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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Legendre Transformation of a Lagrangian in Classical Mechanics

I have some questions about the Legendre Transformation of a Lagrangian in Classical Mechanics to the Hamiltonian: We start with a Lagrangian $L(q,\dot{q})=\frac{\langle \dot{q} , \dot{q}\rangle }{2} ...
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1answer
82 views

Testing for Convexity for a function

Please any one can help figure out if this funcion is concave or convex, any help is greatly appriciated. Any links on how to test fo convexity for such a function is also greatly appriciated. I tried ...
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0answers
152 views

The convex function proof.

Can anyone help me prove convexity conditions? I know convexity conditions. The second derivative function is greater 0 first order convexity conditions. convex function conditions Because my ...
0
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1answer
106 views

Is $f(x,y) = x^\beta/y$ quasi-convex for positive $x,y$ for any real $\beta \geq 1$?

A multivariate function $f:{\mathbb R}^d \to {\mathbb R}$ is quasi-convex on a convex set $S \subset {\mathbb R}^d$ if $f(\lambda z + (1-\lambda)z') \leq \max\{f(z),f(z')\}$ for all $z,z' \in S$ and ...
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2answers
199 views

n-simplex in an intersection of n balls

Consider a $n$-simplex, $n \geq 2$ with vertices $x_i,i=1,...,n+1$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ ...
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219 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 ...
2
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3answers
180 views

A secant inequality for convex functions

Suppose $f(0) =0 $ and $0<f''(x)<\infty (\forall$ $x>0)$, then $\frac{f(x)}{x}$ strictly increases as $x$ increases. I have shown that $f'(x)-\frac{f(x)}{x} = \frac{1}{2}xf''(c)$, for ...
2
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2answers
158 views

Convexity of log sum function

Is $f\left( x \right)=\log \left( \sum_i \beta_i e^{-\alpha_ix} \right)$ a convex function where $\beta_i,\alpha_i\in \mathbb{R}$?
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1answer
40 views

question about positive concave functions.

I am quoting a line from a text: The Laplace exponent $\Phi$ is concave and non-negative, the inequality $\Phi(\lambda)\leq k\Phi(\lambda/k)$ for all $\lambda>0$ and $k>1$ follows. Why does ...
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118 views

Local Extrema and Global Extrema

When we have a convex function we know that a local minimum is a global minimum, and similarly for a concave function. What are some other situations where finding local extrema can yield global ...
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1answer
57 views

Interpreting a theorem about convex sets

I'm studying about linear programming and I bumped into following theorem (I have added my questions into the image): So in 1st rectangle how did we obtain the resulting equation when $\alpha ...
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1answer
93 views

Help with a proof of a theorem about convex sets

I'm studying the following theorem: Theorem 1. Let $C$ be a convex set and let $\textbf{y}$ be a point exterior to the closure of $C$. Then there is a vector $\textbf{a}$ such that ...
0
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1answer
77 views

Equivalent definitions of piecewise affine convex function over a convex set of $\mathbb{R}^n$.

Let $C\subset\mathbb{R}^n$ be a convex set and $f:C\to\mathbb{R}$ a convex function. I want to show that the following are equivalent: $\mathrm{epi}(f)=\{(x,y)\in C\times\mathbb{R}\mid y\geq f(x)\}$ ...
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1answer
115 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 ...
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1answer
277 views

The unit ball in a Hilbert space

I have a request for any ideas to prove: If $H$ is a Hilbert space, then any unit vector is an extreme point of the unit ball of $H$. Every isometry is an extreme point of the unit ball of the ...
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1answer
58 views

Minimisation of Concave Function

$f\left(\mathbf{x}\right):\mathbb{R}_+^n\rightarrow\mathbb{R}_+$ is a concave monotonically increasing function to be minimised over the feasible region $\sum_{i=1}^n x_i=1$ and $x_i\geq ...
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1answer
162 views

closed epigraphs equivalence

Is there a way to prove that the epigraph of any real function $f$ is closed iff $f$ is lower semi-continuous without using limit superior or inferior?
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109 views

Is the function of two strictly concave functions also concave?

This may be a trivial question to most, but here we go: I have two strictly concave functions, say $f(x)$ and $g(x)$. From this can I say that a function of those two functions, $h[f(x), g(x)]$, is ...
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1answer
108 views

Graphics clipping: How can repeated half-space clipping fail?

Hi I am currently going through the past exam problems and I am stuck on this clipping problem. Could you give me some hint on how to solve it? If we clip a polygon to a window, it is inadequate ...
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2answers
245 views

when is the epigraph a convex cone?

The problem is from Stephen Boyd's textbook, which I couldn't solve. The question is "when is the epigraph of a function a convex cone?" The solution says that it is when the function is convex and ...
0
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1answer
327 views

A convex function with a Lipschitz continuous always has a strong convex conjugate function.

A smooth convex functions with $C^1$ has not always a Lipschitz continuous gradient. Please see the answer. If $F$ is convex and has a Lipschitz continuous gradient with modulus L, then $F^*$ is ...
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0answers
118 views

Separating hyperplane condition for complex vector spaces

I am only learning convex analysis properly now for the first time, and most of the references I am using only deal with topological vector spaces over $\mathbb{R}$. Is there any serious stumbling ...
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2answers
202 views

isoperimetric inequality using Fourier analysis

I'm trying to prove an isoperimetric inequality, but I have absolutely no idea how to go about it. let $\Gamma$ be a closed plane curve parametrized by $\gamma(t) = (x(t), y(t))$ on $[-\pi, \pi]$. ...
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897 views

What are the main uses of Convex Functions?

Up till now I have just learned that the concept of convexity in functions of one variable is used to complete the graphs of functions, meaning to locate points of inflexion and see if the graph is ...
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1answer
321 views

Is Positive Semidefinite matrix Same as Positive Number in Convex Optimisation?

Consider the optimisation problem expressed in a crude form $\max_{\mathbf{Q}}\sum w_ir_i$ where $w_i$ are constants, $r_i$ are concave functions of positive semidefinite matrix $\mathbf{Q}$ ...
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1answer
28 views

Optimization problem interpretation

I posted a question in http://math.stackexchange.com/ and got a solution. But the solution is a bit hard for me to understand. The actual question is here : minimizing $\sum_{i=1}^n \max(|x_i - x|, ...
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81 views

Procedure to find generators of the dual cone

Page 11 of Fulton's "Toric Varieties" gives the following procedure for finding generators of the dual of a convex polyhedral cone in $\mathbb{R}^d$: For each set of $n-1$ independent vectors ...
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88 views

Number of supporting hyperplanes

I know that, for any convex set $S$, there is at least one supporting hyperplane at every point in $B$, the boundary of $S$. Also, there can be more than one supporting hyperplane at the same point in ...
0
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1answer
59 views

a function is approximated by a convex function

Let $g$ be a positive function on $(0,1)$ such that $g(x)\to\infty$ as $x\to 0$. Then, there exists a convex function $h$ on $(0,1)$ such that $h\leq g$ and $h(x)\to\infty$ as $x\to 0$. We can find ...
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211 views

Is alpha divergence a convex divergence measure?

Alpha divergence is defined as following : $$ D_\alpha(p||q) = \frac{1}{\alpha (1-\alpha)} \left( 1- \int _x p(x)^{\alpha} q(x)^{(1-\alpha)} dx \right) $$ if the distributions are restricted to ...
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1answer
124 views

A weak version of Markov-Kakutani fixed point theorem

Let $\emptyset \not = X\subseteq \Bbb{R}^n$ be convex and compact and let $\cal{A}$ be a commuting family of affine maps from $\Bbb{R}^n$ into $\Bbb{R}^n$ such that $X$ is invariant under each element ...
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1answer
173 views

Prove convexity of complicated rational function

Can anyone help me prove the convexity of this rational function? The man who proved the convexity of function used these facts. But I don't know this fact is correct or not. Here are the facts and ...
2
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1answer
199 views

Von Neumann's minimax theroem and Carathéodory's theorem

In J.F. Mertens(1986)(Paywall), there's a neat proof of a version of Von Neumann's minimax theroem. But I can't understand how Carathéodory's theorem is invoked. Suppose, in a two-person zero sum ...
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Checking convexity from outside

Is there any method or algorithm to determine convex (or non-convexity) property of a region from outside (perimeter) ? One way is plotting tangent line in each point of perimeter and discuss how ...
3
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1answer
211 views

Find the maximum convex area

My question is very similar to Plow's Question; but with this difference: How can I find the maximum convex area that can fit inside a non-convex region? For an example, consider this non-convex ...
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1answer
64 views

Subdifferential calculus

Let $\phi : H \to \mathbb{R}$ ($H$ is a vectorial space) be a convex function $\mathcal{C}^1$. I have the following inequality, for $\sigma \in H$ fixed, $$\forall \tau\in H, \ (\sigma - \tau \mid ...
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1answer
97 views

Supporting hyperplanes Theorem in Boyd's Convex Optimization

On page 51, the authors applied the separating hyperplane theorem to the sets ${x_0}$ and the interior of $C$ to prove the supporting hyperplanes theorem (assuming the interior of $C$ is nonempty). ...
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269 views

Volume of the projection of the unit cube on a hyperplane

Let $C_n\subset\mathbb{R}^n$ be the $n$-dimensional cube with side $1$, and let $P_k$ be any $k$-dimensional plane, $k\leq n$. What is the maximal $k$-volume $V_{n,k}$ of the projection of $C_n$ on ...
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1answer
53 views

Is this function convex or not?

Is this function convex ? $$ f(\mathbf y) = { \left| \sum_{i=1}^{K} y_i^2e^{-j\frac{2\pi}Np_il} \right| \over\sum_{i=1}^{K}y_i^2} $$ where : $ P = \{p_1,p_2,\cdots,p_K\} \subset\{1,2,\cdots,N\} $ I ...
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1answer
351 views

Convexity of the squared Frobenius norm of a matrix

I was reading this paper where the define an optimization problem as where K and L are kernel matrices and $\pi$ is the permutation matrix. They have explained that the function is convex because ...
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1answer
742 views

Optimization of a function of two variables.

Suppose we have to minimize a function $f(\mathbf x,\mathbf y)$ where $\mathbf x$, $\mathbf y$ are vectors in Euclidean space. The function is convex in $\mathbf y$ when $\mathbf x$ is kept constant ...
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1answer
89 views

Why can we assume WLOG that $x$ is zero?

I am new to mathematics, so I apologize in advance if this question is trivial. I was trying to prove a property of an arbitrary three-point system in $\mathbb{R}^2$ regarding convexity. I tried it ...
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Averages of a Function

Let $(X,M,\mu)$ a measure space and $f:X \rightarrow \mathbb{C}$ a function in $L^{\infty}(\mu)$. Define $A_f$ as the set of all averages \begin{equation} \frac{1}{\mu(E)} \int_{E} f d \mu ...
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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 + ...
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0answers
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Looking for a “Neat” Transform to Yield a Convex Set

Optimizing on a unit sphere $\mathbb{S}^n$ is almost a convex problem (if the function is convex in the new set) if we make our "new" set $\mathbb{R}^n$, via the stereographic projection. Clearly ...
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2answers
122 views

Prove Convexity of Recursively Defined Function

Let $\mathbf{x}=[x_1, x_2, \dots, x_K]\in\mathbb{R}^K_{++}$ and $E_1>E_2>\dots>E_K>0$ are positive constants. If $$f_i:\mathbb{R}^K_{++}\rightarrow\mathbb{R}_{++}\quad\forall1\leq i\leq ...
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1answer
110 views

Convexity of Exponential Composite Function

$f:\mathbb{R}_{+}^M\rightarrow\mathbb{R}_+$ is a convex analytic function. For $\mathbf{x}\in\mathbb{R}_{+}^M$ and $y\in\mathbb{R}_{+}$, consider the function ...
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1answer
215 views

Support function of a convex domain

Let $Q$ be a compact, convex domain in the plane, with smooth boundary $\partial Q$. We further assume that the origin is contained in $Q$. For a concrete example, let's take an ellipse ...
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154 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) ...
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1answer
57 views

Convex Function of Two Vectors

Let $f:\mathbb{R}^M\times\mathbb{R}^N\rightarrow\mathbb{R}$ be a mapping such that for $\mathbf{Y}\in\mathbb{R}^N$ constant, $f(\mathbf{X}, \mathbf{Y})$ is a convex function of $\mathbf{X}$ and for ...