $\alpha>1$.

Fix a real number $\alpha>1$ and an integer $n \geq 1$. Let $T_\alpha$ be the mapping defined on the set $\mathcal{E}$ of closed convex subsets of $\mathbb{R}^n$ by \begin{equation*} T_\alpha(E) = \...
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0answers
8 views

Affine dependence and minimal dependence of a polytope?

I'm trying to follow along in a book with this example for calculating minimal dependencies of an affine dependence of a hexagon. Here $\mathbf{X}$ is the vertex set of a hexagon $$\mathbf{X}=(\...
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0answers
28 views

Efficient Algorithm to Find the Closest Points of Two Sets

Let's assume there are two sets $R$ and $B$ with each having $n$ points in the plane. Is there an Algorithm with $\Theta(n \log n)$ that can find points $r$ and $b$ with minimal distance such that $r \...
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1answer
27 views

References for this statement on convex functions?

Here is the statement : $\forall n \ge 2$, if $f : \mathbb{R}^n \to \mathbb{R}$ is a continuous convex function whose the non empty set $f^{-1}\{(0) \}$ is compact then $\lim \limits_{\Vert x \...
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0answers
21 views

convexity of negative log likelihood under constraints

The negative log-likelihood function \begin{align} \text{min}_B -log|\Sigma^{-1}| + \text{tr}[(Y-XB)\Sigma^{-1}(Y-XB)] \end{align} is a convex optimization with respect to $\Sigma^{-1}$. However, ...
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2answers
47 views

Why isn't this function affine?

I'm studying convex optimization using Convex Optimization (Boyd & Vandenberghe) and had a question from an example used in Chapter 4.2: Convex Optimization. The specific example is as follows: ...
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1answer
17 views

Strict inequality of the H1 seminorm of a curve emerging from a convex set and its convex projection

Let $\gamma(t) \in W^{1,2}([0,1]; \mathbb{R}^d)$ such that $\gamma(0) \in C$, with $C \subset \mathbb{R}^d$ some closed convex set, and $ \gamma(t) \not \in C, \ \forall t\in (0,1]$. Furthermore we ...
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0answers
41 views

Function convex if second derivative not negative

Let $X$ be a finite dimensional Banach space. A function $f:X\rightarrow \mathbb{R}$ is called convex, if $f\left( \left( 1-\lambda \right) x+ \lambda y\right) \leq \left( 1-\lambda \right) f\left( x\...
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1answer
23 views

Minimizing $\ell_p$ norm on a flat - what type of convex programming subproblem is this?

Suppose that we are in a finite-dimensional real vector space $\Bbb R^n$, and we are on a flat $F \subset \Bbb R^n$ (aka, an affine translation of a subspace of $\Bbb R^n$). We want to minimize some $\...
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1answer
29 views

Finite set of linear functional attaining certain values (Brezis functional analysis exercise 1.12)

Problem: Let $E$ be a vector space. Fix $n$ linear functionals $(f_{i})_{1\leq i\leq n}$ on $E$ and $n$ real numbers $(\alpha_{i})_{1\leq i\leq n}.$ Prove that the following properties are equivalent....
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0answers
15 views

Using notation of conjunctive normal forms for multi-objective optimization.

I need to maximize several objective functions $ f_i(x)$, that I have arranged in a vector. $ f(x) = [ f_1(x) , f_2(x) \cdots, f_K(x) ]^T $. Essentially, my question is whether I can represent the ...
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0answers
28 views

Is $K$ is convex?

Consider the sup-normed Banach space $C(S)$, which designates the space of continuous functions over a compact Hausdorff Space $S$. Let $K=\{\mu\in A^{\perp}:||\mu||\leqslant 1\}$. I tried ...
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0answers
18 views

Convex bodies and their convex hulls

Let $K$ and $L$ be two convex bodies that contain the origin in their interior. Show that: $$(K \bigcap L)^\omicron = \text{conv}(K^\omicron \bigcup L^\omicron) > \text{ and } \text{conv}(K\...
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0answers
4 views

Generalization to $n$ dimensions of partition into convex subsets of a function’s domain?

Let $f:\mathbb R\to\mathbb R$ be a smooth and bounded function with a finite number of local minima. Then we can partition $\mathbb R$ into a finite number of sets $\{ A_i\}$, such that the ...
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1answer
14 views

Support function of convex hulls

Show that if $K$ and $L$ are convex bodies, then: $$h_{conv(K \bigcup L)}=max\{h_K, h_L\}$$ First doubt is: $h$ should be the support function, defined as: $h_C(u)=\sup\ \{ u \cdot y:y \in C \} \...
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1answer
45 views

$f$ is convex, increasing then $\lim_{x\rightarrow -\infty }f(x)/x$ exists

I am stuck with the following question: Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be convex and increasing and $\displaystyle\lim_{x\rightarrow -\infty}f(x)=-\infty$. Prove that there exists $\alpha_{...
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1answer
28 views

What is the relationship between strict convex combination and convex hull of a set?

Let $x_i \in \mathbb{R}^n $ where $i=1,...,l$ and $1\leq l \leq n$. Also, let $$ W = \{y = \sum_1^l \alpha_i x_i \mid \alpha_i > 0 \sum_1^l \alpha_i =1 \} $$ be the set of all strict convex ...
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1answer
18 views

convexity about a function of many variable

assume $\mathcal X$ is a convex set and $\mathit f(x,a)$ is a function that $$f:\mathcal X \times \Bbb R \to \Bbb R$$ $f(x,a)$ is convex with respect to $x$ for every $a$ and $f(x,a)$ is convex with ...
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0answers
8 views

What is the difference between strict quasi-concave and quasi-concave function?

Been struggling to differentiate these two concepts. Is there an intuitive/graphical way to understand the difference between strict quasi-concave and quasi-concave functions? I understand the ...
-1
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0answers
14 views

Partial ordering induced by cones

Is the set of nonnegative vectors $V=\{ x\in E|x\geq 0\}$ always a closed convex cone under any partial ordering (in the Euclidean space)? I have so far proven it's a convex cone. Here's my proof: ...
1
vote
1answer
24 views

If section is always contractible, is that convex?

Consider compact set $\Omega\subset\mathbb{R}^d$, whose intersection with any $(d-1)$-dimensional subspace of $\mathbb{R}^d$ is contractible. Then, is such $\Omega$ convex?
5
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1answer
113 views

Problem in convex analysis : Easy or hard one?

I hope your day is going well. This is a problem, I don't know how to solve it since 1 week. It's going to be a relief and a pleasure to get your help. Problem : Let $X = \{x_{1},\ldots,x_{N+M} \}$ ...
0
votes
1answer
48 views

Why are the mollifiers negative in this sequence?

I'm reading a proof that starts the following way: Assume $E$ is open and $u \geq 0$ a.e. Given $K \subset E$ compact, let $\psi\in C_0^{\infty}(\mathbb{R}^n)$ satisfy: $$\psi = 1 \quad \text { ...
0
votes
0answers
40 views

If $f:(a,b)\to\mathbb R$ with $f''(t)\geq0\,\forall t\in(a,b)$, then $f$ is convex

$\textbf{Definition we use:}$ We call a function $f:(a,b)\to\mathbb R$ convex if, whenever $x,y\in(a,b)$ and $0\leq\lambda\leq1$, then $$f(\lambda x+(1-\lambda)y)\leq \lambda f(x)+(1-\lambda)f(y).$$ ...
1
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0answers
19 views

Sufficient conditions to have $\operatorname{co} (\operatorname{epi} f) = \operatorname{epi} (\operatorname{co} f)$

Let $f : \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty,-\infty\}$ and denote $\operatorname{co} f $ to be the convex envelope of $f$ and $\operatorname{dom} f := \{x \in \mathbb{R}^n : f(x) < +...
0
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0answers
29 views

Find $\ell_2$ shortest vector in simplex (related to simplex projection)

Suppose that I have an arbitrary $n$-dimensional simplex in a real vector space, and I want to find the $\ell_2$-shortest vector in the simplex. Is there some simple way to do so? There are two ...