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
1answer
40 views

Prove an inequality through convexity

I'm trying to prove $-hp + ln(1 - p + pe^h) \le (1/8)h^2$ for all $h > 0$ and $0 \le p \le 1$. After moving the term $-hp$ to the RHS and exponentiating we get $1 - p + pe^h \le e^{(1/8)h^2 + ...
2
votes
0answers
120 views

Minimum / Maximum and other Advanced Properties of the Covariance of Two Random Variables [closed]

Are there any advanced results established regarding the behavior of the Covariance of two random variables other than the bounds on the correlation and independence when it is zero etc. which are ...
1
vote
0answers
48 views

Show Direction of Steepest Descent is Unique

If f is a proper convex function and x is in the interior domain(f), how would one go about proving that the direction of steepest descent at x is unique? I intuitively get it, but don't get how one ...
0
votes
1answer
19 views

Functions with convex/concave potential

A function $f:A\to A$ has convex/concave potential if there is $F:\mathbb{R}^n\to \mathbb{R}$ such that $\nabla F=f$, and $F$ is convex/concave. Let $A\subset \mathbb{R}^n$ be a compact set. Are ...
3
votes
1answer
41 views

A convex real function is continuous - can we generalize?

There is a well-known theorem $\newcommand{\RR}{\mathbb{R}}$ Let $f: (a,b) \rightarrow \RR$ be a convex function. Then $f$ is continuous on $(a,b)$ The proof I know makes use of the fact that ...
0
votes
0answers
30 views

Supremum of a Sequence of convex and closed functions {f_i} is also closed [closed]

I feel like this is an obvious question, but I am having difficulty formally proving it. Given a set of convex and closed functions {f_i(x)} and f(x) is defined as the sup f_i(x), then how does one ...
1
vote
0answers
29 views

Conflict with definition of “face”

I am given this definition of face from Convexity: An analytic viewpoint: Definition: A face of a convex set $P$ is a set $F\subsetneq \overline{P}$ such that for every $x,y\in\overline{P}$ and for ...
0
votes
0answers
15 views

Convexity of Maximum Inscribed Ellipsoid Problem

I am confused about the problem of finding the ellipsoid of maximum volume inscribed in a polyhedron as discussed in section 8.4.2 of Convex Optimization by Boyd and Vandenberghe. I've followed the ...
3
votes
2answers
33 views

A caracterization of convexity in $\mathbb R^n$

Let $C$ be a closed subset of $\mathbb R^n$ such that $$\forall x,y \in C, (x,y)\cap C \neq \emptyset$$ where $(x,y)=\{(1-t)x+ty, t\in (0,1)\}$ Prove that C is convex A quick drawing shows ...
1
vote
1answer
16 views

Convex and Symmetric subset of a Banach space

Let X be a Banach space and A be a convex and symmetric subset of X. Is it true then that the closure of A will be a subset of 2A=A+A? I doubt that this always holds, but can't seem to find a ...
0
votes
1answer
35 views

Is there a geometric interpretation for a function's $\alpha$-sublevel set?

In Boyd and Vandenberghe's "Convex Optimization": The $\alpha$-sublevel set set of a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined as $$C_\alpha=\{x \in \mathbf{dom} ...
3
votes
1answer
44 views

Jensen's inequality in measure theory

Here cites its original claim from http://www.math.tau.ac.il/~ostrover/Teaching/18125.pdf. Theorem 3.1 Jensen's Inequality Let $(X,\mathcal{M},\mu)$ be a probability space (a measure space ...
3
votes
4answers
202 views

Why are convex polyhedral cones closed?

Let $V = \mathbb{R}^n$, $v_1, \dots, v_s \in V$ and let $\sigma = \text{Cone}(v_1, \dots, v_s) = \{r_1v_1 + \dots + r_sv_s \mid r_i \geq 0\}$ be the associated convex polyhedral cone in $V$. Why is ...
0
votes
1answer
18 views

Proving inequality from convexity of function

I am having trouble proving the following inequality for all $x,y>0$ from "The Mathematics of Nonlinear Programming" by Pressini, Uhl. The book states that it follows from the convexity of an ...
0
votes
3answers
48 views

What is the solution for this optimization problem?

I have an optimization problem in the form: $$\max (a-\bar{a})(b-\bar{b}) \qquad \text{subject to} \qquad a+b=1.$$ Here $\bar{a}$ and $\bar{b}$ are known values and both of them are positive. Let ...
0
votes
0answers
32 views

What algorithms are applicable to solve a inequality constraint Quadratic Optimization?

Suppose that we have a quadratic optimization problem $$(QP) \qquad \min \lbrace\frac{1}{2}x^TQX+ q^TX\rbrace $$ s.t. $$AX=a;$$ $$BX\le b;$$ $$X \ge 0;$$ where $Q \in \mathbb{R}^{n \times n}$ ...
2
votes
1answer
39 views

Convolution of convex function and Gaussian is convex

Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function, i. e., for all $x_1, x_2 \in \mathbb{R}$ and $t \in [0, 1]$, $$ \qquad f(tx_1+(1-t)x_2)\leq t f(x_1)+(1-t)f(x_2).$$ I want to prove that the ...
0
votes
2answers
50 views

Prove that the set $C = \{x \in\Bbb R : ax\le b\}$ is convex

Prove that if a and b are real numbers, then the set $C = \{x \in\Bbb R : ax\le b\}$ is a convex set. My solution so far: To show that a set $C$ is convex it needs to be shown that for for every ...
1
vote
0answers
41 views

non negativity of an sequence alternating series

Assume we have the positive real number $a_1,...,a_n$, and variable $x \in \Bbb R^+$, I am trying to prove the following to be positive (or at least non negative): $$ \frac{1}{x^3} - \sum_{i=1}^{n} ...
4
votes
1answer
36 views

Classification of boundary points of convex sets in $R^n$

I'm trying to prove the following: Let $P\neq R^n$ be a convex set containing an $R^n$ neighborhood of $0$. Then $x\in\partial P\iff (\lbrace tx: 0\leq t < 1\rbrace\subset P^\circ)\wedge(x\not\in ...
0
votes
1answer
55 views

Given a convex function $f(x)$, is $xf'(x)$ also convex?

Given a convex function $f(x)$, I'm trying to proof that $g(x) = xf'(x)$ is also convex. I have found neither a proof nor a counterexample so far. A function $g(x)$ is convex iff $g''(x) \ge 0$. ...
0
votes
0answers
32 views

Question regarding “distance function” in convex spaces.

Assume that we have a convex space $K < \mathbb{R}^n$ and a point $a \not\in K$. I need to show that there is a function in the dual, $f\in \mathbb{R}^n$ such that $f(a)\geq f(x)$ for all $x\in K$. ...
0
votes
0answers
15 views

The $d$-skeleton of a polytope is strongly connected

A polytope is the convex hull of a finite set of points in $\mathbb R^n$. The $d$-skeleton of a polytope is the set consisting of faces of dimension at most $d$. I would like to show that every ...
0
votes
0answers
31 views

hyperbolic constraint in SOCP

I learned that any hyperbolic constraint can be transformed as second order cone constraints. Such as if the constraint is, $x^2\leq w$ the second order constraint would be $$ ||[2x,w-1]||_2 \leq w+1 ...
0
votes
0answers
12 views

Decomposition of convex functions

Under what conditions is it possible to decompose a convex real-valued function $f$ and write it as an integral of the simple functions $g(x;w)=\max(x-w,0)$ and $h(x;v)=\max(v-x,0)$, where the ...
11
votes
2answers
1k views

Minkowski Inequality for $p \le 1$

I've been trying to prove the concavity of a particular function which I reduced to proving the reverse Minkowski Inequality for $p \le 1$, $p \ne 0$ for arguments in $\mathbb{R}^{n}_{+}$. That is, ...
2
votes
0answers
37 views

Literature suggestion for “strong convexity”

Does anyone know of a reference that discusses strong convexity and strong smoothness of proper convex functions over Banach spaces? All the references I find only deal with the finite dimensional ...
1
vote
0answers
49 views

Proof or disproof of convexity for $f(x,y)=x^2y^2$

I'm trying to prove or disprove the convexity of $f(x,y)=x^2y^2$. This is part of a larger function but I think I proved that the rest of the function is convex using Hessian's. The other term in the ...
1
vote
0answers
27 views

Efficiency of quasiconvex optimization

Summary: Could we minimize quasiconvex objectives in polynomial time? Whenever an objective function of an optimization problem can be formed as a convex function, this is considered as victory. ...
2
votes
0answers
34 views

shrinking a convex hull around a set of polygons

I'm trying to find (Or design) an algorithm that will let me, after I have a convex hull, progressively shrink the hull towards the polygon set via increasing some parameter. I.e., if we use the ...
3
votes
1answer
59 views

Subadditivity of the $n$th root of the volume of $r$-neighborhoods of a set

Let $A$ be a closed subset of $\mathbb{R}^n$. For $r>0$, let $A_r$ be the $r$-neighborhood of $A$, namely the set $\{x:\operatorname{dist}(x,A)\le r\}$. Let $f(r) = \mu(A_r)^{1/n}$ where $\mu$ is ...
5
votes
1answer
108 views

Concavity of the $n$th root of the volume of $r$-neighborhoods of a set

Let $A$ be a closed subset of $\mathbb{R}^n$. For $r>0$, let $A_r$ be the $r$-neighborhood of $A$, namely the set $\{x:\operatorname{dist}(x,A)\le r\}$. Is the function $f(r) = \mu(A_r)^{1/n}$ ...
2
votes
3answers
48 views

Convexity of a non linear optimization problem

I have a non linear optimization problem, namely: $$\min {\sqrt{(x-u)^2 + (y-v)^2 + (z-w)^2)}}$$ How can i show that the above function is convex. Doing via Hessian is a difficult task.
-1
votes
2answers
51 views

Is the function $\frac{f(x)}x$ increasing, if $f(x)$ is convex? [closed]

Suppose $x$ is real and positive. $f''(x) > 0$ (f is convex) is the function $h(x) = \frac{f(x)}x$ increasing? If so, is it strictly increasing? If yes, why? Thank you.
0
votes
1answer
17 views

Collecting terms in an example of checking concavity/convexity

$z=x_1^2+x_2^2$ $u=(u_1,u_2)$ $v=(v_1,v_2)$ Height of arc: $f[\theta u+(1-\theta )v]= f[\theta u_1+(1-\theta)v_1,\theta u_2 + (1-\theta)v_2 ]$ $= [\theta u_1+(1-\theta)v_1]^2 + [\theta u_2 + ...
3
votes
1answer
117 views

A connectivity-preserving function from a connected set onto an interval

Let $C$ be a connected set in the plane and $I$ the unit interval interval. Call a function $f$ from $C$ onto $I$ Connectivity-preserving if the following is true for every subset $I'\subseteq I$: ...
5
votes
2answers
220 views

Set is Convex regardless of b

Let the function $f$ be convex, $f :\Bbb R^n \rightarrow \Bbb R$ and let $$S = \{x : f(x) \le b\}$$ The proposition states that the set $S$ is convex regardless of $b$. Can someone explain to me how ...
1
vote
1answer
42 views

Continous family of $n$-gons

The statement of this problem asks to show that if $A$ and $B$ are two distinct convex $n$-gons there is a continous family of convex $n$-gons such that the first in that family is $A$ and the last is ...
4
votes
1answer
28 views

Which functions are the composition of convex functions?

The composition of convex functions is not necessarily convex or concave: For example, composing $f_1(x) = x^2-1$ and $f_2(y) = y^2$ gives $f_2(f_1(x)) = (x^2-1)^2$. Or consider $f_1(x) = x^2$ and ...
4
votes
1answer
37 views

What class of functions is characterized by the property $f[\operatorname{conv} A] \subseteq \operatorname{conv} f[A]$

It is well-known that the inclusion $f[\overline A] \subseteq \overline{f[A]}$ (for every subset $A$) characterizes continuous functions.1 Asking similar questions for other closure operators seems ...
0
votes
2answers
30 views

SVD with Sparisty Penalties

I am trying to solve a problem like $$\min_{u,v} \Vert X-uv^T \Vert_F^2 + \beta \Vert v \Vert_1 \mbox{ s.t. } \Vert u \Vert_2 \le 1$$ where $X \in \mathbb{R}^{N\times N}$, $u, v \in ...
0
votes
0answers
27 views

Convex optimization of a fractional objective function involving matrix determinants

I am interested in convex representation of the following fractional optimization problem. I have also described my approach in the following. However, as I am new to convex optimization, I am not ...
0
votes
4answers
57 views

for $I = [0,1]$, is $I\times I$ convex in $\mathbb{R} \times \mathbb{R}$?

for $I = [0,1]$, is $I \times I$ convex in $\mathbb{R} \times \mathbb{R}$? The definition of convex seems to be that $Y \subset X$ is convex in $X$ if $\forall a < b $ in $Y$ whole of $(a,b)$ ...
1
vote
1answer
41 views

Prove Jensen Inequality holds for a function

Given function $$f:\mathbb{R}^n_{+} \rightarrow \mathbb{R}, \ f(x) = \sum_{k=1}^K c_k x_1^{a_{1k}} x_2^{a_{2k}} \cdots x_n^{a_{nk}}$$ Show that for any $x, y \in \text{dom} \ f, \theta \in [0,1]$, ...
0
votes
0answers
21 views

Negative eigenvalue of a hessian matrix entails a local decrease in function value?

I was reading up on non-convex optimization, and I can across this sentence: "Since Hessian(f(w)) has a negative eigenvalue, there is always a point that is near w which has smaller function value" ...
0
votes
1answer
23 views

Convexity of an exponential function.

A random variable $Y_i$ is given such that, $\mid$Y$_i\mid$$\leq$ $c_i$ where i ranges from 1,.....,t and t is some constant. Now, $Y_i$ is expressed as : $Y_i = ((Y_i - c_i) + (Y_i + c_i))/2$ $= ...
2
votes
1answer
25 views

Is there a name for dividing a set into pieces, some of which may be empty?

Suppose that $X$ is a set and $V_{0}$, $J$, and $V_{1}$ are pairwise disjoint subsets of $X$ whose union is $X$. If the three subsets were nonempty it would be a partition of $X$. However, I wish to ...
1
vote
1answer
28 views

Inequality with three unknowns

Consider: \begin{equation} \Big(e^x-1\Big)\mathbb{1}_{(x\geq0)} \leq \lambda_1+\lambda_2e^x + \lambda_3x^2 \end{equation} where $\lambda_1$, $\lambda_2$, $\lambda_3$ are three unknown constants. By ...
0
votes
1answer
18 views

Prove that there exists some $y \in \mathbb{R}^n$ in the boundary of $C$ satisfying $\{z \in \mathbb{R}^n : (x-y)^T(z-y) = 0\}$.

Let $C \subset R^n$ with $C \neq \emptyset$ be a closed convex set. Consider some $x \in \mathbb{R}^n$ satisfying $x \notin C$. Prove that there exists some $y \in \mathbb{R}^n$ in the boundary of $C$ ...
2
votes
1answer
46 views

Are posynomial functions convex?

I know that you can transform a posynomial function into an exponential function, which is convex. Does this imply that all posynomial functions are convex?