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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Convexity of a function

Define function $F$ as $F(x,y,x,t)= (xy-zt)^2$ where $x,y,z,t \geq 0$. Question: Is this function Convex? Thanks!
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“Interpolation” of polynomials

I'm dealing with a probability problem and I have to understand the following operation on polynomials: let $F$ and $G$ be any two polynomials of variable $p\in [0,1]$ (to be thought of as a Bernoulli ...
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Banach space Lower semi-continuity (lsc) implying continuity

How to show the following: If the monotone real valued function $f(x)$ whose domain is a subset of $R$ is lower semi continuous on every point of the interior of the domain then it is continuous on ...
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Showing that the set of polynomials is convex

How to show that the set of polynomials of $x^2+bx+c$ having at least one real root, is convex? Let $x^2+b_1x+c_1$ and $x^2+b_2x+c_2$ have at least one real root. Need to show that ...
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directional derivative sublinear of a convex function sublinearity problem to show

How to show the following: If $f:\mathbb R^d \rightarrow \mathbb R$ is convex then its directional derivative is sublinear? Thank you...
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Weak convexity and continuity

For any open interval $(a, b)\subset {\mathbb R}\,$, define a weakly convex function $f:(a, b) \rightarrow {\mathbb R}$ as one for which $$f(q\;x_0 + (1 - q)\;x_1) \leq q\;f(x_0) + (1-q)\;f(x_1)$$ ...
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Possibility of Unboundedness in Least Squares Minimization

Suppose we have the quadratic minimization problem \begin{equation} \min_x \frac{1}{2} x^TPx + q^Tx +r \end{equation} We know that when $P$ is symmetric positive semi-definite, but the optimality ...
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If a vector in a convex set can be extended infinitely to a certain direction, can any vector in that set be extended infinitely to that direction

Assume we have a convex set $U$. Given $x \in U$, assume there exists a vector $y$ such that $\forall t>0, \ \ x+ty \in U$. I wish to prove that $\forall z \in U,\ \ \forall t>0,\ \ z+t y \in U$ ...
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106 views

directional derivative of convex function sublinear proving that fact

How can we show that the directional derivative of a proper convex function on $\mathbb{R}^n$ is sublinear? Thank you!
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How to check the convexity of a constrained set

Say that we have the following matrices: $A\in M_{n,n}$ which is unknown, $Y\in M_{n,m}/\{0_{n,m}\}$ and $X\in M_{n,m}/\{0_{n,m}\}$. I want to show that the following set is convex $\Omega=\{S\in ...
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convex conjugate $f^*$ is proper if both $f$ and $f^{**}$ are

If $f$ and $f^{**}$ on $\mathbb R^d$ are proper functions where $f^*$ stands for the convex conjugate of $f$ why does that follow that $f^*$ is proper, too? Thanks a lot...
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Is the composition of trace inverse and convex matrix product convex?

Is the trace of the inverse of the matrix product $B^TB$, i.e. $\mathrm{trace}((B^TB)^{-1})$, convex where $B\in M_{n,m}$. I know that $S\longrightarrow \mathrm{trace}(S^{-1})$ is a convex function ...
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Convex conjugate of a function triple conjugate

How to show that: f: R^n->R $f^{*} = f^{***}$ where $f^*$ stands for the convex conjugate of the function. Thanks a lot!
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convex lsc function affine minorant theorem proof

How to show the following: f:R^n->R A lower semicontinuous convex function f equals the pointwise supremum of all its affine minorants. Thank you!
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Infimum/Supremum of an intersection of subsets of a vector space.

Consider a collection of subsets $A_i$ of an ordnered vector space or field. I am trying to find out, under what (minimal) conditions the following holds. $\inf \bigcap_{i\in I}A_i \le \sup_{i\in ...
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Convex functions - multi variables

I have $ x_1,x_2,..,x_n\in\left[0,1\right] $. Suppose $ x=(x_1,x_2,..,x_n), F(x,i,j,\epsilon)=g(x_1,...,x_i+\epsilon,...,x_j-\epsilon,..,x_n) $ . We know that $F$ is convex with respect to ...
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Relative interior inclusion of convex sets

Is relative interior(C-D) = relative interior(C) - relative interior(D) where C and D are nonempty convex sets. If so please give the proof If thats not the case could you give a counterexample ...
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epigraph relative interior not included

Let $f(x)$ be a convex function from $R^{d}$ to $R$. Why is the point $(x_0,f(x_0))$ not in relative interior of epigraph of the function. I know it is not in interior but why not in relative ...
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Epigraph supported at some point meaning of the sentence

Can you tell me what does the following sentence mean? Let $z \in \mathbb{R}^d$ and $(-z,1)$ supports epigraph of $f$ at $(x_0,f(x_0))$ Thank you..
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When is a sequentially closed cone, closed?

Let $X$ be a locally convex, Hausdorff topological vector space and $C\subseteq X$ a convex cone, which is sequentially closed. What are criteria, that would imply that $C$ is closed (in the topology ...
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Is the trace of inverse matrix convex?

Hi I would like to know whether the trace of the inverse of a symmetric positive definite matrix $\mathrm{trace}(S^{-1})$ is convex. Actually I know that the trace of a symmetric positive definite ...
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maps from a convex set to itself

Suppose $S\subset \mathbb{R}^n$ be a closed convex set under Euclidean topology (but not necessarily bounded, example a closed cone). Let $\mathcal{E}(S)=\{L:\mathbb{R}^n\rightarrow \mathbb{R}^n\text{ ...
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Weak upper semicontinuity of convex function?

Let $f:X\rightarrow\mathbb{R}$ be a continuous convex function over the banach space $X$. (Note that it is everywhere finite.) In particular, it is lower semicontiuous, and by Mazur's theorem [S. ...
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convex function right left derivatives

Hello how to show the followings for a convex function $f(x)$: Let $f(x_0) \in R$ then $\frac{f(x_0 + \epsilon) - f(x_0)}{\epsilon}$ is nondecreasing in $\epsilon$ Similarly how to show the left ...
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How to generalise a result regarding intersections of cones and other convex sets?

To test for a particular property of positive LTI systems using feasibility problems I've come across the following claim which, intuitively, I believe can be generalised. I think I've (rather ...
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affine function definition

If we define the affine function as $f(\lambda x + (1-\lambda)y) = \lambda f(x) + (1-\lambda)f(y)$ for every $x,y \in R^d$ and $\lambda \in R$ How to show that it is equivalent to the definition ...
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Convex Extremal

I was asked whether this is true in a question paper: If p is a subset of q (where both p & q are convex), then an extreme point of q is also an extreme point of p. Ans.: Yes statement is correct. ...
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Simple proof? If $x$ lies outside a compact convex set, there exists a $y$ closer to every point in the set than $x$.

This seems rather obvious intuitively, but I can't find a simple proof. If $C$ is a compact, convex subset of $\mathbb{R}^n$ and $x \not \in C$, then there exists a point $y$ such that, for every ...
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Is this function convex or non-convex?

Let $$f(a,b,c,d)=\frac{(a-b)*(c-d)}{\sqrt{(a-b)^2+(c-d)^2}},$$ where $a,b,c,d$ are variables. Is this function convex or non-convex?
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Questions regarding internal and interior points for a convex subset of a topological vector space

Suppose that $X$ is a topological vector space, with a convex subset $A$. How do we show that if the vector $u$ is in the interior of $A$, then $u$ is an internal point of $A$ and if the interior of ...
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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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Intersection of two $n$-dimensional quadratic inequalities?

I have two quadratic inequalities of the form $$ a_1x^TAx + b_1^Tx + c_1 \le 0\\ a_2x^TAx + b_2^Tx + c_2 \le 0 $$ where $A\in\mathbb{R}^{n\times n}$ is positive semidefinite, $x\in\mathbb{R}^n$, ...
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Sets whose intersection with line segments have finite components

Certain subsets $A$ of $\mathbb{R}^n$ satisfy the property that given any line segment $L$, $A \cap L$ has a finite number of connected components. For instance, $A$ can be any finitary union of ...
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Convex functions and uniform convergence of derivatives

Let $f_n:[0,1]\to\mathbb{R},\ n\in\mathbb{N}$ be a sequence of convex analytic functions. Consider the sequence of derivatives $(f_n')_{n\in\mathbb{N}}$. Suppose that ...
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A particular ILP where the existence of a relaxed solution implies the existence of an integer solution

This question emerged from a discussion on my previous question Determining quickly whether this Integer Linear Program has any solution at all, which I would like to elaborate separately. I am ...
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Numerical Methods for minimizing a Non-Differentiable Convex Function of Several Variables

I have a multi-variable convex continuous function which is not differentiable. I am interested to know about different numerical techniques, possibly also references to them, used for this. Read ...
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Edges of the convex hull of a finite point set

I'm looking for a formal proof of a statement that seems obvious (and yet may be wrong) to make sure one of my algorithms is correct. Given a set S of N points in $\mathbb{R}^3$, suppose we have a ...
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Direction of recession, convex analysis

Hi how to show the following: Let $C$ and $D$ be two non-empty closed convex sets with no common direction of recession. Then $C - D$ is closed. Thanks a lot...
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Intersection of a 2-Dimensional body and a Line given west-most point and south-most point

I have a 2-Dimensional Closed Convex Compact Body $\mathbb{S}$(a set, for eg, a circular disc)). Assume, it has a non-zero intersection with the all-negative quadrant. Consider the following 2-D point ...
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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 ...
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Convexity of affine function.

Can someone help me with a proof that affine function preserves convexity? Given that $f$ is convex, $A$ is in $\mathbb{R}^{M\times N}$ and $b$ is in $\mathbb{R}^m$ then show that $g(x) = f(Ax+b)$ is ...
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affine set convex set

How to show the following: Let $C$ be a convex subset of $\mathbb R^d$. Then $\operatorname{int} C \neq \emptyset$ if and only if $\operatorname{aff} C = \mathbb R^d$ where $\operatorname{aff} C$ is ...
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affine subset closedness

How to show an affine subset $C$ of $R^d$ is closed (yes convergence in norm sense) I thought the following: Let $x_0$ be an accumulation point $C$ define $C - C + x_0$ then $C-C$ being closed $C - C ...
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Prove every local minimum is a global minimum

Let $Q\in\mathbb{R^{dxd}}$ and $A\in\mathbb{R^{d'xd}}$ be two matrixes and $b\in\mathbb{R^d}$, $c\in\mathbb{R^{d'}}$. Suppose $d'\lt d $. For $x\in\mathbb{R^d}$. Minimize $$f(x)= ...
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proving this inequality related to conjugate functions

For $x \in \mathbb{R}^n$ let us denote $x_{[i]}$ the $i$th largest component of $x$ s.t $$ x_{[1]} \geq x_{[2]} \geq x_{[3]}\ge\cdots $$ The function $$ f(x)= \sum_{i=1}^r x_{[i]} $$ is the sum of ...
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Continuous boundary of a convex set

Is the boundary of a compact convex set in Rn continuous? Seems like the answer should obviously be yes, but I cannot find any such result in the literature. Can somebody provide a reference (or a ...
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The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is ...
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Mid-Point Convexity Implies Convexity in Banach Spaces? [duplicate]

Possible Duplicate: Showing that $f$ is convex Let $X$ be a real Banach space and $f:X\rightarrow \mathbb{R}$ a continuous function. We say that $f$ is Mid-Point convex if for all $x,y\in ...
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Is $f(x_1,x_2)=x_1x_2$, with $x_1,x_2 \in \mathbb{R}_+$, a convex function?

Let $f(x_1,x_2)=x_1x_2$, with $x_1,x_2 \ge 0$. Is $f$ a convex function? Why? $\ \\$ Edit (in view of the comments below) The Hessian matrix is $H=[0\, 1; \,1 \,0]$, which is indefinite (in ...
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Jensen's inequality and a estimate in $L^p$

In problem 3 we have: If $f:\mathbb{R} \longrightarrow\mathbb{R}$ is mensurable, $E:=\mathrm{supp}\ f$ and $$\int_E e^{|f(x)|}dx =1,$$ then $f\in L^p(\mathbb{R})$, for all $p\in(0,\infty)$ and ...