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.

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2
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1answer
27 views

$U$ bisected by all hyperplanes $\implies$ $U$ symmetric?

Let $U \subset \mathbb{R}^n$ be a bounded open convex set, such that every hyperplane passing through the origin divides $U$ into two sets of equal volume ($n$-dimensional Lebesgue measure?). ...
1
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1answer
36 views

Simple question on maximizing a family of concave functions

Given $x \in [0,1]$, let $f_k(x)$ be concave function in $x$ for all $k= 0,1,...,N$. Is the following two maximization formulations equivalent? $$ \max_{0 \le k \le N} f_k(x) =?= \max \{f_0(x), ...
0
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1answer
38 views

Is a maximum function concave?

A function such as $$\max (X, Y)$$ given $X\geq 0$, $Y \geq 0$. Because It would form an inverse L shape on the graph, and that looks like a concave function. However my teacher said that the function ...
5
votes
2answers
104 views

The convex hull of every open set is open

Let $X$ be a topological vector space. Prove that the convex hull of every open subset of $X$ is open. I tried using definition of Convex Hull and Open Set, but I couldn't prove the statement.
0
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1answer
20 views

Convexity of $f(x) = \sum_i x_i^p$ for $x_i \ge 0, p \ge 1$

Let $x = (x_1,\ldots, x_n)$ be nonnegative real numbers and $p \ge 1$, then the function $f(x) = \sum_{i=1}^n x_i^p$ is convex, the following proof is wrong, or not? I have it from here, but then ...
1
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0answers
13 views

Signs and value of higher order terms in the Taylor expansion of a strongly convex function

Say that I have a function $f: S \rightarrow \mathbb{R}^+$ such that: $S \subset \mathbb{R}^n$ is a closed convex set such as $S=[-10,10]^n$ $f$ is continuous and infinitely differentiable at all ...
0
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2answers
32 views

Maximal eigenvalue is convex function

Let $A$ be a symmetric real matrix. let $f(A)=\lambda_{max}(A)$ be it's largest eigenvalue. Why is $f(A)$ convex?
1
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1answer
26 views

Orthogonal projection of an $n-$vector onto the subspace ($m\leq n$)of $\mathbb{R}^n$ containing a convex polytope

Lets say we have an $n \times m$ matrix $A$, whose column vectors are $(\vec{\mathbf{0}},a_1,a_2,...a_j)$ are points in $\mathbb{R}^n$ and the non-zero vectors have unit length. Let ...
1
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1answer
19 views

Multivariate convex function / increasing differences

$\newcommand\Rr{\mathbb{R}}$I am trying to show the following statement. It feels true to me, but I haven't found any reference in the literature so far: Let $\Rr^n$ be ordered component-wise, i.e., ...
0
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1answer
14 views

Differentiability of support function (even for non-convex)

I am reading an economics book (for those who are interested, MWG Microeconomic Theory) and there's a theorem that was just given without proof, but I am interested in the proof - also because I ...
0
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1answer
15 views

Boundary of convex set is piecewise $C^1$

Let $K$ be a convex and compact subset of $\mathbb{R}^2$. Is it true that the boundary of $K$ can be parameterized by a piecewise $C^{1}$ application $\gamma :I\to\mathbb{R}^2$?
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0answers
14 views

how can we check convex or nonconvex feasible?

example if i have 20 constraints functions.These functions cut the objective function and create the feasible region. Their intersections can become edges and create a nonconvex feasible region even ...
0
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0answers
25 views

Optimising a piecewise convex function

I have a function $f(x)$ defined for $x\in \mathbb{R}^+$ that is decreasing, piecewise convex and continuous. The 'pieces' of $f$ are exponential and the rate of decrease for each piece reduces as we ...
1
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1answer
17 views

Quasi concavity and Quasi Convexity-intuitive understanding

I'm having trouble grasping the concept of quasi concavity and quasi convexity. My textbook states that if f is quasi-concave, then f (λx + (1 − λ) y) ≥ min {f(x), f(y)} . Also that is f is quasi ...
0
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0answers
24 views

Is an lsc sublinear function $X^* \rightarrow (-\infty, \infty]$ always a support function for some closed non-empty $C \subset X$?

I can't seem to find any resources on this, even though it seems like an obvious question to ask. The separation theorem implies that, if we have an lsc sublinear function $\phi : X^* \rightarrow ...
0
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1answer
44 views

Isn't every increasing continuous convex function strictly increasing (disregarding $f(x) \equiv 0$)?

Isn't every increasing continuous convex function $f$ strictly increasing (disregarding the trivial case $f(x) \equiv 0$)? I don't see any counterexample!
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2answers
47 views

Is this Function of Product of variable and Ratio of CDF and PDF of Standard Normal Distribution Convex?

Let $G\left(x\right)=x\frac{\phi\left(x\right)}{\Phi\left(x\right)}$. Here, $\phi$ and $\mathbf{\Phi}$ are the standard normal PDF and CDF, respectively. Is $G\left(x\right)$ convex? It has been shown ...
0
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0answers
17 views

Properties of a proper cone

Let $K$ be a proper cone. I need to prove following properties: if $x \preceq_K y$ and $u \preceq_K v$, then $x+u \preceq_K y+v$ if $x \preceq_K y$ and $y \preceq_K z$, then $x \preceq_K z$ if $x ...
0
votes
1answer
21 views

Given Convex Function, Conditions when Variable times Convex Function is convex

Given that say, $f(x)$ is convex for $x>0$. We can arrive at the following conditions for when $xf(x)$ would be convex. Please add anything that I might have overlooked and further simplifications ...
1
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2answers
58 views

Proof that Convex Function with alternate variable is convex

Given that say, $f(x)$ is convex for $x>0$. Would the proof that $f(a-x)$ where $a>x$ is convex proceed as shown below. Please verify the correctness and/or alternative approaches that can ...
1
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1answer
21 views

Convex Hull = Boundary+Segments

If $A\subseteq\mathbb{R}^n$ is an non empty set and $H$ is the convex hull of $A$, how can I prove that the boundary of $H$ consists only of points that lie in the boundary of $A$ and segments that ...
1
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1answer
25 views

Is a twice differentiable function whose only extrema is a minimum automatically convex?

I have a twice differentiable function $H(x)$ for which I have already proven that: $\exists !x^*:H'(x^*)=0$ and furthermore that $H''(x^*)>0$ (the only extremal is a minima). $H''(x)$ is ...
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1answer
31 views

Prove a convex function

I have to prove that if $f:A \to \mathbb{R}$ is convex and $c \ge 0$ then $c \cdot f:A \to \mathbb{R}$ is convex. I know that function $f:A \to \mathbb{R}$ is convex if for $\forall x,y \in A$ and ...
1
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2answers
105 views

Convexity of the ratio of the standard normal PDF by its CDF

Is there some way to show that the following function $\psi$ is concave or convex? Here, $\phi$ and $\mathbf{\Phi}$ are the standard normal PDF and CDF, respectively. ...
0
votes
1answer
25 views

A convex hull of a union of convex sets [closed]

Let ${A_1},{A_2},....{A_n}$ be convex sets in a vector space and suppose $x \in \operatorname{co}({A_1} \cup {A_2} \cup \dotsb \cup {A_n})$. Is it true that $x = {t_1}{a_1} + \dotsb + {t_n}{a_n}$ such ...
0
votes
1answer
40 views

Proving a convex function [closed]

I'm given a function $f:A \to \mathbb{R}$ which is twice continuously differentiable on $A \subseteq \mathbb{R}^n$. $A$ is a convex set. Show that $f$ is convex. Any ideas on how to prove this?
0
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1answer
35 views

What are the extreme points of the closed unit ball of $C$? .

What are the extreme points of the closed unit ball of C(the space of all continuous functions on the unit interval), with the supremum norm?
0
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1answer
24 views

Rectificable curve as a boundary of a convex set

Let $K\subseteq\mathbb{R}^2$ be a convex compact set. Is it true that $\partial K$ (the boundary of $K$) is a rectificable curve (i.e. it has length)?
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37 views

how to prove convexity of the function below?

For a graph $G$ consider the following function, $$f=\sum_{(i,j) \in G ,(i,k) \notin G } \max(0,c+ \left\|e_i-e_j\right\|_2^2-\left\|e_i-e_k\right\|_2^2)$$ where $e_i \in\mathbb R^n$ ($n$ dimension ...
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0answers
20 views

Why there is nonconstant linear funectional $\Gamma $ on $X$ such that $\Gamma (A) \cap \Gamma (B)$ contains at most one point?

Let $X$ is a real vector space(without topology). call a point $x \in A \subset X$ an internal point of $A$ if $A-x$ is an absorbing set.Suppose $A$ and $B$ are disjoint convex set in $X$ and $A$ has ...
0
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1answer
20 views

Is the following function concave? (or log of it)

I have a function $f(x_1, x_2, ..., x_M) = \displaystyle \prod_{i = 1}^N \frac{(\sum_{j = 1}^{M} a_jx_jI_{ij})^2}{\sum_{j = 1}^{M} a_jx_j}$ in domain $\{{\bf x} \in {\bf R}^m \setminus {\bf 0} \ ...
0
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2answers
15 views

Sequence of convex non increasing sets convergence

I have a question for you. I was wondering whether a non increasing sequence of convex set converges to a convex set. Here my question made more precise: Let $\{S_k\}_{k=1}^\infty$ be a sequence of ...
0
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1answer
30 views

KKT conditions for a convex optimization problem with a L1-penalty and box constraints

I am having some trouble deriving / understanding optimality conditions for a convex optimization problem of the form: $$\begin{align} \min_{x\in\mathbb{R}^d}~ & f(x) + C.\|x\|_1 &\\ &x_i ...
0
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3answers
28 views

How to show the optimal condition of $f(\alpha) = \frac{R^2+G^2\sum_{i=1}^k \alpha_i^2}{2\sum_{i=1}^k \alpha_i}$

Consider the following function: ($\alpha>0$) $$f(\alpha) = \frac{R^2+G^2\sum_{i=1}^k \alpha_i^2}{2\sum_{i=1}^k \alpha_i}$$ It is a quadratic (in $\alpha$) over linear (in $\alpha$); therefore, ...
0
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1answer
19 views

Compact convex subset and hyperplanes

Suppose $K$ is a compact and convex subset and $x^*$ a point in $\mathbb{R}^n$. Suppose there exists $y\in \mathbb{R}^n$ such that $$\langle x^*, y\rangle> \sup_{x\in K} \langle x, y\rangle$$ ...
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3answers
302 views

Integral of an increasing function is convex?

Let $f$ be a real valued differentiable function defined for all $x \geq a$. Consider a function F defined by $F(x) = \int_a^x f(t) dt$. If f is increasing on any interval, then on that interval F is ...
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2answers
46 views

Proof that a log-of-sum-of-exponentials is a convex function

It's well known in statistical mechanics that the following is a convex function of the vector $\theta$: $$ A(\theta) = \log \left( \sum_{i=1}^\infty e^{\theta \cdot f(i)} \right) $$ where $f(i)$ is ...
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0answers
13 views

Subhessians for maximum eigenvalue of a matrix

I am trying to solve a non-linear, non-smooth convex optimization problem using a generic convex optimization solver. This solver requires and (sub)gradients of the objective and the constraints, as ...
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0answers
13 views

Good graphic tool for drawing the convex hull of two planes?

If I want to draw the convex hull of two 2D planes, what kind of tool box should I use ? The graph of functions in page 2,3 of the following file are quite nice, anyone can guess what kind of ...
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1answer
69 views

Intersection of a convex polygon and a moving circle

I have a straight line which intersects a convex polygon in 2D plane. There exists a circle with constant radius. The center of circle is moving on this line. So at first the polygon and circle don't ...
2
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2answers
201 views

Crossing of two convex functions with same asymptotic slopes

Suppose you have two continuous, positive convex functions $F(x)$ and $G(x)$, $x\in\mathbb{R}$ such that: $$\lim_{x\rightarrow\pm\infty}F'(x)=\lim_{x\rightarrow\pm\infty}G'(x)=\pm 1$$ and that ...
0
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1answer
33 views

Two duality theorems

Suppose $X$ is a Hilbert space with norm $||.||$ and $K$ is a weak compact and convex subset of $X$. The supporting functional: $$h(x^*)=\sup_{x\in K} \langle x^*, x \rangle$$ The indicator ...
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0answers
17 views

Intuition on primal convergence in dual subgradient method

It is well known that the subgradient method applied to the Lagrange dual of a convex problem may produce a sequence converging to the dual optimum, but the primal iterates produced by this sequence ...
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0answers
21 views

Centers of divisorial valuations on toric varieties

Suppose we are given a divisorial valuation $\mathcal{v}$ on a smooth toric variety $X_{\Sigma}$ (for a fan $\Sigma$), i.e. $\mathcal{v}$ is the valuation induced by a torus invariant divisor ...
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0answers
8 views

Uniqueness of supporting hyperplane for a face of a cone

In William Fulton's 'Introduction to Toric varieties' he says - " When $\sigma$ spans $V$ and $\tau$ is a facet of $\sigma$ then there is a $u \in \sigma ^{\vee}$ unique upto multiplication by a ...
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1answer
51 views

Jensen inequality for convex functions - infinite countable number of weights

Does Jensen inequality for convex functions hold if there is countable infinite number of weights? For example weights are given by sequence $(\frac{1}{2^n})_{n\in\mathbb{N}_1}$ ? If not, where is ...
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0answers
15 views

If the hessian of a function is positive definite everywhere, is it convex everywhere? [duplicate]

G'day! If the hessian of some multivariable function is positive definite everywhere, does that necessarily imply that the function is convex everywhere?
0
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1answer
19 views

Increasing Function and Convex Set Question

Consider a function $0 \le f(x) \le 1$ which is increasing in $x \in [a,b]$, I was wondering can I say that $f(x) \le \epsilon$ for $0< \epsilon <1$ defines a convex set? I think the ...
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0answers
16 views

Directional derivative of difference of two convex functions

I would like to find the references and the proof for the following fact: Let $g,h:\mathbb{R}^n\rightarrow\mathbb{R}$ be two convex functions and $f=g-h$. Suppose that $\bar{x}\in\mathbb{R}^n$ such ...
0
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1answer
49 views

Convex Hull algorithm.

Working on making a Convex Hull algorithm. I need to figure out how to iterate the remaining points to find the shortest angle as marked below in the picture. I am ...