1
vote
2answers
51 views

How to check for convexity of function that is not everywhere differentiable?

I have a question. I have just been introduced to the subject of convex sets and convex functions. I read this in wikipedia that a practical test for convexity is - to check whether the 2nd ...
0
votes
0answers
28 views

Proof of sufficient condition of existence of Lagrange multipliers

Consider the optimization problem $$ (P) \quad \inf\{ f(x) : g_i(x) \le 0, i = 1,\ldots, m, x \in \mathbb R^m \} $$ where $f, g_i : \mathbb R^n \to (-\infty, \infty]$ are convex and $0 \ne ...
0
votes
1answer
34 views

Sufficient condition on convex function such that $f(x) > -\infty$ for all $x$.

Let $f : \mathbb R^n \to [-\infty, \infty]$ convex and let $f(\overline x) > -\infty$ for $\overline x \in \mbox{int}(\mbox{dom}(f)$. Show that $f(x) > -\infty$ for all $x \in \mathbb R$. ...
0
votes
0answers
24 views

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: $$ ...
1
vote
1answer
29 views

Equality constrained Quadratic Program

Consider the QP $$ x^* = \arg \min_{\displaystyle x \in \mathbb{R}^n{\geq 0}} \ \frac{1}{2} x^\top P x + q^\top x \ \text{ sub. to: } A x = b, $$ where $P \succ 0$. Without the non-negativity ...
1
vote
1answer
87 views

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 ...
2
votes
0answers
31 views

Strictly convex self-concordant function

Some definitions: A function $f:R^n\rightarrow R$ is convex[strictly convex] if for every $\lambda\in[0,1]$ [$\lambda\in(0,1)$] and for every $x,y$ [$x\neq y$] in $R^n$ we have $f(\lambda ...
0
votes
1answer
53 views

Strictly positive solution of linear equations

Suppose $A\in\mathbb{R}^{m\times n}$, $b\in\mathbb{R}^m$, and $b\in \mathcal{R}(A)$. Show that there exists an $x$ satisfying $x \succcurlyeq 0$, $Ax = b$ if and only if there exists no ...
0
votes
0answers
14 views

Proximal operator involving frame analysis operator

Combettes and Pesquet (paper) contains a result that the proximal operator of a composite function $f\circ L$, where $L$ is a semi-orthogonal operator $L$ such that $LL^*=\nu\cdot Id$, is given by ...
1
vote
1answer
88 views

Prove that $ri (epi f)=\{(x, y): x\in ri (dom f), y >f(x)\}$

$f: R^n \to (-\infty, \infty]$ is a convex function. Prove that $ri (epi f)=\{(x, y): x\in ri (dom f), y >f(x)\}$ I have used definition of convex functions to prove that the right-side is ...
3
votes
2answers
805 views

Second derivative positive $\implies$ convex

In proof of the following theorem; If $f$ has a second derivative that is non-negative (positive) over an interval then $f$ is convex (strictly convex). $f$ is in real number space., the book I ...
0
votes
0answers
18 views

Homework: compare two complex functions' local maximizers

Suppose $F_i,i=1,2$ and $Q$ are independent CDF with support on $[0,\infty)$, let $\bar F_i=1-F_i$ and $\bar Q=1-Q$, define $$\Psi_1(u) =-c\int_0^u\bar F_1(z)dz+\bigg[1-\frac{C-c}{\bar ...
1
vote
1answer
101 views

Convex functions and Hahn-Banach application

Let $Z$ be a convex subset of a real vector space, and $f:Z \to \mathbb{R}^m$ be such that every component $f_i:Z \to \mathbb{R}$ is a convex function. Let $S:\mathbb{R}^m \to \mathbb{R}$ be defined ...
2
votes
3answers
227 views

A robust convex optimization problem

Consider a function $ f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R} $ such that $\forall x \in \mathbb{R}^n$ the map $f(x,\cdot)$ is convex, and $\forall y\in \mathbb{R}^m$ the map ...
1
vote
1answer
39 views

Robust feasibility with halfspace?

Consider a convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, such that for all $x \in \mathbb{R}^n$ we have $$ a_1^\top x + b_1 \leq f(x) \leq a_2^\top x + b_2 $$ for some given $a_1, a_2 ...
0
votes
1answer
76 views

convex conjugate $f^*$ is proper if both $f$ and $f^{**}$ are

If $f$ and $f^{**}$ on $\mathbb R^d$ are proper functions where $f^*$ stands for the convex conjugate of $f$ why does that follow that $f^*$ is proper, too? Thanks a lot...
0
votes
3answers
71 views

Membership based on maximum of a function over the set.

Let $S\subset \mathbb{R}^n$ and let $f(x)$ be a continuous function over $\mathbb{R}^n$. Furthermore, define $s_{\text{max}}:= \sup_{x\in S} \{f(x)\}$ and let $f(x)$ attain its minimum for at least ...
1
vote
1answer
371 views

Convex Conjugate of Absolute Norm

Let $f:\mathbb{R}\rightarrow[-\infty,\infty]$ be a continuous function. The convex conjugate of $f$ is: $$f^*(p) := \sup_{x\in\mathbb{R}}\{px-f(x)\}~.$$ Furthermore, let us define the subderivative ...