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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Non-subdifferentiable convex function

Is there any convex function $f$ on a norm space $X$ such that $\partial f(x)=\emptyset$ at every $x\in X$? Thanks in advance.
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Exact Line Search in Projected Gradient Descent

How is exact line search adapted for projected gradient descent in convex optimization? One way I think of is that unconstrained exact line search is run, and the new point is projected into the ...
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Relation between sum of a max and max of a sum?

Consider $\frac{1}{T}\sum_{t=1}^{T}\max\{ 0,a_t\}$. Can we say whether this is greater or equal then $\max\{ 0,\frac{1}{T}\sum_{t=1}^{T}a_t\}$?
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Tangent cone of graph and epigraph sets.

Let us first recall the definition of tangent cone $\; T(\bar x; \Omega)$ of a subset $\Omega$ at $\bar x \in \Omega$, where $X$ is a Banach space: $$T(\bar x; \Omega)=\{v\in X:\; \; \exists ...
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Convex Sets and extreme supports

Let the set $S$ in $R^n$ consists of the origin $0$ and $n$ lineary independent vectors $T_1, \ldots, T_n$. Show that $C(S)$, the convex hull of of $S$, is the intersection of its extreme supports, ...
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The closednees in Moreau - Rockafellar Theorem.

One says that $x\mapsto \langle x^*,x\rangle +\alpha\;$ is an affine minorant of $f: \; X\to \overline{\mathbb R}\;$ if $\;\langle x^*,x\rangle +\alpha \leq f(x)\;$ for all $x\in X$. The Moreau - ...
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Show that the graph of a convex function is above any tangent plane

In proving jensen inequality one use that the graph of a convex function is above any tangent plane. I've been reading Property of convex functions and Tangent line of a convex function. But what ...
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20 views

Volume of Minkowski sum of a point and a hypercube

Let $A$ be a single point and $B$ a unit cube in $\mathbb{R}^n$, what is then the volume $\lambda \mapsto \mathrm{Vol}\big((1-\lambda)A + \lambda B\big)$? I am not exactly sure, what the set ...
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Volume of unit n-dimensional ball, definite integal

As a part of an assignment to calculate the volume of unit n-dimensional ball I got to the following expression, which I believe is true: ...
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Linear transformation preserving strict convexity

Consider the functions $f:\mathbb{R}^n\to\mathbb{R}$, $g:\mathbb{R}^m\to\mathbb{R}$ and the non-square matrix $A \in \mathbb{R}^{m\times n}$ with $m>n$. Let $x\in\mathbb{R}^n$ and consider the ...
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Computing subgradients, a basic example.

A vector $v \in \mathbb{R}^n$ is a subgradient of a convex function at $x$ if $$f(y) \geq f(x) + <v, y - x>$$ for all $y \in dom(f).$ The set of all sub gradients is known as the sub ...
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Sums of convex functions strictly convex in one variable

Let $f_i:\mathbb{R}^n\to\mathbb{R}$, $i=1,2,\ldots,n$ be twice continuously differentiable, convex functions in $x = (x_1,x_2,\ldots,x_n)$. Let each $f_i$ be strictly convex in $x_i$. Is the function ...
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Convexity definition confusion

When one writes $$f(\lambda x + (1-\lambda)y) \le \lambda f(x) + (1-\lambda)f(y)$$ for $x,y\in \mathbb{R}^n$, $\lambda\in(0,1)$ what does this mean? 1) Does it mean that the function is jointly ...
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Creating a $C^2$ smooth convex curve which joins circular arcs

Suppose that I have many points $p_1$,..., $p_n$, all very close to origin 0. For R very, very large, I have circular arcs of radius R, $\alpha_i$, about $p_i$. I do not speak about their ...
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L1 ball contained in convex hull of L0 ball

Consider the set $S$: the set of vectors whose $L^0$ pseudo-norm is upper bounded by $s$. Also, consider the $L^1$ ball of radius $\sqrt{s}$. It is apparently a well known fact that the $L^1$ ball is ...
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Impact of constraints on convex optimization

Let me start by saying I know almost nothing about optimization so please bear with me. Basically, I am wondering whether it is possible to solve a problem with two constraints by solving the problem ...
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Question on convexity

If I have a function $f:\mathbb{R}^n\to\mathbb{R}$ that is convex in ${\bf x} = (x_1,x_2,\ldots,x_n)$ and strictly convex in one of the variables, say $x_1$, then is $f({\bf x})$ strictly convex in ...
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Properties of the square norm in Banach spaces

Let $X$ be a Banach space with its dual $X^*$. Consider the mapping $f: X\rightarrow \mathbb{R}$ given by $$ f(x)=\frac{1}{2}\|x\|^2. $$ We have know that when $X$ is a real Hilbert space ($X=X^*$) ...
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Gradient of the optimal point

Hope to ask a Q on S. Boyd's cvx book: (p.139) A point x is optimal iff x in X and My Q is: If x is optimal, the 1st order differentiation at this point should be zero. Gradient is like ...
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Continuity of convex functions at point out of domain

I have been studying the continuity of a convex function and having a trouble below: In some books, the authors defined the the continuity of a convex function $f$ even $f$ is not in $\mbox{dom }f$, ...
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A Simple Algorithm for Imposing Semi-definite Constraints

What is the simplest algorithm to implement, to impose semi-definite constraints? $\min_{X\succeq 0} f(X) $, where $X$ is an $n \times n$ symmetric matrix, and $f$ is a general smooth convex ...
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Properties that guarantee quasiconvexity in $\mathbb{R}^n$

I have an open bounded connected domain $\Omega \subseteq \mathbb{R}^n$ and I would like to say that for every two $x,y \in \Omega$ there is a path $\gamma$ from $x$ to $y$ of length at most $C|x-y|$ ...
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Distance of convex combination of pairs of points in $\mathbb{R}^n$

Given 4 points $w,x,y,z \in \mathbb{R}^n$ define for $t\in [0,1]$ $f(t)=d(wt + (1-t)x, yt + (1-t)z)$. Is this function convex? I have found a proof by differentiating twice and calculating a lot but ...
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A Q about convex optimality criterion

Hope to ask about p. 139 of S. Boyd's cvx book: x is optimal iff x is in X (feasible set) and And the book use the following pic to illustrate: My Q is: why there is a negative sign '-' in ...
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Integral of increasing continuous function is convex

Suppose $g$ is increasing and continuous. Does it follow that $G(x) = \int_0^xg(y)dy$ is convex? Clearly $G'$ is increasing and continuous, and $G''\geq 0$ exists a.e., but I don't see how this ...
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26 views

Partial Ordering of proper cone K

$K$ in $\mathbb{R}^n$, and $K$ is a proper cone. Partial Ordering of $K$ : $x \leq_K y$ iff $y-x\in K$ (S. Boyd p. 43) My questions are: Does it require $x,y\in K$? If $x,y\in K$, it seems ...
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A Q about S.Boyd's CVX book p. 107

A Q about the following: (Come from S.Boyd's note) Note 1: y is a r.v with log-concave pdf "p(y)". My Qs are: What is the f(x) in the proof? It is marginal pdf or cdf or expected value? It ...
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The key step to prove log-convexity is preserved under sums

In S. Boyd textbook p.105 (button): (cvx = convex) Let F = log f & G = log g are convex (i.e. Let f & g are log-cvx) (This guarantees f & g are cvx, since log-cvx is included in ...
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Example of sum of log-concave is not always log-concave

I know that sum of log-concave is not always log-concave. Could anyone provides me with an example to prove this? Like probability distribution fn (pdf) of normal distribution is log-concave; on ...
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Convex function that has a finite limit at infinity

Can someone give me an example for a convex function that has a finit limit at infinity ?
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A bounded subset in $\mathbb R^2$ which is “nowhere convex”?

Let $F : \mathbb S^1 \to \mathbb R^2$ represents a simple closed curve $C$ in $\mathbb R^2$. The Jordan curve theorem says that the curves bounds a interior domain $\Omega$ and $\partial \Omega= C$. ...
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A statement for convex sets

The following statement is true or false? Given a convex set $S$ then for any $y \in S$ and $\theta\in[0,1], \theta \in \mathbb R$ there exist $y_1,y_2 \in S, y_1 \ne y, y_2 \ne y$ such that ...
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Convex in $ \mathbb{R^n}$

Prove that: [A be a convexe part $(A\subseteq \mathbb{R^n})] \implies [\forall x_1,x_2,...x_n\in A ,\forall\alpha_1,\alpha_2,...\alpha_n\ge0 $ $with$ $ \ \alpha_1+\alpha_2+...+\alpha_n=1 ...
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Subdifferential boundary conditions: Testing with $L^2$ or $H^{1/2}$ functions

My question was essentially this: Does it make a difference if I test subdifferential boundary conditions with functions from $L^2(\Gamma)$ or $H^{1/2}(\Gamma)$? In the following, I will phrase the ...
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Convex function almost surely differentiable.

If f: $\mathbb{R}^n \rightarrow \mathbb{R}$ is a convex function, i heard that f is almost everywhere differentiable. Is it true? I can't find a proof (n-dimentional). Thank you for any help
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Show homeomorphism between convex hull and unit ball?

In the proof of Schauder fixed point theorem in Evans' PDE book, he uses a claim that the convex hull $K$ of $N$ points $x_1,\dots,x_N$ in a convex compact subset $A$ of a Banach space $X$ is ...
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Total Variation minimization problem

Thanks for reading this thread. I have a object function, with constraints, I am trying to minimize. The object function is the Total Variation of an image. The Total Variation is defined as: ...
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About the hyperplane conjecture.

I have recently heard about the hyperplane conjecture and I would like to understand better the problematic behind this conjecture. The hyperplane conjecture: There exists a universal constant ...
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Is $f(H)=H^TH$ convex? $H$ is a $m\times n$ matrix

I tried to prove $f(H)=H^TH$ convex, where $H$ is a matrix. We know when $h$ is a vector, then $f(h)=h^Th$ is convex. Can I prove it using the following equation? $[\theta H_1 + (1-\theta) ...
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Is there any version of Jensen's inequality for quasiconvex function

I am looking for some generalization of Jensen's inequality for functions $g:\mathbb{R}^n \rightarrow \mathbb{R}$ where $g(x)$ is quasiconvex (or not convex). We known that for convex functions, ...
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Does one of these conditions for norms follow from the other?

The two conditions are: For all unit vectors $\mathbf{x}$ and $\mathbf{y}\hspace{-0.02 in}$, $\:$ if $\; \left|\left|\hspace{.03 in}\mathbf{x}\hspace{-0.05 in}+\hspace{-0.04 ...
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Folding in $\mathbb{R}^2$ - which area of modern geometry formalises this concept?

For any lines $L_1,...,L_n$ in $\mathbb{R}^2$ and $w_1,...,w_n \in \{-1,+1\}$ we can define a folding operation $F(L_1,...,L_n; w_1,...,w_n): \mathscr{P}(\mathbb{R}^2) \rightarrow \mathbb{R}^2$, ...
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Is this function strictly convex?

I think this function is strictly convex in the vector ${\bf x} = (x_1,x_2,x_3,x_4)$ but the fact that some terms are zero when variables take on the same values leaves me uncertain, i.e. when ...
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Checking convexity by looking at 2-dimensional cross-sections

If I have a closed set of n-dimensional points and I want to know if it's convex just by examining some set of 2-dimensional cross-sections (and checking each cross-section for convexity), how small ...
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Online convex programming: Projection followed by normalization

I have the following projected gradient descent online linear programming problem which has been well studied in www.cs.cmu.edu/~maz/publications/techconvex.pdf‎ $\mathbf{y}_{t+1}=\mathbf{w}_t - ...
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How can I prove this problem is quasiconvex?

I'm doing a convex optimization problem. It requires me to fit a rational function to an exponential function. I assumed the original problem would be a quasiconvex optimization problem and based on ...
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Minimizing a convex cost function

I'm reviewing basic techniques in optimization and I'm stuck on the following. We aim to minimize the cost function $$f(x_1,x_2) = \frac{1}{2n} \sum_{k=1}^n \left(\cos\left(\frac{\pi k}{n}\right) x_1 ...
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Strict convexity of the following function

I have a function that is of the form $C({\bf x}) = c_1\left(a_1x_1 + b_1x_1^2\right) + c_2\left(a_2(x_1-x_2) + b_2(x_1-x_2)^2\right) + c_3\left(a_3(x_2-x_3) + b_3(x_2-x_3)^2\right)$ where each ...
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Solution in general for a seemingly simple problem

Let $\mathbb{S}$ be a closed, bounded, convex set in $\mathbb{R}^N$. Let $\mathbb{x}=[x_1,\dots,x_N]$ be any arbitrary vector in $\mathbb{S}$. Then what can we comment on the problem \begin{align} ...
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Proving convexity of the Schatten 1-norm

Is it possible to show that the Schatten 1-norm is convex by the definition of convexity? I can't seem to find any way to derive an expression of the sum or matrix of singular values of a convex ...