Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
5 views

Prove convexity of the given function on $\mathbb{R}^n$

$f:\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ is given by $$f(x)=\mbox{max}\{ (z_1-x_1)^{+}, (z_2-x_2)^{+}, ..., (z_3-x_3)^{+} \}$$ where $x=(x_2,x_2,...,x_n) \in \mathbb{R}^n$ and ...
1
vote
1answer
22 views

“Norm of norms” is another norm?

Suppose that, for some finite-dimensional real vector space $\Bbb R^n$, that $n_1(v)$, $n_2(v)$, ..., $n_k(v)$ are a set of norms on the space. Given some $v$, then, we can look at the "vector of ...
1
vote
0answers
13 views

Notions of Convexity

Let $(S,g)$ be a complete Riemannian manifold and $M\subset S$ an embedded submanifold (of the same dimension) with non-empty boundary $\partial M$. I am interested in understanding the relation ...
0
votes
1answer
19 views

Inequality for convex functions

If $f$ is a convex function then, for all $a<b$ and $0\le c<b-a$, $$f(a)+f(b)\ge f(a+c)+f(b-c).$$ What is the shortest proof for the inequality?
1
vote
1answer
31 views

if $ A \in R^{n \times n}$ , $A > 0$ and $ b \in R^n$ then the function $\frac{1}{2}\langle Ax,x\rangle - \langle b,x\rangle$ is convex in $R^n$

Show by direct estimates that if $ A \in R^{n \times n}$ , $A > 0$ and $ b \in R^n$ then the function $$\frac{1}{2}\langle Ax,x\rangle - \langle b,x\rangle$$ with $x$ is convex on $R^n$. My ...
0
votes
0answers
12 views

Well-definedness of greatest convex function smaller than function

A proof I am reading relies on the existence in $\mathbb{R}^d$ of a greatest convex function $f_0:\mathbb{R}^d \rightarrow \mathbb{R} \cup \{-\infty\}$ less than or equal to a (continuous) function $f:...
0
votes
1answer
24 views

Equivalence of logarithm-convex.

‎we have: A function ‎$‎f:I‎‎\rightarrow‎‎\mathbb{R}‎^+‎$ ‎is ‎log-convex ‎if ‎and ‎only ‎if‎ ‎\begin{align*}‎‎ ‎f(‎\lambda‎ x + u y)\leq f^{‎\lambda‎}(x) f^u (y) ‎\end{align*}‎‎ ‎for ‎$‎x, y\in I‎$ ‎...
1
vote
1answer
17 views

Legendre-Fenchel conjugate: $\dot x \in \operatorname{dom}(\Psi^*), -\operatorname{D}\mathcal E(x) \in \partial \Psi^*(\dot x)$

Consider $\mathcal E: \Bbb R \to \Bbb R$ lower semi-continuous with $\mathcal E(x) \to \infty$ if $|x| \to \infty$ and the equation $\dot x = -\nabla \mathcal E(x)$. Apparently, this differential ...
0
votes
0answers
7 views

Epigraphical Cones, Fenchel Conjugates, and Duality

I'm trying to derive a result relating cones conceived as epigraphs of convex functions, duality, and Fenchel conjungates. Let me state exactly what I'm looking for: Let $\mathbb{E}$ be an Euclidean ...
0
votes
1answer
42 views

Prove that a half-plane is convex

Prove that the following set is convex. $$S = \{ (x_1, x_2) \in \Bbb R^2 \mid x_1 + x_2 \geq 1 \}$$ I'm still confused about how to prove this.
1
vote
1answer
14 views

Complement of half spaces covering boundary of a convex body is a polytope

I have the following problem about compact convex sets. Let $K\subset\mathbb{R}^n$ be a compact convex set with nonempty interior. Assume that $A_1,\dots,A_m$ are open half spaces that cover $\...
0
votes
0answers
19 views

convex body in R ^ n [on hold]

Let $ K\subset{\mathbb{R}^n}$ convex and compact with $0\in int(K)$, for each  ${t>1} $.Prove that there exist $ s>0 $ such that $s\overline{B}\subset{int(K)} $ and $K\subset{st\overline{B}}$, ...
0
votes
0answers
10 views

Compact convex body

Let $L$ and $K$ be two compact convex bodies in $\mathbb{R}^n$, suppose $L$ is proper subset of $K$ and $0 \in \operatorname{(L)} $. Prove that there exists a point $p\in \operatorname{int}(K)-L$. ...
0
votes
1answer
22 views

Multivariate convex function

Assume $f : \mathbb{R}^n \to \mathbb{R}$ is a function that depends on $x\in \mathbb{R}^m$ and $y\in \mathbb{R}^{n-m}$. If it is known that for any $x_0 \in \mathbb{R}^m$ function $f(x_0,y)$ is convex ...
1
vote
2answers
53 views

How reliable is the linear problems like $ \min \|Ax - b\|^2$?

The following linear optimization is common used $$\min_x \|Ax - b\|^2$$ Here, $A$ is the matrix; $x,b$ are the vectors. I am curious about how reliable is the solution $x$ by solving above ...
1
vote
0answers
15 views

Continuity and Lipschitz Property of Infimal Convolution

If $f$ is convex and Lipschitz-continuous on a real Hilbert space $H$ and $g$ is lsc and convex then is the infimal convolution $f\square g$ Lipschitz?
2
votes
1answer
71 views

Proving that the following inequalities are equivalent

Prove that if ‎$‎‎f:(0,+‎∞‎)→\mathbb R‎$ ‎is a‎ ‎‎continuous ‎function, then the following are equivalent for every $x\in(0,+‎∞‎)$: (1)‎$‎‎‎‎‎\displaystyle\frac{‎‎f(x_4)-f(x_3)}{{x_4}-x_3}‎\leq‎‎‎‎\...
0
votes
0answers
21 views

A function that is convex but not strictly convex

Let $c_1,c_2,...,c_m\in\mathbb{R}^n$ and $b_1,b_2,...,b_m\in\mathbb{R}$. Consider $\mathbb{R}^n\ni x \mapsto f(x)=\displaystyle\max_{1\leq i\leq m}\Arrowvert c_i^Tx+b_i \Arrowvert$. Prove that $f$ is ...
0
votes
0answers
25 views

Is the Legendre transformation continuous?

Is the Legendre transformation continuous on the space of the convex $\mathbb R^{n} \to \mathbb R$ functions for the topology of the simple convergence? First the limit of a sequence of convex ...
0
votes
0answers
14 views

Polyhedra, half spaces and compact

I would like to show that $\{ x ; \forall j \in \{1,...,n\}, \langle x, z_{j} \rangle \le s_{j} \} $ is compact when the convex cone generated by the $(z_{j})_{1 \le j \le n}$ is $R^{d}$. I have been ...
1
vote
0answers
16 views

Propriety of cone function

Ok, I've got this exercise: Let $K$ be a convex body in $\mathbb{R}^n$ and let $\phi: K \to \mathbb{R}$ be a cone function. Set $M := \text{max}_{x\in K}\phi (x)$. Prove that, for $0<t<M$, it ...
0
votes
1answer
12 views

A multivariate function is convex iff it is convex in all axes?

Does the following statement is true? And if so, how can one prove it? Given the function $f:R^n->R$ And it is given that for every $x_i\in \bar{x}$ setting $x_j$ $j\neq i$ to zero The function $f(...
0
votes
0answers
14 views

Show pseudoconvexity of a function

I need to show that a function $f(x): S \to \mathbb{R}$ defined by: $f(x)=\frac{g(x)}{h(x)}$, where both $g(x)$ and $f(x)$ are defined $:S \to \mathbb{R}$ and they are differentiable, it's ...
0
votes
0answers
8 views

Condition for convexity and quasi-convexity for single variables

A function is convex on I if the second derivative is positive on that interval. What is the related condition for quasi-convex (single variable) functions? I understand the difference between them, ...
0
votes
0answers
17 views

condition of convexity when midpoint convex

Let $f:I \rightarrow R$ be a function satisfying the equation $f(\dfrac{x+y}{2}) \leq \dfrac{f(x)+f(y)}{2}$ The question is, 1)Is $f$ continuous when $I$ is closed? 2)Is $f$ continuous when $I$ is ...
0
votes
2answers
24 views

Separation of two disjoint convex closed sets

Assume that $A,B\subset \mathbb{R}^n$ are two disjoint closed convex sets. Without using that $A$ and $B$ are closed sets, it follows already, that there is a non zero element $v$ and a real number $c$...
2
votes
0answers
27 views

Stochastic programming: Is the linear program over the vertices the same as over the simplex?

Suppose we have a random variable $W$ with probability distribution, $\Pr(W = w) = p_w \in [0,1], \quad w \in I = \{1, \ldots n\}$ Consider the maximization problem: $$\max\limits_{w \in I} \...
0
votes
0answers
27 views

The measures which are non-negative on all convex non-negative functions

Let $X$ be a subset of $\mathbb{R}^n_{++}$ (vectors with non-negative coordiantes), and let $M$ be the set of regular Borel measures with finite variation on $X$ and finite first moment: $$M:=\{\mu;\ |...
0
votes
0answers
41 views

Convexity properties of a cone

Let $M$ be a normed space and $A$ a subset (nonempty) of the unit sphere ($S_1$, which is the points of norm $1$). Define, for $\alpha >0$, a $A^{\alpha}=\{x\in S_1 | d\left( x,A\right) \leq\alpha\}...
0
votes
0answers
21 views

KL divergence contraction coefficient - basic question

I'm studying the Blahut Arimoto algorithm using these notes and towards the end of section 6, an interesting quantity arises. The author does not talk about how to compute it so I was hoping I could ...
0
votes
0answers
14 views

Integral of a Function of a Convex Combination bounded above by the integral over the entire ball

I am working on this homework problem, and I went to the professor for help but I don't think his solution is correct (it seems to assume that $W^{1,p} \cap L^\infty$ functions are uniformly ...
0
votes
1answer
28 views

Recession cone characterization.

Let $\mathbb{C} \subset \mathbb{R^n}$ be a convex, closed and not-bounded set. Let $d$ be a vector which $||d||=1$, then show that: $d \in recc(\mathbb{C})$ $\iff$ $\exists$ $\{x_k\}_{k\in\mathbb{N}}\...
0
votes
0answers
12 views

Self-weighted average of the function of a Random Variable

Thanks for your help with the following. For $y>0$ and $n>0$, a random variable $X$ as well as a bivariate function $(y,x)\mapsto f(y,x)$ that is increasing and concave in $y$ and decreasing in ...
0
votes
2answers
92 views
+50

Distance between a cone and a disjoint hyperplane

I seek to prove the following, which I guess is true: Define $A:=\{x \ge 0\} \subset \mathbb{R}^m$ and assume that $U\subset \mathbb{R}^m$ is an affine subspace with $A \cap U=\emptyset$. Show that ...
0
votes
1answer
20 views

How can the infimum of this function be written this way?

From a step in a convex analysis proof: $$\large \inf_{v \ge 0}(\frac{(d - v)^2}{v})= \{4d \text{ if } d \le 0, 0 \text{ if } d \ge 0\}$$ Can someone explain why this is true? I can't see why this ...
0
votes
1answer
66 views

Does there exist a real function whose square is concave? [closed]

Does there exist a function $f : \mathbb{R} \rightarrow \mathbb{R}$ whose square $f^2$ is concave? edit: where $f$ is not a constant function.
0
votes
2answers
32 views

Basic question about convexity

A convex function is defined as one that satisfies the following condition for $p_1 + p_2 = 1$. $$f(p_1x_1 + p_2x_2) \leq p_1f(x_1) + p_2f(x_2),$$ Does this imply that for all $\lambda \leq 1$ $$...
0
votes
1answer
20 views

Strong convexity of Loss function in Logistic Regression

I wonder if the Loss function of a Logistic regression can have strong convexity when the explanatory variables are linearly independent. From a theoretical point of view, if I have a sample of p ...
1
vote
0answers
32 views

Question related to problem 4.6 of Convex Optimization (Boyd & Vandenberghe)

In problem 4.6 of the book they ask under which condition the following problem can be converted into a standard convex optimization problem. $\text{min. } f_0(x)$ $\text{s.t. } f_i(x)\leq 0, ~~ i \...
1
vote
1answer
23 views

Question related to exercise 4.4(d) of Convex Optimization (Boyd & Vandenberghe)

The question is as follows: Suppose $G=\{Q_1,\cdots Q_k\}\subset R^{n\times n}$ is a group. We say that the function $f$ is $G$ invariant or symmetric with respect to $G$ if $f(Q_ix)=f(x)$ holds ...
1
vote
1answer
46 views

Maximum of the root of a quadratic functional with inequality constraint

The function $f(x)$ is the larger eigenvalue of sum of two rank-one matrix. My aim is to get the upper bound of the eigenvalue. The problem can be described by $$ \max_{x} f(x)=b^Tx+\sqrt{x^TAx}\\ ...
1
vote
1answer
49 views

How to prove convexity of an optimization problem?

Consider the following optimization problem. Let $d_3, d_2, d_1 > 0$. Maximize $\log(p_1)+\log(p_2)+\log(p_3)$ Subject to: $p_1d_1 + p_2d_2 + p_3d_3= 1$ $p_1 \geq p_2\geq p_3\geq 0$. I ...
0
votes
3answers
33 views

Proving an inequality with convexity [closed]

Hello i struggle prooving this inequality $$abc\leq\frac13 (a^3+b^3+c^3)$$ $(a,b,c)$ are positive reals I tought about using the fact that the exponential is convex and the jersen inequality but i ...
0
votes
0answers
41 views

Does a convex set look the same near its face?

Recall that a face of a convex set $X\subseteq\mathbb{R}^n$ is a convex subset $F$ of $X$ such that every line segment with endpoints in $X$ whose relative interior meets $F$ lies entirely in $F$. ...
0
votes
0answers
31 views

Convexity of quadratic forms

I do not understand a basic concept about the convexity of a quadratic form. I read that A quadratic form $q(h)=h^{T}\mathbf{A}h$ is convex if and only if $\mathbf{A}$ is positive semidefinite. ...
0
votes
0answers
17 views

How to determine whether the function $ f(x_1,x_2) = \frac{x_1^2-1}{x_2} $ is concave, convex, quasi-concave and quasi-convex? ($x_1,x_2 > 0$)

I know the definition of convex, concave, quasi-convex and quasi-concave functions but I am unable to determine this function as one of them. Based on these definitions, say for convex, I have tried ...
4
votes
1answer
73 views

convexity of a relatively open subset of a compact set

I'm struggling with the following problem: it seems to be true but I'm not able to prove it! Let $C$ be a compact convex subset of a locally convex metric vector space and $\hat{C}$ be a relatively ...
0
votes
0answers
13 views

About polyvariable functions.

Let f: R^n -> R is convex of every variable (if all another are fixed). Prove that f is are continuous. So, it is obviosly if n=1. But I can't do this for n>1. I think that there need induction. It ...
5
votes
1answer
72 views

Convexity cones and polar set.

Let $\mathbb{K} \subseteq \mathbb{R^n}$ be a closed convex cone. Show that: $\mathbb{K}$ is a linear subspace $\iff$ $\mathbb{K}^º \cap (-\mathbb{K})=\{0_n\} $ where $\mathbb{K}^º:=\{x\in\mathbb{...
1
vote
0answers
45 views

Optimisation under constraint of Wasserstein distance

Let $\mathcal P_n = \{P \in \mathbb R^n_{\geq 0}: P^T \mathbb I = 1 \}$, where $\mathbb I = (1,...,1)^T \in \mathbb R^n$ and $f: \mathcal P_n \to \mathbb R$ a convex and differentiable function (or ...