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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Calculating Legendre Transform

Let $\Omega,\Omega^*$ be bounded domain in $\mathbb{R}^n$, $B_r(y_0)$ is a small ball of radius $r$ center at $y_0\in\Omega^*$, in $\Omega^*$ define a function $\psi(y)=-\sqrt{(r^2-|y-y_0|^2)}$ on ...
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Proving convexity

I ask you please some help with this problem: Let A $\subseteq$ R$^n$ be a convex set and $C(A)$ = {$\lambda$x, $\lambda \in \mathbb{R}$, $\lambda \geq 0$, x $\in$ A}. Prove that $C(A)$ is convex. ...
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Regularization vs. Inequality Constraint

For what values of a regularization parameter $\alpha$, there is an equivalent inequality constraint in convex optimization? In particular, in the convex optimization problems below $$ \text{ Problem ...
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Rate of convergence of an exponential function

If I have a function $$f = \exp(\sqrt{n} \cdot \frac{\sqrt{\log{n}}}{\sqrt{n}-\sqrt{\log n}}),$$ I can notice, that $$\lim_{n \to \infty} f = \infty,$$ but also I can notice that it goes very slowly ...
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Supremum of convex lipschitz functions.

Let $f_i:K\to R, i\in I$ be a family of convex, equi-Lipschitz functions on some compact subset $K$ of $\Bbb R^n$. Is it true that $\sup f_i$ is also Lipschitz continuous(assuming that the sup ...
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Linear constraint in convex optimization

Is it true that the solution to a linearly constrained convex minimization problem can only be placed on the boundary of the constraint set, for any nonlinear convex objective, e.g. $$ \min_x f(x)$$ ...
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Is it possible to convexify this cone constraint?

General question An SOCP constraint is given by: $$ \| A_i \mathbf{x} + b_i\| \leq \mathbf{c}_i^T \mathbf{x} + d_i.$$ I have the following constraint: $$ \| A_i \mathbf{x} \| \geq d_i.$$ Is it ...
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Show that $\mathcal{C}(A)$ is the smallest convex of $E$ containing $A$.

Let $E$ be a $\mathbb{R}$-vector space, and $A$ a nonempty subset of $E$. Show that $$\mathcal{C}(A) = \biggl\{\sum \limits_{k=1}^n \lambda_kx_k \biggm| n \in \mathbb{N}^*, (x_1,\dots,x_n) \in ...
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compact and convex set

I recently have worked on compact convex sets in the context of time series and my question is related to that. If we have a set $$ C=\{\beta_1 X_1 +\beta_2 X_2 +\phi H_1 , ...
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Projection onto Polyeder

I know how to projects onto a linear subspace of $\mathbb R^3$, but how to project a point $x$ onto an polyhedron given as the intersection of three halfspaces $$ \langle y_1, x \rangle \ge c_1 ...
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Series and concavity

If $u(x)$ is strictly concave, can I say: $$ \sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^{n+1}\cdot u(n) < \infty. $$ I am having trouble finding counterexamples. Thanks.
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Convexity defined by Karamata inequality

Just as the Jensen inequality is used to define convex functions, can the Karamata inequality be used instead to define convex functions?
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Separate two convex sets with disjoint interior (in $\mathbb{R}^n$)

In $\mathbb{R}^n$, I know that if $A$ is a convex set and $b$ in the boundary of $A$. Then we can separate $A$ and $b$, which means there exists $f \in \mathbb{R}^n$ such that $f\cdot x \ge f \cdot b$ ...
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Properties concave functions

Is is true that if $f(x)$ is a concave function of $x$ with domain $C$, then $f'(a) \leq \frac{f(a)}{a}$ for any $a \in C$, where $f'(a)$ denotes the derivative of $f(x)$ with respect to $x$ evaluated ...
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regarding the concept of dual cone

When studying the covex analysis, I am not clear about the concept of dual cone. In the following graph, $\mathcal{K}*$ was the dual cone. I marked two points, the ...
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if convex or nonconvex function

There is an iteration recurrence relations between the argument. In fact, it is part of my optimization model . The equation F is convex or not convex? thank u
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Polar set and adjoint operator

I'm trying to understand the following statement: A bounded operator between Banach spaces $u:X\rightarrow Y$ satisfies: $$\textrm{inf}\{\;||u(B_X \cap S)||:S\subset X,\,\textrm{codim}S=k \; ...
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Convex optimization: affine equality constraints into inequality constraints

I have the following problem: \begin{equation} \begin{array}{cll} \displaystyle \min_{ \mathbf{x} } & & \displaystyle f(\mathbf{x}) \\ \mathrm{s.t.} & & \mathbf{x} \in \mathcal{C} \\ ...
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Convexity in each argument and directional derivative

Let $f(x,y)$ be a continuous function, convex in each argument separately. Does this imply the existence of one-sided directional derivatives in any direction? For example, does there exist (finite or ...
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Separating convex sets in a tvs $X$.

I got doubt with the proof of this theorem. Let $X$ be a tvs, $A,B \subset X$ with $A$ an open convex set and $B$ convex such that $A \cap B = \emptyset$. Then there exists $f \in X^*$ (where ...
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Union of all sets of optimal solutions to a perturbed linear programming problem

Please let me know if you have some ideas on how to approach this proof? I got stuck part-way through. The following linear program is a function of $\theta$, $ \begin{array}{ll} \min & c^\top x ...
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Proving function defined by algorithm is convex

I'm working on my bachelor thesis and I'm trying to prove a conjecture, but I seem to miss the hint that helps me. I have an algorithm that defines a function $f:\mathbb{R}_{\geq ...
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Prove that function is convex

How can it be proved that the function $f(x) = \ln \bigl(\sum\limits_{i=1}^{n} e^{x_{i}}\bigr) $ is convex?
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To prove the existing and uniqueness of a solution

Let function $f$ be differentiable and convex in $R^{n}$. How can it be proved that $\forall \lambda > 0$ solution of system equations $f'(x) = -\lambda x$ exists exclusively ($\exists \hspace{3mm} ...
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What is the smallest possible angle of this polygon?

A convex polygon contains a square with side-length 1 and is contained in a parallel square with side-length 2 (which is its smallest containing square). What is the smallest possible angle of the ...
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Concavity of a function

While I am reading a book I couldn't follow the following step. " By concavity of the function $x \sqrt{\log\frac{1}{x}}$ for $x \in (0,1)$ we have that " $O(x \sqrt{\log\frac{k}{x}})$ = ...
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Sufficient conditions for convexity using the right derivative

We have a function $f:[0,1] \rightarrow \mathbb{R}$ that is continuous on $[0,1]$ with a non-decreasing right derivative everywhere in $(0,1)$. Is this definitely sufficient to show that $f$ is convex ...
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A a.e. strongly convex function

Suppose that $f=f(x)$ is strongly convex a.e. for $x\in\mathbb{R}$, i.e. there exists $\epsilon>0$ such that $f''(x)\geq\epsilon>0$ a.e. for $x\in\mathbb{R}$. Then there exists ...
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Proof that a set is convex

$$k\in \mathbb{R}_{+}$$ $$M=\left \{ (x_1,x_2)\in \mathbb{R}_{++}^2 \mid x_1 x_2\geq k\right \}$$ Prove that the set $M$ is convex. A hint is given (quoted from the text): We could choose to ...
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Begin study of convex algebraic sets in complex projective space

Where should I begin the study of convexity of (semi-)algebraic sets? In other words, projective varieties defined by polynomials of complex variables. The long-term goal is to study optimization in ...
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Dual convex pairs

I am currently trying to understand a certain proof. The author uses the term dual convex pair for a pair $(\phi,\psi)$ of convex functions defined on subsets $X,Y$ of $\mathbb R^n$ satisfying: $$ ...
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A question about norm for bounded linear transformations

Let $H$, $K$ be Banach spaces, and let $A: H \rightarrow K$ be a bounded linear transformation. Its norm is defined by: \begin{equation} \|A\| = sup\{\|Ah\|_K: \|h\|_H \le 1\} \end{equation} How to ...
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Convex subsets, Normed spaces, Separating hyperplane

I'm trying to understand the proof of a theorem taken from a textbook. Theorem: If $S_1,S_2$ are disjoint non-empty convex subsets of a real vector space $X$ (which may be infinite dimensional) then ...
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Deriving projection operator for an affine set

Given an affine set $Ax=b$, the Projection operator to this set is $$P(z) = z - A^{T}(AA^{T})^{-1}(Az-b)$$ which is also affine. How is this derived?
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Robust LP question using box uncertainty model

I am trying to solve this robust LP problem by writing it as a QP $$\min_x x^TSx : \mu \leq r^T x , Ax \leq b$$ Under Box uncertainty model: $$R = \{r : \| r - \hat{r}\|_\infty \leq \rho\}$$ Here ...
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What is the shape of all convex combinations of $\geq$ five vectors in $\mathbb{R}^3$?

The convex combinations of two linearly independent vectors in $\mathbb{R}^3$ span a line. The convex combinations of three linearly independent vectors in $\mathbb{R}^3$ span a solid triangle. The ...
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Proving upperbound using convexity

The original question is to prove $$\frac{1}{n}\sum_{i=1}^n x_i \leq \log{(\frac{1}{n}\sum_{i=1}^n e^{x_i})} \leq \max_{1 \leq i \leq} x_i$$ I show that $$x_{max} = \max_{1 \leq i \leq} x_i$$ ...
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Is it always possible to partition a fat shape to fat shapes?

The slimness factor of a geometric 2-dimensional shape is defined (for this question) as the ratio of the side-length of its smallest enclosing square to the side-length of its largest enclosed ...
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Prove that a given function is convex

Consider the following convex set: $$S = \{m \in \mathbb{R}^N : m_i \geq 0 \text{ }\forall i=1, \ldots, N \wedge \sum_{i=1}^Nm_i = 1\}$$ and following function $f : S \rightarrow \mathbb{R}$: ...
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Gaussian measure of a convex hull with respect to scaling.

consider the following problem: Let $m\geq 1$ and consider $a_1,\ldots,a_{m+1}\in \mathbb{R}^m$. Let $A=(a_1,\ldots,a_{m+1})$ be the matrix having the $a_i$ as columns. Let ...
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property of Convex body on the plane

Let $K$ be a convex body on the plane with smooth curve. Observe the triangle $\triangle ABC$ that contains $K$ with minimal perimiter and let $X,Y,Z$ the points on $BC,AC,AB$ that belong to $K$. I ...
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Increasing convex-like function in Hilbert space

I am intersted with the differential equation $$x'(t)=f(t,x(t)),\ t\in \mathbb{R}.$$ Can we find an example of a Hilbert space $H$ and a function $f:\mathbb{R}\times H \to H$ which satisfy the two ...
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Simplex - Help with a proof

I am trying to prove the following (basic) claim about simplex. If you check my proof and help me with the part where I stuck, I would appreciate it very much. Let $X$ be a finite set and define ...
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Difference of concave functions

Suppose that there are two concave functions $f_1(x)$ and $f_2(x)$ defined on $x\geq0$. In addition, the functions are positive, smooth, bounded ($|f_2|\leq b_2,|f_1|\leq b_1$ such that $b_2 = ...
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What functions are support functions of convex sets

Given a closed, convex, non-empty set $K\subseteq\mathbb{R}^n$ the support function $h_K:\mathbb{R}^n\to (-\infty,\infty]$ is defined as $$h_K(y) := \sup_{x\in K} \langle x,y\rangle$$ It is easy to ...
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Is the volume of a convex polytope efficiently computable from the vertices

Is there an efficient method to compute the exact volume of a bounded full-dimensional convex polytope, given the coordinates of its vertices (V-representation)?
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Convexity and Jensen's Inequality for simple functions

Suppose $\varphi$ is convex on $(a,b)$. I want to show that for any $n$ points $x_1,\dots,x_n \in (a,b)$ and nonnegative numbers $\theta_1,\dots,\theta_n$ such that $\sum_{k=1}^n \theta_k = 1$ we are ...
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Composition of non-monotonic convex function

Given the following composition of functions: $h:\Bbb R^k\rightarrow\Bbb R$ $g:\Bbb R^n\rightarrow\Bbb R$ $f(x)=h(g_1(x),g_2(x),...,g_k(x))$ There are known rules which guarantee ...
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How to prove that the following function is convex?

I want to prove convexity of the following function: $$f(x) = log_x \left(1 + \frac{(x^a-1)(x^b - 1)}{x-1}\right)$$ for any fixed $a, b \in (0, 1)$ and: $x\in(0,1)$ $x\in(1, \infty)$ I'm trying ...
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Decreasing Function Projected onto Simplex

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$, defined as $f(x) := a x + b$, where $a<0$ and $b \in \mathbb{R}_{\leq 0}^{n}$. Note that $f$ is decreasing: $$ x \geq y \Longrightarrow f(x) ...