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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The dual of a cone with every face exposed

Let $C \subseteq \mathbb{R}^n$ be a closed convex cone with the property that every of its faces is exposed, i.e. the intersection of $C$ with a supporting hyperplane. Does then the dual cone $C^*$ in ...
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A conjecture on increasing, convex functions

Suppose that $F:\mathbb R_+\to\mathbb R_+$ is twice-continuously-differentiable and satisfies $F(0)=0$, $F'\geq0$, and $F''\geq 0$. Is it necessarily true that $$F''(x)x^2-F'(x)x+F(x)\geq0$$ for all ...
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Convex Hull given set of planes.

If I have some finite amount of planes, for example \begin{equation} z_1=2x, \\ z_2=2y, \\ z_3=3+x+y, \\ z_4= 2+x, \\ z_5=2+y \\ z_6 =3 \end{equation} And I wish to find the convex hull in order to ...
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26 views

Subgradient at the boundary of a closed set

Suppose I have the convex function $f(x) = |x|$ over the domain $x \in [-1,1]$, and I wish to find the subgradient. It is easy to find the subgradient in the interior of the domain. At the ...
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Assuming that $f$ is convex and continuous from the right at zero show that $∂f(0)$ is empty using the definition of the subdifferential.

Suppose that $$ f(x)= \begin{cases} +\infty,\;\;\;\;\;\;\;if\; x<0\\ 0,\;\;\;\;\;\;\;\;\;\;if\;x=0\\ x\ln(x)\;\;\;\;if\;x>0 \end{cases} $$ Assuming that $f$ is convex and continuous from the ...
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How to show that a set is not convex in the following problem? [closed]

I need to show that $x^2>y$ is not convex where $z=x+iy$. I know what it means to be convex but I cannot figure out how to show it rigorously. A methodological answer would be helpful.
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1answer
45 views

Can something that is not a level curve be quasi-convex/concave?

Something is quasi convex (concave) if the lower (upper) contour is convex. I don't know if we can talk about an upper/lower contour for a function, not a function's level curves/sets. Therefore, can ...
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40 views

Bound functin using convex function

Given a continuous function $f$ on the interval $[a,b]$, I want to prove that there is a convex function $ g$ such that: 1. $g(x) \ge f(x) , x\in [a,b]$ 2. $g(a) = f(a)$ 3. There is $p\in (a,b]$ such ...
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84 views

Show that a real-valued function with non-empty subdifferential is convex

Let $f:X \to \mathbb{R}$ be a function such that $\partial f(x)\neq \emptyset$ for all $x \in X$. Show that $f$ is convex. I would appreciate some help with getting started on this problem. Thanks
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Is the function characterized by $f(\alpha x+(1-\alpha) y) \le f^{\alpha}(\alpha x)f^{1-\alpha}(y)$ convex?

Is a non-negative function $f(x)$ convex ? If for $x \ge y$ it satisfies for any $\alpha \in [0,1]$. \begin{align} f(\alpha x+(1-\alpha) y) \le f^{\alpha}(\alpha x)f^{1-\alpha}(y) \ \text{ ...
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68 views

Suppose that $f = ι_{\mathbb{R}_+}$. Show that [closed]

Suppose that $f = ι_{\mathbb{R}_+}$. Show that $0$ is in the boundary of dom f and that $∂f(0)$ is nonempty using the definition of the sub differential. Any hints or suggestions is greatly ...
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31 views

Relative interior commutes with cartesian product

I have been reading convex analysis by Rockafellar. One of the proofs made use of the fact that $ri(A\times B) = ri(A) \times ri(B)$, where $A$ and $B$ are convex sets, $\times$ denotes the Cartesian ...
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76 views

Examples of tangent cone

In http://users.ece.utexas.edu/~cmcaram/EE381V_2012F/Lecture_2_Scribe_Notes.final.pdf The definition of a tangent cone is defined as the closure of the feasible directions. Definition 9. (Tangent ...
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66 views

The derivative of a uniformly convex function is onto

My question stems from Evans' PDE book (p.142 in 2nd ed). Given that $F:\mathbb{R} \rightarrow \mathbb{R}$. Assume that $F$ is uniformly convex ($F'' \geq \theta > 0$ for some constant $\theta$). ...
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Inequality involving a convex function

Do the points that satisfy an inequality involving a convex function constitute a convex set? Specifically if $x \in \mathbb R^n$ and I have a function $f(x)$ then is the set $\{x \mid f(x) \le 0\}$ ...
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Convex risk measures

What is the intuitive explanation for convex risk measures represented as: $$\rho(X)=\sup_{P\in Q}\{E_{P}(-X)+\alpha(P)\}$$ where $\alpha(P)$ is a penalty function depending on the plausibility of P. ...
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33 views

Show a function is convex on the given domain

$f(x,y,z)=\frac{(y+2z)^2}{(x-3y)}$,$\quad$ on $\{(x,y,z)\in R^3\mid x-3y>0\}$. I tried to figure out the Hessian matrix of $f(x,y,z)$ to see whether it is positive definite , but it's very ...
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52 views

The number of facets of an affine image

I have a full dimensional polyhedron $P_1 \subseteq \mathbb{R}^d.$ Now i define another polyhedron as follows: $$P_2 = AP_1 \oplus B$$ with $A \in \mathbb{R}^{(d-1) \times d}, \,\, B \in ...
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Suppose that f : X → ]−∞, +∞] is convex and proper, that x ̄ ∈ dom f

Suppose that $f : X → ]−∞, +∞]$ is convex and proper, that $x ̄ ∈ dom f$ , and that $λ > 0.$ Show that $$∂(λf)(x ̄) = λ∂f(x ̄).$$ I am looking for a little help with this one. I am a bit unclear ...
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Optimality criterion for unconstrained convex optimization problems

Consider a general convex optimization problem: \begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} & & f_0(x) \\ & \text{subject to} & & f_i(x) \leq 0, \; i = 1, ...
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Minimum-norm Projection on Convex Sets

The general method of Projection on Convex Sets (POCS) can be used to find a point in the intersection of a number of convex sets i.e. $$ \text{find } x \in \mathbb{R}^N \text{ s.t. } x \in ...
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Convex combination

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuos function. Let $x_1,x_2,x_3\in\mathbb{R}$ and $\alpha_1,\alpha_2,\alpha_3\in[0,1]$ such that $\alpha_1+\alpha_2+\alpha_3=1$. I am trying to prove ...
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Is $x^TAy\geq 0$ when A is positive semidefinite and $x,y \neq 0$

I know that when A is positive semidefinite then for any $x \in R^n,x\neq 0$ then $x^TAx\geq 0$ What about I have two different vectors $x,y$? Is $x^TAy\geq 0$ when A is positive semidefinite and ...
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Subgradient of a composition w/ affine $f$

Let $f:\mathbb{R}^n \to \mathbb{R} \cup \{ \infty \}$ be convex, w/ subgradient at x in its domain $\partial f(x):=\{ d:f(y)\geq f(x)+d^T (y-x),\forall y\in \mathbb{R}^n \}$. Let $h(x'):=f(Ax'+b)$, ...
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43 views

Proving a convex upper contour set implies a quasi-concave function

Could someone confirm whether this proof is correct, or give advice on improving it? I want to prove that if the upper contour set of a function $U$ at $\overline{u}$ is convex, then $U$ is ...
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Lipschitz Continuity of the Solution Set of LP with a single additional Convex Constraint

Consider the following Linear Program with interval bounds on the decision variables: \begin{equation} \begin{aligned} S(\mathbf b) = \ & \arg \min_{\mathbf x \in \mathbb R^n} && \mathbf ...
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Is the function $f(x)=x^TAx$ convex when $A \in S^n,A\geq0$,$x\in R^n$?

Is the function $f(x)=x^TAx$ convex when $A \in S^n,A\geq0$,$x\in R^n$? Notation $S^n$: Symmetric $n$ x $n$ matrix. $R^n$: Column vector $n$x$1$ $A \geq 0$: $A$ is positive semi-definite matrix ...
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Picturing/Graphing (quasi-)concave/convex functions?

I understand the definitions, and can do work with them, but when I try to picture them I get confused (picture simple concave/convex functions that is, not some very complex ones obviously). ...
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Quasiconvexity analog for function with an integer domain.

Suppose I have a function that is not quasiconvex, as in the graph below, but would be quasiconvex if we cared only about integer points. That is, $f:X \subset \mathbb{Z}\rightarrow \mathbb{R}$ ...
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Affine-regular hexagon in convex body

An affine-regular $n$-gon is a non-degenerate affine image of the regular $n$-gon. It seems to be a standard fact in combinatorial geometry that inside every convex compact set $K\subseteq \mathbb ...
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How is the Lagrangian related to the perturbation function?

Given a convex programming problem $$\begin{align*} \text{minimize} &\quad f(x) &\\ \text{such that} &\quad g_i(x) \leq 0 & i=1\dots k\\ & \quad h_j(x) = 0 & j= k+1\dots n ...
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Product of log-concave densites

I have a density function $f(a,b,c,d)$ for random variables $A,B,C,D$ which factors as $f(a,b,c,d)=f_{A}(a)f_{B}(b|a)f_{C}(c|b,a)f_{D}(d|a,b,c)$. where $f_{A},f_{B},f_{C},f_{D}$ are log concave in ...
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Why is this an example of second order cone program?

In https://inst.eecs.berkeley.edu/~ee127a/book/login/exa_ell_sep.html We are trying to separate two ellipses using a hyperplane The resulting "SOCP" is given as: $$\min\limits_a \|R_1^Ta\|_2 + ...
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Convex functions, prove another definition.

I have the next problem: Suppose $f$ is continuos then $f:\mathbb{R}^n\to R$ is convex iff $\forall x,y \in \mathbb{R}^n \left( \int_0^1 f(x+\theta (y-x))d\theta \leq \dfrac{f(x)+f(y)}{2}\right)$. The ...
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Is always a convex function two times differentiable

Assume that f is twice differentiable, that is, its Hessian or second derivative $\nabla^2f$ exists at each point in dom$f$, which is open. Then f is convex if and only if dom$f$ is convex and its ...
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is the Superellipse function convex or not?

I'm trying to solve an optimization problem and i need to know if the following constraint of Superellipse is convex or not $\left|\frac{x-x_o}{a}\right|^n + \left|\frac{y-y_o}{b} \right|^n \geq 1$ ...
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Under what conditions does a convex objective function have a concave value function?

Suppose that $u:\mathbb{R}^{n} \to \mathbb{R}$ is a continuous, (weakly) convex function. Now define the value function $\phi$ to be: $$\phi(p,w) = \max_{x>>0} u(x)$$$$ \text{ subject to: } p ...
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Optimization function convex or not

I need to comment whether my optimization function is convex or non-convex. My optimization function is in the form of $(y-y_{cap})^2$. y is know. $y_{cap}$ comes out of a MATLAB pfile. So, ...
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Is the space R^N convex?

My question is if $\mathbb{R}^{N}$ is convex. The definition I have for a set S to be convex is that if any convex combination of any two elements of S is in S, where a convex combination is defined ...
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When is a halfspace a subset of another halfspace?

Let's say we have the following halfspaces: $$H_1=\{x\mid a^Tx\leq b\}$$ and $$H_2=\{x\mid\tilde{a}^Tx\leq \tilde{b}\}$$ I want to find the conditions that need to hold such that $H_1\subseteq H_2$. ...
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Sub-gradients of non-Convex Functions

I was wondering, why are sub-gradients only mentioned for convex functions? What happens when the assumption of convexity is relaxed? EDIT: I get that for functions which have local/global maxima, ...
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Let $K$ be a nonempty, closed and convex cone in $X$. Define $k^{\ominus}=N_k(0).$ Show that $k^{\ominus\ominus}=k.$ [duplicate]

Let $K$ be a nonempty, closed and convex cone in $X$. Define $$k^{\ominus}=N_k(0).$$ Show that $k^{\ominus\ominus}=k.$ So, what I think this question is asking me to do is show that the normal cone ...
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Geometric Interpretation of the Separation Theorem

I am struggling to see the meaning behind the theorem's statement, which is: Let $K\subset \mathbb R^n, K\neq \emptyset$ be a convex set and $x\not\in \text{clo}(K)$. Then there is $\gamma \in ...
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Volume of convex polytope the vertices of which are vertices of the unit hypercube

I have an infinite family $P_{a,b}$ of non-degenerate convex polytopes of dimension $ab$. Each polytope is given explicitely by a list of its vertices and all of these vertices are elements of ...
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51 views

Projection of a hyperplane

Let $\alpha$ be a vector in $X$ such that $||\alpha||=1$ and let $\beta\in\mathbb{R}$. Consider the hyperplane $$C= x\in X:\langle a,x\rangle=\beta.$$ Prove that $$P_c(x)=x-(\langle a,x ...
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1answer
22 views

An inequality for divided differences of an increasing concave function

Take $0<a<b<c<d$ and consider $f$ strictly increasing and strictly concave (with $f(0)\geq 0$). I would like to prove that $$\frac{f(d)-f(b)}{f(c)-f(a)}<\frac{d-b}{c-a}$$ without any ...
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24 views

Linear transformation (Ax) and the solution set

given that $$A \in \mathbb{R}^{mxn} , S \subseteq \mathbb{R}^n, b \in \mathbb{R}^m$$ Where all elements (x) in S satisfy the inequality $$Ax \le b$$ Then S must be a convex set I was tasked to show ...
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1answer
37 views

How to determine if this particular set is convex?

I am a beginner in convex analysis and optimization, and am teaching myself the basics using Boyd's archived lectures(CVX101/Stanford). I've run into a problem statement described here : [is this set ...
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1answer
41 views

Is this set of functions convex? [closed]

$$ \alpha\leq f(t)\leq\beta,\quad \forall t\in[0,1]\\ f(0)=f_0,\quad f(1)=f_1 $$ where $f_0$ and $f_1\in[\alpha,\beta]$. Is the feasible set of $f$ convex in $f$? Thanks in advance!
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Degree of nef toric divisors which are not big

Let $X$ be a complete toric variety of dimension $n$. It is a classical result that if $D$ is a toric nef divisor, then its degree $D^n$ can be computed as the Volume of the corresponding polytope ...