0
votes
0answers
10 views

Firm non-expansiveness in the context of proximal operators

$\newcommand{\prox}{\operatorname{prox}}$ Probably the most remarkable property of the proximal operator is the fixed point property: The point $x^*$ minimizes $f$ if and only if $x^* = \prox_f(x^*) ...
1
vote
1answer
23 views

Convexity on a direction

In "Ming-Jun Lai, Larry L. Schumaker. Spline Functions on Triangulations. Cambridge University Press, 2007, p.72." we have: A function $f$ defined on a triangle $T$ is said to be convex in the ...
1
vote
1answer
32 views

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. $$ \min_x f(x)$$ ...
2
votes
2answers
46 views

Convexity / Concavity --> Formal Definition

How do I show that $f(x, y)=(x + y)^2$ is convex/concave using the formal definition of convexity/concavity?
0
votes
0answers
45 views

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 ...
1
vote
1answer
37 views

A characterization of convexity for functions with vectors as domain.

Let $f:\mathbb R^n \rightarrow \mathbb R$ be a continuously differentiable function. By $df(w)$ I denote the Frechet derivative of $f$ at $w$ Prove that $$f \:\text{is convex} \Leftrightarrow ...
0
votes
1answer
40 views

Concavity / concavity in one direction and the cross partials

I just have a question with regards to convexity and concavity (in one direction) in relation to its cross partial derivatives. Suppose we have a smooth function $f(x,y)$ on well defined domains. And ...
1
vote
2answers
51 views

Prove that $f(x+h)-f(x) - \langle\nabla f(x), h\rangle\geq 0 \Rightarrow f $ convex

At this link there is a demonstration that for $f$ continuously differentiable on $C \subseteq \mathbb{R}^n$ convex, $f(x+h)-f(x) - \langle\nabla f(x), h\rangle\geq 0 \Rightarrow f $ convex. This ...
0
votes
0answers
153 views

Convex curve in R^2 pass through two point with fixed normal direction

Given two distinct point in $\mathbb{R}^2$ and two distinct normal directions what are some explicit convex $C^2$ curve (e.g. a map $[0,1] \to \mathbb{R}^2$ which satisfy many equivalent property) ...
1
vote
2answers
73 views

Is a continuously differentiable function convex if all its partial second derivatives are non-negative?

I'm having trouble understanding the relevant Wikipedia article which begins with a convex set $X$ and then uses functions of single variables for succeeding examples; the MathWorld article seems to ...
4
votes
2answers
177 views

Maximizing a ratio of convex matrix functions by minimizing a difference?

Given that $g(.),h(.)$ are twice-differentiable convex quadratic real functions whose domain is the set of all real matrices while the range is the set of positive real numbers, then: Is maximizing ...
0
votes
0answers
116 views

Concavity: composition of multivariate functions

i have to study the convexity of a function F(x,y) = z The function is VEEEEEERY UGLY however i can write the function as a composition F(x,y) = G(H(x,y)) Where $H: R^2 \rightarrow R^2$ and $G: ...
7
votes
2answers
200 views

How prove this inequality generalized from 1969 IMO problem 6

Let $x_{1},x_{2},y_{1},y_{2},z_{1},z_{2},w_{1},w_{2} $ are all positive numbers, and such $$x_{1}y_{1}z_{1}-w^3_{1}>0,\; \text{ and }\;x_{2}y_{2}z_{2}-w^3_{2}>0.$$ show that ...
1
vote
2answers
305 views

How can it be proved that the geometric mean function is concave?

A function $f: \mathbb R ^n \rightarrow \mathbb R $ is said to be concave if $\forall x,y \in \mathbb{R}^n, \forall \lambda \in [0,1]$ we have $ \lambda f(x) + (1-\lambda)f(y) \le f( \lambda x + ...
1
vote
1answer
136 views

Intervals where a function is convex and/or concave

I find myself in need of the solution of the following problem for my work. An help is appreciated. Let $a$ be a real such that $0 \le a \le 1$. For what real values of $y$ is the function $$ f(x) ...
3
votes
3answers
154 views

Euclidean Metric and Convexity

Question: Consider the Euclidean metric space $(\mathbb{R}^n , \Vert\cdot \Vert)$. Let $X\subset \mathbb{R}^n$ and $f\colon X \to \mathbb{R}$. $X$ is said to be a convex set if for every $x,y \in X$ ...
2
votes
1answer
105 views

Does setting derivative to zero suffice always for minimization of convex functions?

I have this convex function in $X$, given by $Trace(AX^TBX)$ where $A$, $B$ are p.s.d and all entries are real. Now if I had a linear function $l(X)$ that prevents a trivial zero-matrix solution for ...
2
votes
0answers
33 views

Is the following function of several variables concave?

Suppose that $w_1,\dots,w_p\in(0,1)$ satisfy the condition $\sum_i w_i=1$, and let $$F(w_1,\dots,w_p) = \frac{1-\sum_i w_i^2}{\sum_i(1-w_i)^2 \left[\sum_i \frac{w_i}{1-w_i}\right]^2}.$$ Is function ...
2
votes
1answer
723 views

Is $f(x,y) = x^2y + x y^2$ (quasi-) concave or convex?

I should analyze whether the function $$f(x,y) = x^2y + x y^2 \text{ where } x,y > 0$$ is (quasi-) concave or convex. Thus, as usual, I set up the Hessian as $$ D^2f(x,y) = \left( ...
0
votes
1answer
385 views

Pointwise supremum of a convex function collection

In Hoang Tuy, Convex Analysis and Global Optimization, Kluwer, pag. 46, I read: "A positive combination of finitely many proper convex functions on $R^n$ is convex. The upper envelope (pointwise ...
6
votes
2answers
273 views

Can one define the derivative of a function using tangent cones? Does such a notion already exist?

I'm interested in finding an analogue of a derivative that applies to functions which are defined more general subsets of $\mathbb{R}^n$ than open subsets. In particularly, I'm looking at functions ...
5
votes
1answer
1k views

The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is ...
1
vote
1answer
70 views

A consequence of convexity

Let $f:\mathbb{R}\to\mathbb{R}$ a convex decreasing function. Let $x_0 < x_1 < x_2$. Studying the behaviour of the difference quotient, it is clear that $$f(x_0)-f(x_2) \leq M (f(x_0)-f(x_1))$$ ...
0
votes
1answer
277 views

Global Min-Max Optimization

When is \begin{equation} \min_X \max_Y f(X,Y) \end{equation} globally solvable? (i.e. we can find global solution for the optimization problem?) I am not looking for reformulations. Is it only when ...
6
votes
0answers
241 views

Behaviour at infinity of a function in terms of first and second derivatives

In a paper (dealing with spectra of certain Schrodinger operators) I found the following assumption for a function $f\in C^\infty(\mathbb R^n;\mathbb R)$: there exists a constant $C>0$ and a ...
2
votes
0answers
362 views

Quasiconcavity for the sum of specific quasiconcave functions

I want to show that a function $ψ(a_1,a_0)$, which is separable (additively decomposed) in two quasiconcave functions, is also quasiconcave (QC). I know that the sum of QC functions is not generally ...
0
votes
1answer
722 views

How to prove $f(x,y)=\sqrt {xy}$ is concave?

How can I prove (preferably elegantly) that $f(x,y)=\sqrt {xy}$ where $x≥0$ and $y≥0$ is concave in both $x$ and $y$?
0
votes
0answers
64 views

Relaxing strict convexity

Suppose $F: \mathbb{R^2} \rightarrow \mathbb{R}$ is convex. If $F$ is moreover strictly convex, then $$ \frac{1}{2} F(x_1,y_1) + \frac{1}{2} F(x_2, y_2) - F\left( \frac{x_1 + x_2}{2}, \frac{y_1 + ...
1
vote
0answers
165 views

Convexity of a Set

Consider the following function, $$ f(x, y) = e^{m e^{-y}+n e^{-x}-x-y} \left(a x e^y+b e^x y+c x y\right) $$ where $a, b, c, m$ and $n$ are positive constants. I want to show $f(x, y)$ is ...
2
votes
1answer
185 views

How to compute the subdifferential of a variational represention of the trace norm?

Let $f : {\cal S}_+^n \mapsto \mathbb{R}$ be a function defined as $f(Q) := {\rm tr} WQ^{-1}W + {\rm tr} Q$ where $W \in {\cal S}_{+}^{n}$ is a symmetric positive definite matrix. How to compute a ...