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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converting a equation to convex form which can be given to cvx solver to solve it.

Can anybody tell me how to convert this to quadratic programming format so that CVX could solve it...?? I am not asking the whole solution but need only conversion. objective is:- minimize {sum ( ...
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Extending a convex function

Suppose $f:(a,b) \to \mathbb R$ is twice differentiable with the property that $c_1 \leq f''(x) \leq c_2$ for every $x \in (a,b)$, where $c_1, c_2$ are positive constants. Is it possible to extend $f$ ...
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Identity of the Operator Norm applied to the differential $df$ for convex $U \subset \mathbb{R}^n$ and $f \in C^1(U, \mathbb{R}^k)$

In my Analysis II Script they often use 'special' norms such as the Operator norm to make proofs more 'elegant' or just shorter. There is also the following statement (without a proof) which I can't ...
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Tangent plane of a convex set.

Given a convex set $\Omega$ in $\mathbb{R}^n$ with smooth boundary, is $\Omega$ contained completely on one side of it's tangent plane at any point on the boundary? If so, how can we prove this? ...
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4answers
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Euclidean norm second derivative

I really need Your help. I need to prove that Euclidean norm is strictly convex. I know that a function is strictly convex if $f ''(x)>0$. Can I use it for Euclidean norm and how? ...
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Log-convexity preserved by sum?

I'm currently reading Boyd's book on Convex Optimization (free book which can be found here : http://stanford.edu/~boyd/cvxbook/) , and there is one proof I do not understand : p. 105, the book says ...
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Are the coefficients of a Taylor series bounded when the function is?

Say that I have three real functions $f(x)$, $g(x)$, and $h(x)$ such that $f(x)\le g(x)\le h(x)$ for all real $x$. Additionally, $f(x)$ and $h(x)$ are logarithmically convex. Can I make any definite ...
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Is this function of matrix F convex?

A scalar function $f(F)=vec(F)^{H}\Big(\Omega^{H}\big(M\otimes F(I+F^{H}F)^{-1}F^{H}\big)\Omega\Big)^{-1}vec(F)$ convex? $\Omega$ is arbitrary matrix and $M$ is positive definite. F can be reviewed ...
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Relationships between affine closures and convex closures

Let $V$ denote a vector space. Then the following concepts make sense: affine subset of $V$ affine closure (affine "hull") of a subset of $V$ Suppose $V$ is in fact a real vector space. Then the ...
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Can $\sin (x)$ be represented as difference of two convex functions?

In my optimization homework, I am supposed to prove that every differentiable function with Lipschitz continuous gradients can be represented as difference of two convex functions. I think I have come ...
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30 views

Pointwise convergence implies uniform convergence under concavity?

Suppose that $X$ is a subset of a Euclidean space and $f_n:X\to\mathbb{R}$ converges pointwise to $f:X\to\mathbb{R}$. A book I'm reading seems to say that: if $X$ is convex and compact and each ...
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Maximizing xy subject to x+y <= a x>0 and y>0, using Hessian to check concavity of f [on hold]

I wish to maximize $f$ subject to $x>0,\quad y>0,\quad x+y=a$ where $$f(x,y) = xy \quad x,y\in \mathbb{R}^n_+ $$ The Hessian did not help me so I was wondering whether I can make modifications ...
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Hint on how to proof that $x^2$ is convex

Note: I can't differentiate 2 times and prove that $f''(x) > 0$ The exercise requires me to prove that the function $f(x) = x^2$ is convex by using the following Theorem: $f(x) \ge f(x^*) + ...
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Normal Cone of a Convex Set

Is there a proof that shows that if a convex set $C \subset \mathbb{R}^n$ has smooth boundary, then the classical normal vector at a given point at the boundary is an element of the normal cone? That ...
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26 views

Convex optimization problem: linear equality and inequality constraints

When linear equality constraints can be converted in an inequality constraints for a strongly convex optimization problem? I mean, I got the same solution for both the following problem: 1) $\min_x ...
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1answer
19 views

Continuity of subdifferential mapping

I'm reading Cedric Villani's book topics in optimal transportation, and I have a problem on page 53: If $\varphi$ lower semi-continuous, then the subdifferential mapping $\partial\varphi$ is always ...
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47 views

Proof of some inquality on convex sets in $\mathbb{R}^n$

Suppose $\Omega$ is a strictly convex bounded set with smooth boundary in $\mathbb{R}^n$. If $y \in \partial\Omega$, how can we show that $$\langle(x-y),\nu_y\rangle < 0$$ for all $x \in ...
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24 views

Let S ⊂ R 2 . Show that if cl(S) is convex, then S is convex.

Let $S ⊂ R^2$. Show that if $cl(S)$ is convex, then S is convex. This seems intuitive but, I am having trouble thinking of a proof or counter example.
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How to prove convexity of a given set

I have the set $$ C_c = \{(x,y,z) \epsilon \mathbb{R}^3 : (2x-x^2+y)(2y-3z)(5x-z) > 1, |x| < 1, y > 3, z < 2\} $$ and I need to prove whether it's convex or not. I know that the ...
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36 views

Existence of complicated convex functions

In Stochastic Finance: An Introduction In Discrete Time (by Follmer, Schied), page 400, I found the following proposition: Proposition A.4. Let $I\subseteq\mathbb R$ be an open interval and ...
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Set of points at which a function coincides with its convexification is compact?

Let $f:[0,1]\rightarrow\ \mathbb{\bar{R}}$, and let $\tilde{f}$ be the convexification of $f.$ (i.e., $\tilde{f}$ is the pointwise supremum of all affine functions that lie everywhere below $f$.) Let ...
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connections between polar set, polar cone?

Given a set $S$, its polar cone http://en.wikipedia.org/wiki/Dual_cone_and_polar_cone and its polar set http://en.wikipedia.org/wiki/Polar_set are defined. Could some please tell me the ...
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Proximal operator of spectral norm of a matrix

How can I calculate the proximal operator of spectral norm for any general matrix, $X\in R^{m\times n}$ i.e., $X^* = \arg \min_X ||X||_2 + \frac{1}{2\tau} ||X-Y||_F^2$ I understand that the proximal ...
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Hahn Banach theorem and supporting hyperplane theorem

The question is out of Rudin Functional analysis Chapter 3 problem 1. Call a set $H \subset \mathbb{R}$ a hyperplane if there exists real numbers $a_1,\ldots, a_n, c$ (with $a_i \neq 0$ for at least ...
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Krein Milman Property

If a closed bounded (not compact) set $X$ in a Banach space $B$ (like $L^1$) has extreme point(s), must the max of a linear functional defined on $X$ occur at one of them? I suppose it depends on $B$. ...
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Book recommendation for Choquet theory

Assuming a good background in basic functional analysis and operator algebras, what is an appropriate text for self-study in Choquet theory?
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15 views

How to solve this entropy optimization problem with gradient projection method?

The problem is defined as $$ \min_{w} = \sum_{i=1}^{n} \sum_{j=1}^{n}\left\{ \left(\sum_{k=1}^{3}w_kP_{ij}^{(k)}\right) \log \left(\sum_{k=1}^{3}w_kP_{ij}^{(k)}\right) \right\} + \gamma \|w\|_2^2\\ $$ ...
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How to prove a set is 'norm-closed convex' [closed]

I have to prove that the given set is 'norm-closed convex.' I proved the 'convex' part. But I don't know how to prove a set is 'norm-closed' Can you tell me if there's a general approach to proving ...
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Convex optimization problem [closed]

Can anybody tell me how to convert this to quadratic programming format...?? objective is:- minimize {sum ( $(x(i)-x(j))^2 + (y(i)-y(j))^2$ ) for $j>i$; constraints:- $x(i)+r(i) \leq (1/2)w$; ...
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Intersection of two convex lattices polygons

A convex lattice polygon is a polygon whose vertices are points on the integer lattice. Let P and Q two convex lattice polygons with n ,(resp. m) vertices. Let R be the convex lattice polygon ...
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Convex hull of set of sparse vectors?

I am trying to understand how one can define the convex hull of sparse vectors. I understand that for k sparse vectors can be described as a union of subspaces (such as in: ...
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regularity of a bivariate function

Consider a bivariate function $f(x,y)$ which is concave in $y$. Moreover, for any given $y$, let $x^*(y)$ be the solution to $f_x(x,y)=0$, and there is $f_x(x,y)>0$ for $x<x^*(y)$ and ...
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Proximal operator for the nuclear norm of Hankel (x)

I have a problem in hand for which I need to compute the proximal operator of the composite function $||Hankel(x)||_{nuc}$ where $x \in R^N$ and $||.||_{nuc}$ denotes the matrix nuclear norm. For a ...
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42 views

How to use convexity in this step?

I am trying to fill in the details of a proof about the following statement: If $f:\mathbb{R}^n\to \mathbb{R}$ be a convex function, if subdifferential of $f$ at $x$ is singleton, then $f$ is ...
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Proving that the second derivative of a convex function is nonnegative

My task is as follows: Let $f:\mathbb{R}\to\mathbb{R}$ be a twice-differentiable function, and let $f$'s second derivative be continuous. Let $f$ be convex with the following definition of ...
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Continuity of convex function [duplicate]

Let $f$ be a proper convex lower semi-continuous function on $\mathbb R^n$, how can we prove $f$ is continuous in the interior of $Dom(f)=\{f<\infty\}$ ?
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If $f$ is midpoint convex, continuous, and two times differentiable, then for any $a, b \in \mathbb{R}$, there exists $c$ such that $f''(c) \geq 0$

One of my analysis texts states this as an exercise If $f$ is midpoint convex, continuous, and two times differentiable, then for any $a, b \in \mathbb{R}$, there exists $c \in [a, b]$ such that ...
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Non-trivial lower bound approximation of a convex function using the second derivative at the minimum

Say that I am given an infinitely differentiable convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$. I am wondering if I can construct a meaningful lower bound approximation of $f$ using it's ...
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Expectation of an increasing, bounded concave function of a non-negative random variable

Let $h:[0,\infty)\to [0,1)$ be a strictly increasing and strictly concave function. Let the argument of this function be a random variable $C$ with probability density function (pdf) $f_{C}(c)$ with ...
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Notion of positive third derivative for nondifferentiable functions?

Are there any notions that generalize the idea of a positive third derivative of a univariatve function to those for which the function is not necessarily differentiable? For example, a function ...
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Intersection numbers on blow ups of toric varieties

Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...
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Constructing a quasiconvex function

Let $C\subset\mathbb{R}^2$ be a nonempty convex set. A function $f:C\rightarrow\mathbb{R}$ is called convex if $$ f(\lambda u+(1-\lambda)v)\leq\lambda f(u)+(1-\lambda)f(v), \quad\forall u,v\in C, ...
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How can I prove the hypersurface $M$ is neither convex nor concave?

The question is so simple: how can I prove that $M$ (as defined below) is neither convex nor concave (in the real sense)? The point here is to define a new notion of convexity, the complex convexity, ...
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finding discrète coordinate of Intersection of two convex polygon?

I seek for cartésien coordinate of vertex's of the intersection area between two polygons ? We have two convex polygon's P & Q such that : all vertex of P (resp. Q) are in 2D cartésien plane. I ...
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An improvement of Jensen's inequality - help please!

It would nice if someone could help me with this problem. I am looking at an improvement to the classical Jensen's Inequality: $$\int_\limits{}^{} \phi(x) \mu \mathrm{d}x \geq ...
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Is the optimum of this problem unique?

I have a convex optimisation problem: $$\min_{x_i} \sum_{i=1}^N a_i \exp(-x_i/b_i)\quad\text{ s.t. }\sum_{i=1}^N x_i=x\text{ and } x_i\ge 0$$ The KKT conditions are: $$\lambda=\begin{cases} ...
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How to calculate Jensen's Inequality

How would one show that $\mathbb{E}\Big[f(X)\Big] \geq f\Big[\mathbb{E}(X)\Big]$ for the convex function f(x)=x$^{2}$ ?
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(Updated) Geometric Illustration of Monotone and Maximal Monotone Maps

I am writing a note about the Monotone and Maximal Monotone maps from the following book http://link.springer.com/book/10.1007%2Fb97594 In this book we read a map ...
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Nonlinear discontinuous convex function

Let $X$ be a normed vector space. Construct nonlinear discontinuous convex function $f:X \rightarrow \mathbb{R}$. I tried something with $\frac{-1}{\|x\|}$ but had no success.
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Is this function jointly convex in its variables?

I have a function which I suspect is jointly convex, but have a difficult time proving it, especially since the Hessian is messy. The function is $f(y_i,i=1.2,\ldots,N)=\sum_i l_i w_i + y_i$, where ...