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.

learn more… | top users | synonyms (3)

0
votes
2answers
35 views

Is the logarithm of sum of multiple variables with the constraint on sum of them Concave?

I know that without any constraint $log \sum_{i=1:1:m} \alpha_i C_i $ is not Concave but I am wondering is this function Concave when we have the constraint that $ \sum_{i=1:1:m} \alpha_i =1 $ and ...
1
vote
2answers
43 views

Strictly increasing, strictly convex function: is the second derivative positive?

Consider a twice continuously differentiable function $f \colon \mathbb{R} \to \mathbb{R}$. While $f''(x)>0\ \forall x$ implies strict convexity of $f$, the converse is not true (e.g. $f(x)=x^4$, ...
6
votes
3answers
180 views

Show that $\left(\int_{0}^{1}\sqrt{f(x)^2+g(x)^2}\ dx\right)^2 \geq \left(\int_{0}^{1} f(x)\ dx\right)^2 + \left(\int_{0}^{1} g(x)\ dx\right)^2$ [closed]

Show that $$ \left( \int_{0}^{1} \sqrt{f(x)^2+g(x)^2}\ \text{d}x \right)^2 \geq \left( \int_{0}^{1} f(x)\ \text{d}x\right)^2 + \left( \int_{0}^{1} g(x)\ \text{d}x \right)^2 $$ where $f$ and $g$ ...
0
votes
1answer
36 views

Quasiconvexity of linear-fractional composition

In Boyd and Vandenberghe Section 3.3.4, it is stated that compositon of a quasiconvex function with an affine-fractional transformation is quasiconvex. In specific, if $f(x)$ is quasiconvex, then ...
0
votes
3answers
70 views

Why is the empty set convex?

Why is it the empty set, trivially convex? I see this results stated into a proof as something known, but I do not understand what's the idea idea behind it. How could I reason about convex ...
2
votes
0answers
33 views

Proof $f(x,y)=x_1+e^{x_{2}}$ is strictly convex

I am trying to show that $f(x,y)=x_1+e^{x_2}$ is strictly convex. I can show this using the Hessian Matrix which is positive definite. However for some reason i can not put it together using algebra ...
0
votes
1answer
23 views

Convex hull of $\{ \Vert x \Vert = 1 \}$ is closed in strictly convex space

I'm trying to show that the convex hull of $\{ \Vert x \Vert = 1\}$ is closed in a strictly convex Banach-space. I don't know how to tackle the problem. Are there any nice characterizations for a ...
0
votes
1answer
46 views

Functional analysis - check that a closed subspace of a Hilbert space is convex

Suppose that V is a Hilbert space over $F$ and $W$ is a closed subspace of $V$ . Then for every $x \in V$ , there exist unique $y \in W$ and $z \in$ (the orthogonal compliment of $W$) such that $x = y ...
1
vote
0answers
19 views

Core points of a convex set

In the book of Gamelin "Unifrom Algebras" I found the following definition: Let $K \subset V$ be a convex set in a vector space. An element $x \in K$ is called core point of $K$ if whenever $z ...
6
votes
1answer
106 views

Inf-convolution, two basic questions?

Let $E$ be a normed vector space. Given two functions $\varphi$, $\psi : E \to (-\infty, +\infty]$, one defines the inf-convolution of $\varphi$ and $\psi$ as follows: for every $x \in E$, ...
2
votes
1answer
62 views

Hint on how to proof that $x^2$ is convex

Note: I can't differentiate 2 times and prove that $f''(x) > 0$ The exercise requires me to prove that the function $f(x) = x^2$ is convex by using the following Theorem: $f(x) \ge f(x^*) + ...
1
vote
0answers
20 views

Analytical algorithm to obtain solution to convex optimization problem.

Assume a vector $\vec{P}$ with N elements $\in \mathbb{R}^+$ and constants $T_P$ and $\epsilon$. The vector $\vec{P}$ is arranged in a column $(N\times 1)$. Consider the problem: $$ \begin{aligned} ...
0
votes
0answers
37 views

is this function an ill-shape convex function?

I have a function with parameter $\vec{{\alpha}}$ where it is formulated by the formula: $$ f(D|\alpha)=n_1{\alpha}_{1}+...+n_m \alpha_m -Nlog \sum_{i=1:m} exp(\alpha_{i}+g_i(D)) $$ where $g_i{(D)}$ ...
0
votes
1answer
28 views

How large can the set $\partial ( \bigcup_i B_i ) \setminus \bigcup_i ( \partial B_i )$ be, where the $B_i$ are open balls in $\mathbb{R}^n$?

Suppose that $$E=\bigcup_{i=1}^{\infty}B_i,$$ where the $B_i$ are open balls in $\mathbb{R}^n$ and for $i\ne j$, $B_i \cap B_j = \emptyset$. We know that generally $$\bigcup_i \partial B_i \subsetneq ...
3
votes
1answer
29 views

First order condition in constrained optimization: Alternative characterization via normal cones

Consider the following constrained optimization: Min $f(x)$, $x\in C\subset R^n$, where C is convex. We know that one characterization of a local minimum (necessary condition) is the following: ...
-1
votes
1answer
30 views

The sum of two strongly convex functions is strongly convex

Assume that $f$ is $a$-strongly convex and $g$ is $b$-strongly convex. Is the sum $f+g$ strongly convex, and with what constant? Definition: $f$ is $a$-strongly convex if $$ f(x)-f(y) \le ...
0
votes
1answer
23 views

integer valued outer normal vectors

Suppose a bounded polyhedra $C$ is given by $$x\in \mathbb R^n: Ax\leq b$$ The matrix $A\in\mathbb R^{m\times n}$ contains only elements from $\{-1,0,1\}$, which implies the outer normal vectors of ...
1
vote
2answers
42 views

Minimising a convex set. Is set of solutions convex?

We are minimising a convex function on a non-empty set defined by linear constraints (equalities and inequalities). $X^O$ is the set of all optimal solutions and we assume it is non-empty. Is it true ...
1
vote
0answers
46 views

How to find the tangent cone to a set in a point?

Let $S\in R^{n}$ is a set and $x\in S$. We define tangent cone of $S$ in $x$ as: $$T_{S}(x)=\{z\in R^{n}:\exists (x_{k}), x_{k}\in S, x_{k}\rightarrow x, \exists (y_{k}), y_{k}>0, ...
3
votes
0answers
17 views
1
vote
1answer
35 views

sum of two cones

A non-empty set $K$ of a vector space is called a cone if it satisfies the following: $ K +K \subseteq K,$ $\alpha K \subseteq K$ for all $\alpha \ge 0,$ $K \cap (-K) ={0}$. Let $K_{1}$ and ...
0
votes
0answers
21 views

All facets' coefficients contain only integers -1,0,1

Suppose a polytope $C\subset \mathbb R^{kl}$ is the $l$-product of $k-1$-simplex with extreme points containing coordinates $0$ or $1$ in each coordinate. A linear transformation is given by ...
3
votes
1answer
16 views

Regularization by inf-convolution

Let $E$ be a n.v.s. and let $\varphi: E \to (-\infty, +\infty]$ be a convex l.s.c. function such that $\varphi \not\equiv +\infty$. Let$$\varphi_n(x) = \inf_{y \in E} \{n\|x - y\| + \varphi(y)\}.$$ ...
3
votes
3answers
57 views

Is finding the second derivative of $\sqrt[3]{\vert x\vert}$ the best method to determine if it is convex?

I have an exercise where I have to tell on which intervals a function is concave or convex. I usually do it using second derivative, but I would like to know if there is a simpler way of doing so, ...
0
votes
1answer
23 views

For any two disjoint convex open sets there is a hyperplane that strictly separates them

How to prove the affirmation?: If $K_1$ and $K_2$ are nonempty, nonintersecting, convex and open sets, there exists a closed hyperplane $M$ such that $K_1$ and $K_2$ are strictly on opposite sides ...
0
votes
1answer
31 views

Convexity of the Riemann-zeta function without derivative

Proving that the $\zeta$ function is convex on $(1,+\infty)$ is pretty simple if we use the derivative, but is there a proof without using derivative? I'm allowed to use just the definition of the ...
1
vote
2answers
24 views

preservation of extreme points under linear transformation

Suppose $\{e_1,...,e_N\}$ is the set of all extreme points of a compact convex subset $X\subset\mathbb R^n$. $L: \mathbb R^n\to \mathbb R^m$ is a linear transformation. $L$ is surjective but is not ...
12
votes
2answers
143 views

Is $\frac{1}{2}\|x\|^2$ the only function that is equal to its convex conjugate?

Is $\frac{1}{2}\|x\|^2$ the only function that is equal to its convex conjugate? The convex conjugate is defined as $$ f^{*}(x) = \sup_y\{\langle x, y\rangle - f(y)\}. $$
0
votes
0answers
49 views

How to reshape a nonlinear inquality into a linear matrix inequality?

We have these two nonlinear inequalities (I): $$x^2+y^2>0$$ $$3x^2+3y^2-4y^6>0$$ We want to represent this problem as a Linear Matrix Inequality Problem, i.e, we want to derive a positive ...
8
votes
2answers
74 views

Balanced cutting of a convex polygon

Given a convex polygon $C$ and a number $R\geq 1$, say that a point $x$ is an $R$-balance-point of $C$ if every line through $x$ divides $C$ to two parts $C_1,C_2$ such that: $$1/R \leq ...
2
votes
2answers
54 views

Convexity implies absolute continuity?

The following is taken from an exam: $f:[a,b]\rightarrow\mathbb{R}$ is convex implies $f$ is absolutely continuous (recall $f'$ exists a.e.) One has local Lipschitz-ness by convexity, but how to ...
4
votes
0answers
36 views

Is $A$ convex if and only if $-\ln(i_{A})$ is convex?

Is it correct that we have : $A \subset \mathbb{R}^n$ is convex if and only if $-\ln(i_{A})$ is a convex function? where here $i_A$ takes the value 1 at $x \in A$ and $+\infty$ elsewhere. I have ...
0
votes
2answers
17 views

Convex Set with Empty Interior Lies in an Affine Set

In Section 2.5.2 of the book Convex Optimization by Boyd and Vandenberghe, the authors mentioned without proving that "a convex set in $\mathbb{R}^n$ with empty interior must lie in an affine set of ...
1
vote
1answer
31 views

Tangent lines for convex functions

In theorem 1 here, the author says that if $\phi$ is a convex function on $(a,b)$ then for every point $c\in (a,b)$ there exists a line $L$ that passes through $c$ such that the graph of $\phi$ lies ...
0
votes
1answer
52 views

an example of a non convex ideal [closed]

As an example of a non convex ideal we have in Gillman and Jerison, Rings of Continuous Functions, 1976, Exercise 5E(1), the ideal $I= (|\operatorname{id}_{\mathbb R}|)$ in $C(\mathbb R)$. I need to ...
3
votes
1answer
61 views

Conjugate of a function

Yes this is homework. Given $f(x) = 1^{T}(x)_+$ where $(x)_+ = \max\{0,x\}$, what is $f^*$? I know that the conjugate of a function $f$ is $f^*(y) = \sup (y^Tx - f(x))$ but I do not know how to show ...
3
votes
1answer
195 views

Prove that a convex function on $\mathbb{R}^n$ is continuous [duplicate]

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a convex function on $\mathbb{R}^n$. How to prove that $f$ is continuous?
3
votes
0answers
704 views

Is a convex function defined on a convex open subset of $\mathbb R^n$ continuous? [duplicate]

Let $K$ be a convex open set in $\mathbb R^n$ and $f$ a convex function defined on $K$; how to show that $f$ is continuous?
2
votes
1answer
45 views

In the Krein-Milman theorem, can the weak closure of the convex hull be replaced by norm-closure?

I have a question on the following formulation of the Krein-Milman theorem: Consider a vector space $X$ equipped with the weak topology induced by a separating space $X^*$ of functionals on $X$. ...
1
vote
0answers
32 views

translate of a homothet of a convex body

Suppose we have given a convex body $K \subset \mathbb{R}^2$. How can we prove that it contains a translate of its homothet $-\frac{1}{2} K$? hint: take three vertices $A, B$ and $C$ of the convex ...
1
vote
2answers
46 views

How can we find the area of the triangle which covers a finite point set in $\mathbb{R}^2$ by using the interior triangles with specified area?

Suppose we have given a finite point set $X \subset \mathbb{R}^2$ in a way that any triangle made by vertices of $X$ has area at most 1. How can we prove that there is a triangle of area 4 which is ...
1
vote
0answers
58 views

Taylor Expansion of Power of Cumulative Log Normal Distribution Function - Show Lagrange Remainder tends to Zero

QUESTION I am looking to find a simplification of the expression below. I have attempted this using the Taylor series. The question then remains if we can show the Lagrange remainder goes to zero. I ...
1
vote
0answers
24 views

Proving concavity of a function

Let $f:[a,b]\rightarrow\mathbb{R}$ be twice-differentiable. Show that $f$ is concave if and only if $f''(x)\leq0$ for all $x\in[a,b]$. Moreover, if $f''<0$ for all $x\in[a,b]$, $f$ is strictly ...
0
votes
1answer
14 views

Upper bound for slope and the finiteness of Legendre transform

Let $f$ be a convex function and denote with $f^{*}$ the legendre transform. Is it true that if $f^*(p)=+\infty$, then $p$ is an upper bound for the slope of $f$? Do you have that if the slope of $f$ ...
0
votes
1answer
35 views

Properly or strictly separated sets

Let $A=\{ x,y,z: x,y,z\in[0,1] \}$ and $B=\{(x-2)^{2}+(y-2)^{2}+(z-2)^{2}\le 1\}$. Show if the sets $A$ and $B$ can be properly or strictly separated. Does anyone know the solution of this problem?
1
vote
0answers
37 views

Eigenvalues of a convex sum of two positive-definite matrices (with a lot of collateral information about them)

I have two positive-definite matrices written as a sum of rank-1 matrices $P_i$ (not projectors) $$ S_1=P_1+P_2+P_3+P_4,\\ S_2=P_5+P_6+P_7+P_8. $$ It is not an eigendecomposition ($P_i$s are not ...
4
votes
1answer
56 views

Is it sufficient for convexity?

Is it true that the sufficient and necessary condition for a real-valued function $f$ to be convex on an open interval $I$ is that i) $f$ is continuous on $I$ and ii) $\bar{D}_2 f \geq 0$, where $$ ...
0
votes
2answers
34 views

Convexity and local maxima

If a continuous function $f$ on $(a,b)$ is not convex, there is some choice of number $m$ so that $g(x)=f(x)+mx$ has a local maximum at a point $z$ inside the interval $(a,b)$. Is that true?
1
vote
0answers
29 views

Everything about Legendre transform

The Legendre transform, or transformation, seems to have many properties which are useful in different fields. For example: It switches between Lagrangian and Hamiltonian formalism in mechanics / ...
0
votes
1answer
41 views

Is this combination of convex functional is still convex?

Let $u$, $v\in C_c^\infty$ and $\Omega\subset \mathbb R^N$ is open bounded, smooth boundary. We also assume that $0\leq v\leq 1$. Define $$ F(u,v):=\int_\Omega |\nabla u|^2v^2dx. $$ Do we have ...