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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Locally convex spaces - is any space that contains a locally convex space as a subspace, also locally convex?

Given $E$, a locally convex space (l.c.s.) and $E\subseteq F$ where $F$ is another subspace of a larger vector space. The inclusion is strict since I know there exists a $y\in F\backslash E$. I have ...
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Characterization of projective convexity

Let $K$ be a closed set in projective space $\mathbb P^n$. Is it true that $K$ is "projectively convex", i.e., its intersection with every line is connected, if and only if it is the projectivization ...
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Solution for $\min_{x^Tx=1} x^TAx-c^Tx$

$\min_{x^Tx=1} x^TAx-c^Tx$ looks like a simple QPQC problem. If $A$ is positive semi-definite, can I get the solution by first getting $x=A^{-1}c$ and then projecting $x:=\dfrac{x}{||x||}$ to make ...
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Proof of property of unique nearest point equal to convex.

In the picture below, I really don't know how to use the compactness to get there will be a ball $C$ with maximal radius. I report this book for my classmate and teacher. I tell they the $r$ has a ...
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Can someone confirm this sketch of a proof for the existence of smooth partitions of unity on an unbounded convex open euclidean set?

EDIT!! The problem originally described (see below) has been reduced to the correctness of a simple extension of an argument from Rudin's PMA. Feel free to skip to the proposed solution, below. As ...
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How to determine whether a function of many variables is convex or non-convex?

A function $f$ is convex if $$f(\theta x + (1 − \theta)y) \leq \theta f(x) + (1 − \theta)f(y)$$ for all $x, y \in \mathcal{D}(f)$, the domain of $f$, and $\theta \in [0, 1]$. How do I determine ...
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If $f,g:\mathbb{R} \rightarrow \mathbb{R}$ are positive, non-increasing and convex functions, then $F(x,y) = f(x)g(y)$ is quasiconvex.

Hypothesis: $\forall x_{1},x_{2}\in \mathbb{R}, \forall \lambda \in [0,1], f(\lambda x_{1} + (1- \lambda) x_{2}) \leq \lambda f(x_{1}) + (1- \lambda) f(x_{2})$ $\forall x_{1},x_{2}\in \mathbb{R}, ...
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Extreme points of unit ball of $l_1(\mathbb{N})$

Let $K$ be the closed unit ball of $l_1(\mathbb{N})$ over real numbers. Show that $$ Ext(K)= \{\pm e_n: e_n=(0,\ldots,1,0,\ldots)\}. $$ My attempt: I could prove that $\{\pm e_n: ...
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Solve convex optimization problem with objective function having power \alpha >1

I am not able to figure out how to get the explicit solution of the following minimization problem: $$\min_{\mathbf{w}\in \mathbb{R}^n} = \sum_{i = ...
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Need to prove that convex property is the intersection of an increasing and decreasing property for graph

I need to prove that any convex property for graphs can always be expressed as the intersection of an increasing property and decreasing property for graph, specifically that: $\forall A\subset ...
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general framework for proof of convexity in least square problem

Let $f(x_i,\theta) : X \times \Theta \rightarrow Z$ be a parametric function with parameter $\theta$ that we wish to fit to set of samples $S =\{(x_i,z_i)\}_{i=1}^N$, where $(x_i, z_i) \in X \times ...
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Convex set: extreme points and distance to the origin

I'm fairly sure the following is true, although I wouldn't mind being proven wrong. If true, I would like to see an elegant proof, as my attempts are kind of messy. Let $K\subset\mathbb R^2$ be a ...
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66 views

Convex set without zero

Let $\emptyset \neq A \subset \mathbb{R}^n$ be a convex set with $0 \notin A$. Then there exist a $v \in \mathbb{R}^n$ such that $v \cdot a \geq 0$ for all $a \in A$ and there exists $a_0 \in A$ with ...
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convexity of composite function

There is a composite function F(x)=f(g(x)): R -> R where f(v): R^2 -> R and g(x): R -> R^2. It is given that f(v) is convex on v and v = g(x) gives a map R -> R^2. If x is in a convex set, is F(x) ...
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Taylor's series question on bounds

$f(x) = f(y) + f'(y)(x-y) + \frac{f"(y) (x-y)^2}{2} + ....$ is the taylor's series. I am aware of bounds which specify the error in approximating $f$ with a polynomial. I want sufficient conditions ...
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An inequality of a multivariate function: $f(x_1,…,x_n) \geq \frac{1}{n}\sum_i^n f(x_i,x_i,…,x_i) $

Let us assume we have a non linear function $f : \Bbb R^{n+} \to \Bbb R ^+$, and let $x = \{x_1, x_2 , ..., x_n\}$, $x_i \in \Bbb{R}^+$, further define $\bar{x} = \frac{1}{n}\sum_i^n x_i$. My ...
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A necessary and sufficient condition for $f(x,y) = \phi(x²+y²)$ be a convex function

Let $f(x,y) = \phi(x²+y²) , \phi \in C^2$ and $\phi$ non-decreasing. Proof that $f$ is convex in the disk $x²+y² \leq a² \iff 2u \phi''(u) + \phi'(u) \geq 0 $ $\forall u \in [0,a]$ Here is my ...
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Show non-convexity of a function with vector input

How does one go about proving non-convexity of the function d? $$ d(v) = 1/2*||F(v)- p||^2 $$ $$ F(v)=\sum_{i=1}^n l_i*\begin{pmatrix} cos(\sum_{j=1}^i v_j) \\ sin(\sum_{j=1}^i v_j \end{pmatrix} $$ ...
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convex hull function in matlab

Is there any way to compute the convex hull of a finite set of points in Matlab and gives the half-space representation as its result? I usually use a toolbox called MPT developed at ETHZ Zurich, but ...
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Subdifferential optimality conditions

I need help with subdifferential optimality. Let $f(x_1, x_2)=x_1^2 + x_2^2 + |x_1 -x_2 - y|$. Find: \begin{align} \min_{x_1, x_2} f(x_1, x_2) \end{align} This is convex, so must have unique ...
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minimal representation of convex hull

Here is a question about the convex hull. Let $S$ be a set and $\bar{S}$ be the closed convex hull of $S$, i.e., $\bar{S}$ is the smallest convex set that contains $S$. Then is the following claim ...
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396 views

Simplex - Help with a proof

I am trying to prove the following (basic) claim about simplex. If you check my proof and help me with the part where I stuck, I would appreciate it very much. Let $X$ be a finite set and define ...
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LASSO constrained and penalised forms: first - order conditions?

The LASSO problem can be formulated in two ways: 1) Constrained formulation: $$ \|Xw-y\|^2\to\min_{w}\\ \text{s.t. } \sum_{i}|w_i|\leq{t}. $$ 2)Penalised formulation: $$ ...
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rconvex image under nonlinear function

Let $X\subset R^3$ be a compact and convex set, and let $f: X\rightarrow R^3$ be a nonlinear function, with $f\in C^k$. What are the tools to investigate if the image $K=f(X)$ is also convex, in the ...
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Required conditions for product of two submodular functions to be submodular?

3.32 of Boyd's Convex Optimization book says: Products and ratios of convex functions. In general the product or ratio of two convex functions is not convex. However, there are some results that ...
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Is this function really not concave or convex in any range?

Consider the function $f(x,y)=\frac{y}{1+e^x}$ where $0<y<1$ and $x \in \mathbb{R}$. If you plot this function, it looks like this: Also note that for a given value of $y$ the function $f$ ...
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Sum of convex and concave functions when one is greater than the other

Given two $C^\infty$ functions $f,g:\mathbb{R}\to\mathbb{R}$ such that $f(x)>g(x)\text{ }\forall x\in\mathbb{R}$. Moreover, we know that $f(x)$ is convex while $g(x)$ is concave. Now, let's define ...
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Comparison between secant and derivative in a convex function

Imagine that we have a function $f:\mathbb{R}\to\mathbb{R}$ which is convex, that is $f''>0$. We also know that $f'''<0$, that is its first derivative function is concave. Now, we can define its ...
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Properties of convex function with Lipshitz continuous gradient (Prof. Nesterov's textbook)

I am reading the Prof. Nesterov's textbook: Introductory lectures on convex optimization - a basic course p.57 I have problem in the following: My ...
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If $f$ is log-convex then $f$ is convex

Here's my attempt: $f$ is log-convex. Then: $\log f(\lambda x + (1-\lambda)y )\leq \lambda \log f(x) + (1-\lambda) \log f(y)$ As $e^x$ is increasing, we can apply it to the inequation without ...
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Quasiconvexity (in the sense of Morrey) implies Rank-One convexity

I am trying to understand why Quasiconvexity implies Rank-One convexity. In a standard proof of this fact a sequence of functions is constructed, which converges weakly to zero in $W^{1,p}.$ in ...
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A function is convex if and only if its gradient is monotonous.

Let a convex $ U \subset_{op} \mathbb{R^n} , n \geq 2$, with the usual inner product. A function $F: U \rightarrow \mathbb{R^n} $ is monotonous if $ \langle F(x) - F(y), x-y \rangle \geq 0, \forall ...
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What are the main uses of Convex Functions?

Up till now I have just learned that the concept of convexity in functions of one variable is used to complete the graphs of functions, meaning to locate points of inflexion and see if the graph is ...
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Composition of convex and concave functions

I had a homework question: "Show that the function f(x,u,v) = -log(uv-xTx) is convex on domain {(x,u,v)| uv-xTx,u,v > 0}". EDIT: x,u,v are Real No.s One pdf I found online says: "We can express f as ...
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Determine whether f(x1,x2) is convex, quasi-convex, concave or quasi-concave if x1, x2 > 0 where f(x1,x2) = (x1^2 - 1)/x2

The title says it all. Can someone help me with this please? I have no clue how to go about it. I read something about the Hessian matrix and it's relation to convexity. I computed the Hessian matrix. ...
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Minkowski sum and Polygons

The problem:.. Given two convex polygons $A$ and $B$, we can define Minkowski sum as A + B = {a + b: a $\in$ A, b $\in$ B}, where $a + b$ vector sum. Prove that: every vertex $p \in A + B$ is a ...
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Proximal-type support function properties - nonnegative & strongly convex (proof)

I am reading the paper of Prof. Yurii Nesterov: Primal-dual subgradient methods for convex problems The following part confuses me: $\\$ $\\$ ...
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How to prove a two variable set is convex

$X=\{(x,y)\in R^2\ :\ 3\le 2x+3y\le 8\}$ i tried to solve it as: Let set $X$ is convex for $x_2,y_2\in X$ such that $\alpha x_1+(1-\alpha)x_2$,$\alpha y_1+(1-\alpha)y_2\in X$ Now, $3\le 2x+3y\le 8$ ...
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Proof of unique solution of strongly convex function (Prof. Nesterov Paper)

I am reading the paper of Prof. Yurii Nesterov: Primal-dual subgradient methods for convex problems I am confused about the green part of the following: $\\$ ...
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How to prove this function is quasi-concave? [closed]

Consider the function $f(x,y) = x(1-y)\log(1+y/x^2)$, where $0\le x, y\le1$. Is this function quasi-concave?
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Generalization of Brouwer’s fixed-point theorem

Perhaps the most widely known version of Brouwer’s famous fixed-point theorem reads as follows: For any $n\in\mathbb N$, let $A\subseteq\mathbb R^n$ be a compact (with respect to the Euclidean ...
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626 views

Dual cone of a L1 norm cone?

I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...
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Open ball around a point in the hyperplane in a locally convex space

Let $f$ be a continuous linear functional in a locally convex space $X$ and let $z\in X$ so that $f(z)=\alpha$. Let $V_z$ be an open neighborhood around $z$. It is my intuition that $V_z$ should ...
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Probability that a point lies in an uncertain convex hull

Given $n+1$ independent random vectors $X_i \sim N(\mu_i,\Sigma_i)$, where each $\mu_i \in \mathbb{R}^n,$ let $C$ denote the random region formed by taking the convex hull of a realization of the set ...
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Can we solve minimax in this way?

I am working to use proximal operators for solving a minimax optimization problem. It is known that if you use alternative optimization, the algorithm cycles, see an answer to this question ...
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Minkowski sum and vectors

Problem: Given two convex polygons A, B, we can define Minkowski sum, as A + B = {a + b: a $\in$ A, b $\in$ B}, where a + b vector sum. Prove that: for every external perpendicular u to an edge of ...
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Which is bigger $e^{(a+b)}$ vs $e^a + e^b$?

I understand that exponential function is a convex function so for any convex function $\theta(a+b) > \theta(a) + \theta(b)$, but can someone provide a more formal proofs ?
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A convexity argument

Let $(\alpha_n)$ be a sequence of positive real numbers s.t. $\sum \alpha_n=1.$ Consider a sequence of complex numbers $(\beta_n)$ s.t $|\beta_n|=const$ for all $ n \in \mathbb{N}.$ Suppose that ...
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convexity of a nonlinear function

I tried to search similar answers to my problem here, but unfortunately I'm a little bit lost on this subject, originally from a physical problem, but can be stated as: Let $A\subset R^3$ be a ...
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Does there exist regular planar pentagon?

How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ (treated as a surface) includes 5 points which form a regular planar pentagon?