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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Subgradient of the optimal value of a linear program with respect to its parameters

Consider the linear program $f(c)=\min\{c'x\mid x\in\mathbb{R}^n,Ax=b,x\geq0\}$. Are there any results on what is $\partial f/\partial b$?
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How to pivot to an adjacent vertex in simplex method

In the simplex method, we need to move from one vertex of the polyhedron to an adjacent one. Suppose the polyhedron is $P=\{x\in\mathbb{R}^n\mid Ax=b,x\geq0\}$ with rank$A=m<n$. For a ...
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Question on relation between convex sets

The book I am referring regarding bell inequalites has the following two things it mentions about two convex sets which I am unable to understand. There are two convex sets $C$ ($C$ is a polytope ) ...
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Is the expectation of log-concave function still log-concave?

I know the expectation preserves the concavity (or convexity), but I was wondering is it still true that the expectation of log-concave function still log-concave; to be more precise, Let ...
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Are differentiable and strictly decreasing functions always concave?

If a demand function is continuously differentiable and strictly decreasing in price, does that mean it will be always concave?
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Convex signal reconstruction for convex generator function?

Let $f : \mathbb{R} \mapsto \mathbb{R}$ be a convex (not affine) function and suppose that $y = f(x)$. We want to reconstruct the input $x$ for a given $y$, and the standard approach would be to ...
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38 views

The Legendre-Fenchel transform of $BV$ semi-norm

I am reading a numerical paper in which it calls some "easy" facts from convex analysis but I can't justify it... Let $\Omega\subset \mathbb R^2$ be open. Define for a function $u\in L^1(\Omega)$ and ...
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Convex envelopes of bivariate functions

In order to convexify my nonlinear non-convex program I need convex envelopes for the function $(x/y)^2$, both x,y are positive. I am only aware of the convex envelopes of the type $xy$ from here ...
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Prove the concavity of the transformation from a concave function to another

Let's say we have $f_1$ and $f_2$, both strictly increasing and strictly concave on $[0,+\infty)$. $f_1(0)=f_2(0)=0$ and the difference $f_1-f_2$ is strictly positive and strictly increasing. That is, ...
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Convex functionals, min-max of linear functionals, etc

I was reading in this book on the well-known Courant-Fischer min-max Theorem, which states that if $\lambda_1(A)\geq \lambda_2(A)\geq \lambda_n(A)$ are the $n$ eigenvalues of a $n\times n$ Hermitian ...
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An affine set $C$ contains every affine combinations of its points

Show that an affine set $C$ contains every affine combinations of its points. Proof by induction: From the definition of an affine set, we know that $\forall x_1,x_2\in C \text{ and } \theta_i\in ...
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Minimizing the max function

Suppose we have the single-variable function $$f(x) = \max_k \{f_k(x)\}$$ where each $f_k$ is convex and smooth (and known beforehand). We want to minimize it over some bounded interval. We can, in ...
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Find $U \subset D $ with $|U|$ minimal s.t $v = (1/2,1/2,1/2)$ belongs to the convex hull of $U$.

Consider the convex hull $\text {conv} \{e_1,e_2,e_3,(1,1/2,1/2),(1/2,1,1/2), (1/2,1/2,1)\}$ in $\mathbb R^3$ and the vector $v = (1/2,1/2,1/2)$. I want to compute $U \subset ...
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Function dominated by convex function is eventually convex

Suppose that we have a twice-differentiable function $f$ on $x\in [0,\infty)$ such that $f(x)>0$ on $x\in [0,\infty)$ (i.e. strictly positive) $f'(x)<0$ on $x\in [0,\infty)$ (i.e. strictly ...
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25 views

Every convex function is locally Lipschitz ($\mathbb{R^n}$)

I know that if $f$ is convex function so $f$ is continuous. And I know too that partial derivatives exists. What can I do?
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Prove a closed ball in $\mathbb R^3$ has an infinite number of extreme points.

How do I show that a closed ball in $\mathbb R^3$ has an infinite number of extreme points ? (Closed ball is written as $S = \{(x,y,z) \in \mathbb R^3 | \sqrt {x^2 + y^2 + z^2} \le R \}$) I know ...
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Convex combination of quasiconvex functions.

Is a convex combination of two quasiconvex functions necessarily quasiconvex? If not, what can be said about the convex combination?
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Let $S = \{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 \le 1\}$. Prove $(1-\lambda)x + \lambda y \in S$ for $x=\lambda'y$, $\lambda' < 0$.

Let $S = \{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 \le 1\}$. I've verified that $x,y \in S$ implies $(1-\lambda)x + \lambda y \in S$ when $x,y$ are linearly independent using Pythagoras and when ...
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Compute the extreme points of a polyhedron $P$ and write $(1/2,1/2,1/2)$ as a linear combination of these.

Compute the extreme points of a polyhedron $P$ and write $(1/2,1/2,1/2)$ as a linear combination of these. I want to compute all the extreme points of the set $P$ (polyhedron) in $\mathbb R^3$ ...
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Iterative algorithms to approximate unknown multivariate convex real-valued functions by a set of linear upper/lower bounds

I am looking for iterative algorithms that can approximate a multivariate convex real-valued function $f(\vec{x})=y,\, \Bbb{R}^n\rightarrow \Bbb{R}$. The function is not known beforehand, but it is ...
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Suppose $P = \text{conv}(\text{ext}(P))$. Is it possible to find $S \subset P$ of cardinality $< \text{ext}(P)$ s.t $\text{conv}(S) =P$?

Suppose that $P$ is a polyhedron and that $P = \text{conv}(\text{ext}(P))$. Suppose further that $ \text{ext}(P)$ has cardinality $r$. How do I see that I cannot find a set $S \subset P$ of ...
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Let a polyhedron $P = \text{conv}(S)$ where $S$ extreme points. Can $S' \subset S$ (proper) be a generator?

Let $P$ be a polyhedron and let $S=\{ v_1, \ldots, v_r\}$ its extreme points. Suppose further that $\text{rec}(P)={0}$ so $P=\text {conv}(S)$. How do I see that I cannot remove any points from $S$ and ...
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Can $\text{conv} \{ e_1,e_2,e_3, (1/2, 1/2, 1) , (1/2, 1, 1/2) , (1, 1/2, 1/2) \}$ be reduced to a convex hull of a subset of these?

How do I see whether $$\text{conv} \{ e_1,e_2,e_3, (1/2, 1/2, 1) , (1/2, 1, 1/2) , (1, 1/2, 1/2) \}$$ can be reduced to a convex hull of a subset of these vectors? That is, if $D = \{ e_1,e_2,e_3, ...
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Estimate the volume of a convex body given a uniform random sample of points inside it?

Let $K$ be a convex, full-dimensional, bounded region of $R^n$. More precisely, there exist two balls of radiuses $0<r<R$ such that the ball of radius $r$ is fully contained inside $K$ and the ...
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1answer
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Is the ratio of a decreasing function and an increasing function, a quasi-concave function?

$f(x)$ is a strictly decreasing function and $g(x)$ is a strictly increasing function and positive. Is $h(x) = f(x)/g(x)$ quasi-concave?
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How do I show that the set $S$ in $\mathbb R^3$ defined by linear inequalities is a $3$-simplex?

Consider the set $S$ in $\mathbb R^3$ defined by the inequalities: $x+y+z \ge 1$ $-x+y+z \le 1$ $x-y+z \le 1$ $x+y-z \le 1$ How can I show that $S$ is a $3$-simplex ? (Convex hull of $3 + 1$ ...
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How do I show that $B$ has an infinite number of extreme points, but no faces (sides) of dimension $1$ or $2$?

Let $B = \{ (x,y,z) \in \mathbb R^3 \mid x^2 + y^2 + z^2 \le 1\}$. How do I show that $B$ has an infinite number of extreme points, but no faces (sides) of dimension $1$ or $2$ ? I've been thinking ...
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Let $B=\{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 \le 1\}$. Prove existence of a unique point $v_0 \in B$ that is closest to $v \notin B$?

Suppose $B=\{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 \le 1\}$. Consider a point $v \notin B$. How do I prove that there exist a unique point $v_0 \in B$ that is closest to $v$. Also, how do I ...
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Prove that a functional is convex

Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. We define the functional $\Phi$ as: \begin{equation} \Phi(x)=\frac{1}{2}(Tx,x) \end{equation} My exercise says that $\Phi$ is ...
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Prove the existence of minimal height of a convex polygon

Suppose we have a polygon in $\mathbb{R}^n$. Obviously, we can always trap the polygon into two parallel hyperplanes perpendicular to any given direction. Now the question is, how to prove the ...
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$U$ bisected by all hyperplanes $\implies$ $U$ symmetric?

Let $U \subset \mathbb{R}^n$ be a bounded open convex set, such that every hyperplane passing through the origin divides $U$ into two sets of equal volume ($n$-dimensional Lebesgue measure?). ...
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Simple question on maximizing a family of concave functions

Given $x \in [0,1]$, let $f_k(x)$ be concave function in $x$ for all $k= 0,1,...,N$. Is the following two maximization formulations equivalent? $$ \max_{0 \le k \le N} f_k(x) =?= \max \{f_0(x), ...
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Is a maximum function concave?

A function such as $$\max (X, Y)$$ given $X\geq 0$, $Y \geq 0$. Because It would form an inverse L shape on the graph, and that looks like a concave function. However my teacher said that the function ...
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The convex hull of every open set is open

Let $X$ be a topological vector space. Prove that the convex hull of every open subset of $X$ is open. I tried using definition of Convex Hull and Open Set, but I couldn't prove the statement.
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Convexity of $f(x) = \sum_i x_i^p$ for $x_i \ge 0, p \ge 1$

Let $x = (x_1,\ldots, x_n)$ be nonnegative real numbers and $p \ge 1$, then the function $f(x) = \sum_{i=1}^n x_i^p$ is convex, the following proof is wrong, or not? I have it from here, but then ...
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Maximal eigenvalue is convex function

Let $A$ be a symmetric real matrix. let $f(A)=\lambda_{max}(A)$ be it's largest eigenvalue. Why is $f(A)$ convex?
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Orthogonal projection of an $n-$vector onto the subspace ($m\leq n$)of $\mathbb{R}^n$ containing a convex polytope

Lets say we have an $n \times m$ matrix $A$, whose column vectors are $(\vec{\mathbf{0}},a_1,a_2,...a_j)$ are points in $\mathbb{R}^n$ and the non-zero vectors have unit length. Let ...
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Multivariate convex function / increasing differences

$\newcommand\Rr{\mathbb{R}}$I am trying to show the following statement. It feels true to me, but I haven't found any reference in the literature so far: Let $\Rr^n$ be ordered component-wise, i.e., ...
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Differentiability of support function (even for non-convex)

I am reading an economics book (for those who are interested, MWG Microeconomic Theory) and there's a theorem that was just given without proof, but I am interested in the proof - also because I ...
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Boundary of convex set is piecewise $C^1$

Let $K$ be a convex and compact subset of $\mathbb{R}^2$. Is it true that the boundary of $K$ can be parameterized by a piecewise $C^{1}$ application $\gamma :I\to\mathbb{R}^2$?
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how can we check convex or nonconvex feasible?

example if i have 20 constraints functions.These functions cut the objective function and create the feasible region. Their intersections can become edges and create a nonconvex feasible region even ...
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28 views

Optimising a piecewise convex function

I have a function $f(x)$ defined for $x\in \mathbb{R}^+$ that is decreasing, piecewise convex and continuous. The 'pieces' of $f$ are exponential and the rate of decrease for each piece reduces as we ...
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Quasi concavity and Quasi Convexity-intuitive understanding

I'm having trouble grasping the concept of quasi concavity and quasi convexity. My textbook states that if f is quasi-concave, then f (λx + (1 − λ) y) ≥ min {f(x), f(y)} . Also that is f is quasi ...
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Is an lsc sublinear function $X^* \rightarrow (-\infty, \infty]$ always a support function for some closed non-empty $C \subset X$?

I can't seem to find any resources on this, even though it seems like an obvious question to ask. The separation theorem implies that, if we have an lsc sublinear function $\phi : X^* \rightarrow ...
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Isn't every increasing continuous convex function strictly increasing (disregarding $f(x) \equiv 0$)?

Isn't every increasing continuous convex function $f$ strictly increasing (disregarding the trivial case $f(x) \equiv 0$)? I don't see any counterexample!
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Is this Function of Product of variable and Ratio of CDF and PDF of Standard Normal Distribution Convex?

Let $G\left(x\right)=x\frac{\phi\left(x\right)}{\Phi\left(x\right)}$. Here, $\phi$ and $\mathbf{\Phi}$ are the standard normal PDF and CDF, respectively. Is $G\left(x\right)$ convex? It has been shown ...
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Properties of a proper cone

Let $K$ be a proper cone. I need to prove following properties: if $x \preceq_K y$ and $u \preceq_K v$, then $x+u \preceq_K y+v$ if $x \preceq_K y$ and $y \preceq_K z$, then $x \preceq_K z$ if $x ...
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1answer
21 views

Given Convex Function, Conditions when Variable times Convex Function is convex

Given that say, $f(x)$ is convex for $x>0$. We can arrive at the following conditions for when $xf(x)$ would be convex. Please add anything that I might have overlooked and further simplifications ...
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60 views

Proof that Convex Function with alternate variable is convex

Given that say, $f(x)$ is convex for $x>0$. Would the proof that $f(a-x)$ where $a>x$ is convex proceed as shown below. Please verify the correctness and/or alternative approaches that can ...
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1answer
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Convex Hull = Boundary+Segments

If $A\subseteq\mathbb{R}^n$ is an non empty set and $H$ is the convex hull of $A$, how can I prove that the boundary of $H$ consists only of points that lie in the boundary of $A$ and segments that ...