2
votes
1answer
28 views

Affine to linear like conversion of a concave function

Is the following true: $$\log \left( \frac{1}{f(x)+K}\right)\mathrm{is\;concave}\Longleftrightarrow \log \left( \frac{1}{f(x)}\right)\mathrm{is\;concave},$$ where $K\in\mathbb{R} $ and ...
0
votes
0answers
26 views

Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
0
votes
2answers
32 views

Convexity of functions

I'm trying to read up on convex/concave functions (is that the same as concave up, concave down?) If I were asked to prove a convexity of a function, what are the general steps to follow? (So far I ...
0
votes
2answers
30 views

Composition of a convex function and a convex decreasing function is quasi-concave

Let $h$ be a convex decreasing function and $g$ a convex function. Is it true that $h(g(x))$ is a quasi-concave function?
1
vote
2answers
37 views

How to check the function is convex or not?

How one can check the function whether it is convex or not. I know one method by using Hessian Matrix but I think it did not fit for the following example. I think Hessian matrix method cannot be ...
0
votes
1answer
18 views

Question about strongly convexity and affinity

For a function $f$, it is said to be strongly convex if for all $x,y$ \begin{equation} (\nabla f(x) - \nabla f(y) )^T (x-y) \ge m \|x-y\|_2^2 \end{equation} for a constant $m \ge 0$ Is it called ...
1
vote
0answers
17 views

mid-point convex but not a.e. equal to a convex function

I recently stumbled upon Wikipedia's page on convexity (http://en.wikipedia.org/wiki/Convex_function#Properties) and there's reference to Sierpniski's theorem from which we can deduce that for ...
1
vote
1answer
79 views

Is $\sqrt{x}$ concave?

I have function $f(x)= \sqrt(x)$. To check is it concave or convex i am checkin $f''(x). $ Which is $ -\frac{1}{4x^{\frac{3}{2}}} < 0$ So the $f(x)$ is concave. Is it correct ? And is is the same ...
1
vote
1answer
17 views

Properties concave functions

Is is true that if $f(x)$ is a concave function of $x$ with domain $C$, then $f'(a) \leq \frac{f(a)}{a}$ for any $a \in C$, where $f'(a)$ denotes the derivative of $f(x)$ with respect to $x$ evaluated ...
1
vote
1answer
37 views

Difference of concave functions

Suppose that there are two concave functions $f_1(x)$ and $f_2(x)$ defined on $x\geq0$. In addition, the functions are positive, smooth, bounded ($|f_2|\leq b_2,|f_1|\leq b_1$ such that $b_2 = ...
1
vote
2answers
56 views

Composition of non-monotonic convex function

Given the following composition of functions: $h:\Bbb R^k\rightarrow\Bbb R$ $g:\Bbb R^n\rightarrow\Bbb R$ $f(x)=h(g_1(x),g_2(x),...,g_k(x))$ There are known rules which guarantee ...
0
votes
1answer
16 views

Establishing convexity of a function

Let $\theta \in \Theta \subset \mathbb{R}^k$. I have the following objective function $$ F(\theta):=||\max(0,f_1(\theta)),...,\max(0,f_n(\theta))||^2 $$ where $||\cdot||$ is the Euclidean Norm and ...
0
votes
1answer
36 views

Convex function that has a finite limit at infinity

Can someone give me an example for a convex function that has a finit limit at infinity ?
0
votes
1answer
42 views

Intersection of strictly monotone, convex functions on interval with two given intersections

Given two functions $f,g:[0,1]\rightarrow[0,1]$ that are smooth, strictly convex, strictly monotone s.t. $f(1)=g(1)=0$, $g(0)=f(0)=1$ and $f(x_0)>g(x_0)$ holds for some point $x_0$, does it follow ...
0
votes
1answer
14 views

Problem with convex function

In Papadimitriou book I found a problem. If I know that function $f$ is a convex function, and I have values $x_2,...,x_n$, is function $g(x_1) = f(x_1,x_2,...,x_n)$ also a convex function? I know ...
0
votes
1answer
93 views

Matlab - Generate square convex function with positive definite Hessian Matrix

So, I have to generate a square convex function in Matlab and it's Hessian Matrix must be positive definite but I can't find any function that can help me do that. Is there anything I should search ...
0
votes
3answers
133 views

Concave function divided by a convex function. What is the result?

Let us say that I have a function $f(x)$ that we know is a concave. And let us also say that we have another function $g(x)$ that is a convex. If I make a new function, $h(x) = \frac{f(x)}{g(x)}$, ...
1
vote
1answer
191 views

Lipschitz Smoothness, Strong Convexity and the Hessian

I'm working with the following two concepts: Lipschitz Smoothness - a function $f$ is Lipschitz smooth with constant $L$ if its derivatives are Lipschitz continuous with constant $L$, in other words ...
1
vote
3answers
86 views

Convex functions - two questions

I have two questions regarding convex functions: First question: Let f be convex function on closed interval [a,b]. Prove that f has maximum in x=a or x=b. I understand that $\forall ...
0
votes
2answers
347 views

How to prove that the product of a decreasing monotonic function and a strictly increasing monotonic function is a concave function?

Given that I have a strictly increasing monotonic function $f$ and a decreasing monotonic function $g$, are there any nice properties to show that the product function $h(x) = f(x)g(x)$ is a concave ...
0
votes
2answers
107 views

Is the function of two strictly concave functions also concave?

This may be a trivial question to most, but here we go: I have two strictly concave functions, say $f(x)$ and $g(x)$. From this can I say that a function of those two functions, $h[f(x), g(x)]$, is ...
2
votes
1answer
88 views

Why can we assume WLOG that $x$ is zero?

I am new to mathematics, so I apologize in advance if this question is trivial. I was trying to prove a property of an arbitrary three-point system in $\mathbb{R}^2$ regarding convexity. I tried it ...
0
votes
0answers
480 views

sum of concave and convex function

Suppose $f$ is the sum of a concave and convex function, i.e. $$f=f_1+f_2$$ where $f_1$ is a concave function and $f_2$ is a convex function. I wonder if $f$ can be written as the following: ...
2
votes
0answers
33 views

Is the following function of several variables concave?

Suppose that $w_1,\dots,w_p\in(0,1)$ satisfy the condition $\sum_i w_i=1$, and let $$F(w_1,\dots,w_p) = \frac{1-\sum_i w_i^2}{\sum_i(1-w_i)^2 \left[\sum_i \frac{w_i}{1-w_i}\right]^2}.$$ Is function ...
0
votes
1answer
39 views

Special type of convexity

Does anyone know if there is a name for a function $f:W\rightarrow\mathbb{R}$ ($W$ is a vector space) that satisfies $$ f\left(\theta x + \left(1-\theta\right) y\right) \leq \theta^\alpha ...
2
votes
1answer
80 views

Is there a common name for the property $f(\lambda x+(1-\lambda)y)\leq f(x)+f(y)$?

For $X$ a vector space and $f:X\to\mathbb{R}$, I'm considering the property $$f(\lambda x+(1-\lambda)y)\leq f(x)+f(y) \qquad\forall x,y\in X, \;\forall\lambda\in[0,1].$$ So it's like convexity ...
3
votes
0answers
45 views

Need $f:[0,\infty)\to[0,\infty)$ such that $f$ is not convex but $f(x^p)$ is for $p>1$.

To be more specific I need to find an $f:[0,+\infty)\to[0,+\infty)$ which satisfies the following: (somewhat trivial stuff) The function $f$ is continuous, nondecreasing, there exists $k>0$ such ...
4
votes
1answer
120 views

Is this matrix function convex or non-convex?

Given, $g(Z)=Tr(Z^Tf(Z)Z)$ , where $f(Z)=h(Z)-ZZ^T$ is a p.s.d matrix formed using entries in $Z$, where again $h(Z)$ is a diagonal matrix with its $i$'th diagonal entry being ...
2
votes
0answers
558 views

Log-concave functions whose sums are still log-concave: possible to find a subset?

Rationale: I am puzzled by a problem of log-concavity, which arises in population dynamics where the curvature of the logarithm of sums is a quantity of interest. It is well-known that sums of ...
3
votes
4answers
152 views

Sufficient condition for a function to be affine

If for a function $f:\mathbb{R}\to\mathbb{R}$, I can prove for any real $x,y$, that $f(\frac{x+y}{2})=\frac{f(x)}{2}+\frac{f(y)}{2}$, can I say for sure that it is affine, as in of the form ...
0
votes
2answers
135 views

Convexity of $\log \det(I+(P-\text{tr}(X))\cdot X)$

where $P-\text{tr}(X)>0$ and $X$ is a diagonal matrix with diagonal elements are $(x_1,x_2,\dots x_m)$
5
votes
2answers
2k views

How $x^4$ is strictly convex function?

I hear that $f(x)=x^4$ is a strictly convex function $\forall x \in \Re$. However, strict convexity condition is that the second derivative should be positive $\forall x \in \Re$. For the mentioned ...
1
vote
1answer
157 views

Is this function convex or concave on $(x,y,z)$?

Is this function convex or concave on $(x,y,z)$? $A$, $B$, $a$, $b$, and $c$ are positive constants. $$f(x,y,z) = A\exp\left(\sqrt{(x-a)^2+(y-b)^2+(z-c)^2}\right) + ...
10
votes
1answer
262 views

Approximating the area below average of a concave function

Given a non-decreasing concave function $f:[0,1]\rightarrow \mathbb{R}^+$. Define \begin{align*} F(n)=\displaystyle\sum_{i=1}^n \min\left\{\frac{1}{n},\frac{f\left(\frac{i}{n+1}\right)}{n+1}\right\} ...
3
votes
1answer
62 views

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 ...
2
votes
0answers
203 views

A sequence of differentiable convex function converges uniformly to a convex function?

Let $f$ be a convex function. How to prove that there is a sequence of differentiable convex function that converges uniformly to $f$?
4
votes
2answers
196 views

How to verify the following function is convex or not?

Consider function $$f(x)=\frac{x^{n_{1}}}{1-x}+\frac{(1-x)^{n_{2}}}{x},x\in(0,1)$$ where $n_{1}$ and $n_2$ are some fixed positive integers. My question: Is $f(x)$ convex for any fixed $n_1$ and ...
1
vote
1answer
821 views

Intersection of two (related) concave functions

Question: In general, two concave functions intersect at at most two points. True or False? If false, can you please provide an example. If true, can you please provide a proof. Proving or disproving ...
3
votes
1answer
130 views

Properties of nearly concave functions

Is there any existing literature on the properties/applications of the following class of functions? $$\frac{f(E[x])}{E[f(x)]}\geq c$$ where $c< 1$ is a constant. Note that for $c=1$ these are ...
1
vote
3answers
192 views

Generating an N-Dimensional Convex Quadratic Function

I am currently working on a research project that is trying to show that some numerical integration techniques that do well on separable convex functions will not necessarily do well on non-separable ...
4
votes
3answers
145 views

Is concavity of a real-valued function on a Euclidean space implied by concavity of its restriction to every lower dimensional affine subspace?

Consider a function $f$ over $\Re^n$ to $\Re$. Suppose it is true that for every affine subspace with dimension strictly lesser than $n$ the function $f$ is concave. Is the function $f$ concave over ...
15
votes
6answers
2k views

can any continuous function be represented as a sum of convex and concave function?

I read that any continuous function can be represented as a sum of convex and concave function, meaning for all $f(x)$, $f(x) = g(x) + h(x)$ where $g$ is convex and $h$ is concave. There could be ...