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 (1)

1
vote
3answers
200 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 ...
6
votes
1answer
5k views

Convexity of the product of two functions in higher dimensions

Exercise 3.32 page 119 of Convex Optimization is concerned with the proof that if $f:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto f(x)$ and $g:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto g(x)$ are both ...
1
vote
2answers
656 views

Converse supporting hyperplane theorem

Exercise 2.27 in Boyd and Vanderberghe: Suppose the set C is closed, has nonempty interior, and has a supporting hyperplane at every point in its boundary. Show that C is convex. Seems to me one ...
7
votes
3answers
1k views

Disjoint Convex Sets That Are Not Strictly Separated

Question 2.23 out of Boyd and Vanderberghe: Give an example of two closed convex sets that are disjoint but cannot be strictly separated. The obvious idea is to take something like unbounded sets ...
2
votes
2answers
207 views

What is the right way to prove that the intersection of an infinite number of convex sets is convex?

I am wondering how to prove that the intersection of an infinite number of convex sets is convex. I can prove that the intersection of two convex sets is convex, and I believe that I can simply do ...
0
votes
0answers
206 views

Curvature and intersection of convex functions

Take two weakly convex, weakly increasing non-negative functions, $g(x)$ and $h(x)$, domain of $x$ is $[a,d]$, $g(x)=0$ for $x \in [a,b]$, $h(x)=0$ for $x \in [a,c]$, and $g(d)=h(d)$. So $g$ and $h$ ...
2
votes
1answer
300 views

Applications of Helly's theorem to problem solving

Helly's Theorem states the following: Suppose that $X_1,X_2,...,X_n$ are convex sets in $\mathbb{R}^d$, such that for any $|I|\leq d+1$, $\cap_{i\in I}X_i \neq \emptyset$. Then $\cap_{i=1}^{n}X_i ...
4
votes
3answers
147 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 ...
1
vote
1answer
286 views

convexity of inner product of elementwise powers

For $x \in \mathbb{R}^n$ and $A,B \in \mathbb{R}^{m \times n}$, $f(x) = ((Ax)^{2})^T((Bx)^2)$ where $^2$ denotes the power of 2, element-by-element of vector Ax or Bx. (I wasn't sure how to notate ...
1
vote
2answers
118 views

Radius of a hypercube at a given angle

For a ray from the origin with a given angle in $R^n$, I am trying to find the radius at which that ray intersects the frontier of the unit n-cube. In two dimensions, the picture is this: Given ...
5
votes
1answer
375 views

Is a smooth function convex near a local minimum?

I would like to know if every sufficiently differentiable function is convex near a local minimum. The background to my question is that I became curious if one could motivate the usefulness of convex ...
4
votes
1answer
113 views

Describing a set of normals

In a finite dimensional (think Euclidean) ambient space, let $S$ a compact, convex set and $x$ not in $S$. The two sets can be (weakly) separated, i.e. there exists a vector (normal) that defines a ...
17
votes
2answers
565 views

For what functions $f(x)$ is $f(x)f(y)$ convex?

For which functions $f\colon [0,1] \to [0,1]$ is the function $g(x,y)=f(x)f(y)$ convex over $(x,y) \in [0,1]\times [0,1]$ ? Is there a nice characterization of such functions $f$? The obvious ...
1
vote
1answer
269 views

How close are star-convex sets to convex sets?

What interesting properties of convex sets are retained by star-convex sets?
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 ...
2
votes
1answer
928 views

Convex hull has the smallest perimeter

How do you show that the convex hull of a given set of points S, always has the minimum perimeter ? By perimeter i mean the length of the boundary of the hull
-1
votes
2answers
125 views

convex with generic function

Given: $f(px+(1-p)y)\le pf(x)+(1-p)f(y)$ Suppose $f(x)$ is defined on the interval I. If $x_{1}<x_{2}<x_{3}$ are in I and $f(x_{1})<f(x_{2})$ and $f(x_{3})<f(x_{2})$, then show that ...
1
vote
2answers
849 views

show that function is not convex

I've never had to do this before, so I'm not really sure how to do it. These problems also don't even really relate to what the subject of the book is as well. Given: $f(px+(1-p)y)\le ...
1
vote
1answer
147 views

Weak Continuity of Affine maps

Let $C$ be a convex subset of a Banach space $X$ and $T:C\to C$ a (norm) continuous affine map, i.e. $$T(tx+(1-t)y)=tT(x)+(1-t)T(y)$$ for $0\le t\le 1$. Is $T$ weakly continuous, i.e continuous as a ...
14
votes
2answers
1k views

Farkas’ lemma: purely algebraic intuition

Here is a statement of Farkas Lemma from the Wikipedia. Let $A$ be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the following two statements is true: There exists ...
12
votes
2answers
479 views

A tree of convex sets?

This was suggested by a problem on FreeNode's #math a little while ago... Construct a directed graph $\Gamma$ with vertex set the set of compact convex sets in $\mathbb R^2$, and an arrow $A\to B$ if ...
13
votes
4answers
3k views

Why does a convex set have the same interior points as its closure?

Let $C$ be a convex subset of $\mathbb{R}^n$. I've been trying for hours to prove that $\dot{\overline{C}}=\dot{C}$. Somehow my intuition completely fails me. I found a proof in a textbook, but just ...
1
vote
1answer
217 views

Show convex combination

If I have a bounded set $F$ in $N$ dimensional space and another set $G$ where every element $g$ in $G$ has $h'g=c$ and also must exist in $F$. $H$ is a vector in the $N$ dimensional space and $c$ is ...
2
votes
1answer
186 views

How to compute the subdifferential of a variational represention of the trace norm?

Let $f : {\cal S}_+^n \mapsto \mathbb{R}$ be a function defined as $f(Q) := {\rm tr} WQ^{-1}W + {\rm tr} Q$ where $W \in {\cal S}_{+}^{n}$ is a symmetric positive definite matrix. How to compute a ...
5
votes
1answer
674 views

Lower hemicontinuity of the intersection of lower hemicontinuous correspondences

I have been stumped for long by this exercise (3.12(d)) from Stokey and Lucas's Recursive Methods in Economic Dynamics. Would greatly appreciate any hints. Let $\phi: X \to Y$ and $\psi: X \to Y$ be ...
2
votes
1answer
184 views

study objects of convex analysis and optimization

In the area of convex analysis and the area of optimization in their general sense, are convex subsets assumed to be in vector spaces or topological vector spaces? Are convex functions defined to be ...
6
votes
7answers
3k views

Operations research book to start with

for somebody having a quite strong background in Mathematics, which are some good books for the domain of Operations research? I guess there are textbooks covering topics like linear and nonlinear ...
4
votes
1answer
3k views

Positive definite Hessians from strictly convex functions

Let $f: D \to \mathbb{R}\ $ be a function on non-singular, convex domain $D \subseteq \mathbb{R}^d$ and let us assume the second-order derivatives of $f$ exist. It is well known that $f$ is convex if ...