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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KKT conditions for nonsmooth convex problems

What are the KKT conditions for a non-smooth convex function? Is the vanishing gradient of Lagrangian, replaced by $0$ in sub-differential of the Lagrangian, and all other things remain the same? I ...
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Sets defined by distance to a convex set

Let $Y \subset \mathbb R^n$ be a bounded convex set, let $R>0$, and let $$Z := \left\{z \in \mathbb R^n : d(z,Y) > \dfrac12R \right\}$$ where $$ d(z,Y) = \inf_{y\in Y}|y-z|. $$ If you like, ...
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Convex Function defined on a closed convex set, Semi-Definiteness of the Hessian Matrix

I have a real-valued function that is defined on the subset of the non-negative (possibly zero) n-dimensional real numbers that add to one, i.e., a closed and convex set. Moreover, the Hessian matrix ...
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Lipschitz in R^1 implies Lipschitz along any line in R^k (convex function)

I'm trying to solve a homework problem (baby rudin course) and realized that Lipschitz continuity really makes the problem easier for me. However, since it is not defined in Rudin I have to prove the ...
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244 views

A beautiful inequality for convex functions

Let $f\in \mathcal{C}([0,1],\mathbb R_+)$ increasing. Prove that there exist $g,h\in \mathcal{C}([0,1],\mathbb R)$, convexs, such that $g\leqslant f \leqslant h$ and : $$\displaystyle ...
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Level set of convex functions

Let $f:\mathbb R^n \to\mathbb R \cup\{+\infty\}$ be a proper convex function, assume that there exists $c\in\mathbb R$ such that the $c$-level set $L_{\leq c}=\{x\in R^n: f(x)\leq c\}$ is nonempty and ...
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19 views

Generalisation of circumscribed circle for polygones instead of triangles.

Given a triangle, there exists a unique circle passing through its $3$ vertices. I wonder if more generally for any $n\geq3$ we can define a family $\mathcal{C}_n$ of convex closed curves in the ...
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1answer
57 views

Explain the convexity by looking the hessian matrix of a function

The hessian matrix of a function is given by, $$ H = \begin{bmatrix} a & b & c \\[0.3em] b & b & 0\\[0.3em] c & 0 & c \end{bmatrix} $$ where, ...
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Proving the Obvious

Is it normal that I have the hardest time when I'm trying to prove statements that are blatantly obvious on a visual and/or intuitive level? For instance, how does one go about formally proving the ...
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33 views

A characterization of convexity for functions with vectors as domain.

Let $f:\mathbb R^n \rightarrow \mathbb R$ be a continuously differentiable function. By $df(w)$ I denote the Frechet derivative of $f$ at $w$ Prove that $$f \:\text{is convex} \Leftrightarrow ...
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Monotonically increasing maximum eigenvalue

Let a matrix $A \in \mathbb{R}^{n \times n}$ be the convex combination of two matrices as $A = qB + (1-q)C$. Define $B$ as unit anti-diagonal. Define $C_{i,j} = \delta_{i,i+1}$. Consider $A$ for ...
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28 views

Computation of convex conjugate

I am learning convex analysis by myself and I need help. How to show that if $X=U=\mathbb{R}$ and $f\left(x\right)=\frac{|x|^{p}}{p}$ then the convex conjugate ...
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How to handle concave-concave constraints?

there. I have an optimization problem, which takes the following form ...
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1answer
59 views

Proof of the Moreau decomposition property of proximal operators?

Given the prox operator i.e. $ prox_h (x) = arg min_u (h(u) + 1/2 ||u-x||^2_2) $ the Moreau decomposition property says that $ x = prox_h (x) + prox_{h^*} (x) $ where $h^*$ is the conjugate of ...
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31 views

Express a function as difference of convex functions (DC)

is there a way to express the function $$1-\exp \Big( \frac{-\max(0,x)^2}{\alpha} \Big)$$ as the difference of two convex functions (DC)? Thanks
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convexity of inverse function

I have a question on the reverse of a convex function. Let $f(x)$ be a convex function. Is the reverse function, say $g(x)=f(x)^{-1}$, is necessarily a concave function ? Considering that such ...
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Proving that $f(x) = (1+x)^{-\theta}$ is convex using only definition

How can I prove that the function $f(x) = (1+x)^{-\theta}$ (where $\theta>0$ is a parameter) is convex using only definition, i.e. without using derivatives? I actually want to prove convexity of ...
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Inequality related with concave property

Assume that $f>0,f'<0$ and $f$ is logconcave(the log of $f$ is concave) and twice differentiable. Can we prove, or give a counter example to the following claim: there exists $\bar x>0$ such ...
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Does Jensen's inequality become stricter with respect to the right boundary point?

Let $f(x)>0$ for all $t\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ be a density. I will truncate the density to the finite interval $[a,b]$ and will eventually be taking ...
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14 views

2 questions - Convexity of a function

I have 2 questions about Convex functions Let f(x) be a convex function. Is the reverse function, say g(x), is necessarily a concave function ? (Considering that such function g does exist (f is ...
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Uniqueness of a convex linear decomposition

Let $X$ be composed of $d$ different vectors of $\mathbb{R}^n$ : $X=\{x_1,\ldots,x_d\}$ and $H$ be the convex hull of $X$. Each vector $y\in H$ can be expressed as $$y=\sum_{i=1}^d a_i x_i,$$ with ...
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Relax equality into inequality in convex problem

Let $\mathbf{x}, \mathbf{z}, \underline{\mathbf{x}}, \overline{\mathbf{x}} \in \mathbb{R}^{I}$, where the first two are variables and the last two are given data. I have the following problem: ...
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Convexity of support function

Let $C$ be a closed non-empty set, but not necessarily convex. The support function of $C$ is given by $$S(z) = \sup_{c \in C} \langle z,c\rangle. $$ Prove that this is a convex function. ...
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Minimizing difference and individual variables in convex problem

Let's say I have the following optimization problem: $$ \begin{align*} \min_{\mathbf{x},\mathbf{y}} & \sum_i x_i-y_i \\ \mathrm{s.t.} & \{\mathbf{x},\mathbf{y}\} \in ...
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Prove that the intersection of convex sets is convex using the following three points…

I want to prove each point, then, use points (1) and (2) to prove (3). $C_{1} = \lbrace x \in \mathbb{R}^{n} \mid h(x) = 0 \rbrace $ is convex iff $h(x)$ is affine in $C_{1}$ $C_{2} = \lbrace x ...
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64 views

Matlab - Generate square convex function with positive definite Hessian Matrix

So, I have to generate a square convex function in Matlab and it's Hessian Matrix must be positive definite but I can't find any function that can help me do that. Is there anything I should search ...
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Why is this weighted least squares cost function a function of weights?

Here is a picture from my book regarding weighted least squares: Totally lost here, so I extracted the main nested issues confusing me: First Question: I know that in any LSE we want to minimize ...
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48 views

Largest eigenvalue of symmetric matrix

I am trying to understand why the $\lambda_{\max}$ function is convex given an $n\,x\,n$ symmetric matrix, let's call it $A$. I know from elementary property of eigenvalues that all the eigenvalues of ...
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1answer
21 views

Polytope - Convex Hull

After doing some reading on the V-representation of a convex polytope (finite set of extreme points, also the convex hull?), it's often simply stated that the convex hull is compact. Can anyone show ...
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“Support function of a set” and supremum question.

I have already learned about what a supremum means from wikipedia and from another answer here. However I am not quite sure what 'supremum over a set of functions' means exactly. As an example, my ...
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45 views

Caratheodorys lemma proof

I have to proof caratheodorys lemma for my oral exam. The proof is given here. I dont get the last part. "This process can be repeated until x is represented as a convex combination of at most d + 1 ...
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Inequality involving max is confusing me.

I am trying to understand one line in a derivation here. Simply put, the statement is that: $$ max\{\theta \ f_1(x) + (1-\theta) \ f_1(y) \ , \ \theta \ f_2(x) + (1-\theta) \ f_2(y)\} \leq \theta \ ...
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Examples of affine functions and convex sets

I'm just learning about convexity and affineness, and I've read over some similar questions asked here, but those were more about general properties. I need some help applying those properties to a ...
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Hausdorff metric and convex hull

Let $X$ be a normed linear space and $A, B \in P(X)$. We define $\overline{co}(A) =$ the closure of the convex hull of $A$. Let $h$ denote the usual Hausdorff metric. We need to show that: $$h( ...
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The equivalent definition of denting point

How i can prove that If $K$ is a subspace of Banach space $X$, $x$ is denting point of $K$,when for every $\varepsilon>0$,there is a unit vector $x^{*}\in X^{*}$ and $\delta>0$ such that ...
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Convex Functions are Continuous

In W. Rudin's Principles of Mathematical Analysis, we read in Chapter 4 that real-valued functions defined on an open interval $(a,b)\subseteq\Bbb R$ are continuous (specifically, Exercise 23). I am ...
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Calculation of the set for the polar tangent cone?

I have the following theorem in my book. Assume that $\tilde{x}$ is a local minimum from a minimization problem and that f(.) is differentible at $\tilde{x}$ Let $T_X(\tilde{x})$ be the tangent cone ...
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The convexity of $f(x)/x$ if $f(x)$ is concave/convex?

I was wondering if you know a theorem that states that the function $f(x)/x$ is convex in $x$ if $f(x)$ is concave or convex in $x$. $f(x)$ is convex and increasing in $x$. when $\lambda>0, ...
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Is the geometric-to-arithmetic function convex or concave?

Consider a vector $\mathbf{x} \in \mathbb{R}_{++}^N$. Also consider two functions, $g(\mathbf{x}): \mathbb{R}^N \rightarrow \mathbb{R}$, and $a(\mathbf{x}): \mathbb{R}^N \rightarrow \mathbb{R}$, ...
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Concave function divided by a convex function. What is the result?

Let us say that I have a function $f(x)$ that we know is a concave. And let us also say that we have another function $g(x)$ that is a convex. If I make a new function, $h(x) = \frac{f(x)}{g(x)}$, ...
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An extreme point that is not strongly exposed

I want to construct an example that a point is extreme but is not exposed. This example can be in the following : A compact convex subset $K‎\subset‎\mathbb{R}^{2}$ and a point $u\in K$ such that ...
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Convex set for a set of points in 2d plane

There are set of five points $A(0,0) ,B(1,1) ,C(2,0) ,D(2,2).E(0,2) F(1.5,1.5)$ $S=\{A,B,C,D,E,F\}$ Please tell me whether my understanding is correct or not! The points $A,C,D,E$ forms a convex ...
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On convex hulls and intersections of chains of compact sets

Let $V$ be a topological vector space, let $\{ C_i \}_{i \in I}$ be a set of compact subsets of $V$ which forms a chain with respect to inclusion. For now, assume the following stronger properties: ...
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computation of subdifferential

This question mainly deals with subdifferential of a convex function with respect to the cost function $c(x,y)=\frac{|x-y|^2}{2}$ I want to compute the cost-subdifferential $\partial^{c}\phi$ of the ...
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A characterization of differentiability of a convex function

Let $\phi : \mathbb R^n \to \mathbb R$ be a convex function. For all point $x\in \mathbb R^n$, define the subdifferential as $$\partial \phi(x) = \{ y\in \mathbb R^n | \ \phi(z) \geq \phi(x) + ...
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How to verify correctness of a Fenchel conjugate derivation

Suppose I derived Fenchel conjugate of a function. My goal is to check if my solution is right. Suppose the steps are not available any more and only the final solution is present. Is there any ...
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Show that the Rosenbrock function is strictly convex for a specific region

So we know that the Rosenbrock function is a test function of sorts, but can anyone prove that a specific region is strictly convex? Rosenbrock eqn: $ f(x_{1},x_{2}) = 100(x_{2} - x_{1}^{2})^{2} + ...
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What are the extreme points of the set of probability measures on $(\mathbb N, \mathcal{F})$?

Say, we have some $\sigma$-algebra on $\mathbb N$ and let $\mathbb P$ be the set of all probability measures on it. We know that $\mathbb P$ is convex, so I wonder how do the extreme points look like. ...
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Tangent Cone is a cone?

I first give two definitions. Def1: A set $S$ is a cone if $x \in S, \lambda \geq 0 \implies \lambda x \in S$. Def2: Let $S$ be any set (we may assume $\mathbb{R}^n$ with the usual Euclidean ...
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is convex hull of intersection equal to intersection of convex hull

is $convexhull(S_1\cap S_2)=convexhull(S_1)\cap convexhull(S_2)$ where $S_1$ and $S_2$ are finite sets.