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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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 ...
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1answer
39 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 ...
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3answers
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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, ...
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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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312 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
48 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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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
84 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 ...
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2answers
202 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 ...
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1answer
36 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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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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23 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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11 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
52 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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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?
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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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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 ...
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1answer
54 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 ...
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18 views

Classify extreme points of multivariate implicit functions when cross derivative is not available

I have the following problem: Let $f(x,y)$ be a function defined on $[0,1]^2$ I want to prove that $f(x,y)$ has no local minimum for $x>y$. I have no idea about the sign of the cross derivatives ...
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Calculating the convex conjugate of the function $f(x)=\lim_{n\to \infty}\left(-\frac{1}{n}\log \sum_{k=1}^n e^{a_k\cdot x+b_k}\right)$.

The convex conjugate also known as Legendre–Fenchel transformation of a convex function $f:\mathbb{R}\to\mathbb{R}\cup\{+\infty\}$ is function $f^\ast:\mathbb{R}\to\mathbb{R}\cup\{+\infty\}$ definite ...
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1answer
48 views

Extreme points of the set of positive regular borel measures on a compact Hausdorff space

I have some troubles with a specific proof of a (Bochner-type) theorem in Rudin's book "Functional Analysis". More specifically, let $X$ denote a compact Hausdorff-Space and let $M$ denote the set of ...
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Legendre–Fenchel transformation $ f^{\ast}(x^\ast)=\sup_{x\in\mathbb{R}^n}\{\langle x,x^\ast\rangle -f(x)\} $

The convex conjugate also known as Legendre–Fenchel transformation of a convex function $f:\mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$ is definite by $$ ...
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41 views

When is this set convex and compact?

$ S= \{(f_1,f_2)\in L^2(I)\times L^2(I)| f_1(x)+f_2(x)\leq 1, a.e.; 0\leq f_1(x)\leq a(x)\leq 1, a.e.; 0\leq f_2(x)\leq b(x)\leq 1, a.e. \}$ To make $S$ to be convex and compact, does $a,b$ need ...
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Relation between the number of halfspaces and the number of vertices of a convex polytope

Suppose we have an $n$-dimensional convex polytope $\mathbf{P}$ represented by an intersection of half-spaces as the following: \begin{equation} \mathbf{P} = \{ x \in \mathbb{R}^n \mid x \in ...
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$L_+^p(X,\mu)$ is a closed and convexe subset of $L^p(X,\mu)$.

I have a problem with an exercise: Let $(X,A,\mu)$ a measure place, $p\in[1,\infty)$ and $\mu(X)<\infty$.Prove that the set $$L_+^p(X,\mu):=\{f\in L^p(X,\mu):f(x)\geq 0\ \mu-\text{a.e.}\}$$ is a ...
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24 views

Convex quadratic problem solver gives different answers?!!

I'm not a mathematics girl but I'm pretty sure that the variance of a vector X should be a convex quadratic problem. my objective function is as follows: arg min var(sum(L) + X*L) x>0 vector X is ...
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23 views

Equivalent implications about convex functions

Consider $U\subset\mathbb{R}^n$ open and convex with $f\colon U\rightarrow\mathbb{R}$, $f\in C^1(U)$. Show that the following are equivalent: (i) $f$ is convex; (ii) For all $a, a+v \in U$, one has ...
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194 views

Solution of an equation and a system of inequalities

Consider an integer $n \geq 1$, a positive real number $A$ and a collection of nonnegative real numbers $\{a_{i,j}\}$ defined for $(i,j) \in \{1,\cdots,n\}^2$. I want to find necessary and sufficient ...
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31 views

Proof that f is convex

Consider $D\subset\mathbb{R}^n$ a convex set and $f_i:D\rightarrow\mathbb{R}$ convex functions in $D$, $i \in I$ is a any set of indexes. Suppose there is $\beta \in \mathbb{R}$ such that ...
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1answer
34 views

Understanding the subdifferential sum rule

A previous question asked: Given: $f$ and $g$ are lower-semicontinuous proper convex functions, $x \in \text{ri dom}(f) \cap \text{ri dom}(g)$, $h = f+g$, $p \in \partial h(x)$, Prove that there ...
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Is the ratio of a convex and linear function pseudoconvex?

Both functions are differentiable. I know from Chandra1 that the ratio of a nonnegative convex and a strictly positive concave function is pseudoconvex. Does this hold when the denominator is a ...
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Solution to a nonlinear problem at an extreme point

I have a convex optimization problem of the form: $$ \begin{aligned} \operatorname*{minimize}_{\mathrm{x} = (\mathrm{x}_1, \dots, \mathrm{x}_m) \in \mathbb{R}^{nm}} &\quad f(\mathrm{x}) = ...
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1answer
90 views

Convex Function

$f: U\subset\mathbb{R}^m \to \mathbb{R}$ is a convex function if $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$, for all $x,y \in U$ and all $t \in [0,1]$. If $f$ is convex and continuous function, and $f$ has ...
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Concavity of a positive homogeneous one function

Let $f:(0,\infty)\times(0,\infty)\rightarrow(0,\infty)$ be a twice continuous differentiable funcion such that (1) $f$ is homogeneous one, i.e. $f(tx,ty)=tf(x,y)$, for all $t,x,y>0$; (2) $\log f$ ...
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1answer
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Convex function when λ∉[0,1].

f :R→R is convex, Prove, for every x,y∈R, and λ∉[0,1] f(λx+(1−λ)y)≥λf(x)+(1−λ)f(y). In definitoin of convex funcion λ belongs in [0,1], but here not.
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Logarithm of Gaussian function is whether convex or nonconvex?

I have a gaussian distribution such as $$P(x)=\frac {1}{\sqrt {2\pi}\sigma}e^{-\frac {(x-\mu)^2}{2\sigma^2}} $$ As my knowledge, $P(x)$ is non convex function interm of $x$. However, if I map it to ...
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Describing convex hulls in purely metrical terms

Let $X$ denote a Euclidean space; take $X = \mathbb{R}^n$ for concreteness. Now consider $x,y \in X$. Then the line segment joining $x$ and $y,$ hereafter denoted $[x,y]$, can be described in ...
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Convex functions-Comparison of derivative and second derivative

Let $\phi:(0,\infty) \to \mathbb{R}$ be a function with second derivative, strictly incresing and concave. Suppose that $f(t)=\phi(e^t)$ is convex. Then one can prove that $$ \lim_{x \to \infty} ...
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When is the convex hull of two space curves the union of lines?

I am interested in the convex hull of two space curves. Let $A,B\subset \mathbb R^3$ be two spaces curves. I am interested in when $\mathrm {con} (A \cup B)$ equals to $$\bigcup _{a\in A,b\in B} ...
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Minimizing a Concave Function over a convex set

Here is the optimization problem that I am trying to solve. Thanks in advance for all help/insight provided. Let $T:[-2,2]^N\to\mathbb{R}_{-}$ be a concave function of its arguments. Given ...
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solving the primal problem via dual

On pp. 248 of Boyd and Vandenberghe: suppose 1) strong duality holds, 2) the dual optimal is attained at $(\lambda^*, \nu^*)$, 3) the dual function $L(x, \lambda^*, \nu^*)$ has the unique minimizer ...
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2answers
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Convexity and monotonicity

Let $f(n)$ be non-negative real valued function defined for each natural number $n$. If $f$ is convex and $lim_{n\to\infty}f(n)$ exists as a finite number, then can we conclude that $f$ is ...
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1answer
177 views

DE solution's uniqueness and convexity

I am lost and don't know how to prove the following: If $M$ is a positive definite symmetric square matrix and if $\overrightarrow {v}(t)$ is a solution of: $$\overrightarrow {v'}(t) = ...
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1answer
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Convexity of the complex ellipsoid

Let $p_1,\dotsc,p_n$ be positive integers. Define the complex ellipsoid $$\Omega(p_1,\dotsc,p_n)=\left\{(z_1,\dotsc,z_n)\in\Bbb C^n:\sum\limits_{i=1}^n{\left|z_i\right|^{2p_i}}<1\right\}.$$ I ...
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1answer
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Prove boundedness of 2nd derivatives

Let $f \, \colon \mathbb{R}^d \rightarrow \mathbb{R}^d$ be a smooth and convex function. Assume $f$ behaves asymptotically as a cone at infinity, i.e., $ \lim_{R \rightarrow \infty} \frac{f(R x)}{R} ...
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1answer
42 views

How to prove a point in a set is an extreme point of the set ?

Def: an extreme point of a set $K$ is the point that cannot be expresssed as a convex combination of other points in $K$. Apart from the definition, what else arguments can we use to prove that a ...
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Open embedding of affine toric varieties implies Cone is face of the other

let $\tau, \sigma \subseteq N_{\mathbb{R}}$ be two rational, strongly convex polyhedral cones with $\tau \subseteq \sigma$. Now we get an inclusion $S_{\sigma} \to S_{\tau}$ inducing an inclusion ...
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45 views

Prove that $\int_{0 \le u \le 1,\Omega}g^2(x)udx$ in term of $u$ is convex

I am having a cost function and I want to know whether convex or not. Could you explain help me my problem? My problem is that given a cost function such as $$F(u)=\int_{0 \le u(x) \le ...
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36 views

N-Functions (Nice Young functions)

A mapping $\Phi:[0,\infty)\to[0,\infty)$ is termed an N-function (nice Young function) if (i) $\Phi$ is continuous on $[0,\infty)$; (ii) $\Phi$ is convex on $[0,\infty)$; (iii) $\lim\limits_{t ...
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39 views

How to check whether the following function is concave or convex or neither.?

Let $\pi$ be a vector such that all its elements sum to 1. i.e, $\sum_1^n \pi(i) = 1$ where $\pi(i)$ denotes the i$^{th}$ component and $n$ is the length of the vector. Let $D$ be a diagonal matrix ...