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11
votes
0answers
153 views

How large are the smallest-area projections of a high-dimensional convex body?

Let $B$ be a convex body in $\mathbb{R}^d$, equipped with its standard Euclidean form, and assume that $$\intop_B x \, dx = 0$$ $$\frac{1}{|B|_d} \intop_B x_i x_j \, dx = \delta_{ij},$$ a ...
6
votes
2answers
128 views

On the convexity of element-wise norm 1 of the inverse

Question first asked on math.stackexchange here: http://math.stackexchange.com/questions/317209/on-the-convexity-of-element-wise-norm-1-of-the-inverse On the convexity of element-wise norm 1 of the ...
5
votes
2answers
243 views

Efficient computation of “discrete infimal convolution”

This question arises from an application to graphical models in probability theory, but I have abstracted that part out so only algebra remains. Let $\mathbb{R}$ denote standard field of real numbers ...
3
votes
3answers
180 views

infinite dimensional polyhedra

I have a reference request which I hope some reader here can help me with. I have encountered a set that has all the properties that one would expect from a polyhedral set (in the sense of finite ...
3
votes
3answers
318 views

When is a sequentially closed cone, closed?

The following question I also posed here, but still got no answer. Let $X$ be a locally convex, Hausdorff topological vector space and $C\subseteq X$ a convex cone, which is sequentially closed. What ...
3
votes
1answer
237 views

Possible to find a set of log-concave functions with log-concave sums?

While the set of log-convex functions is closed under addition, the set of log-concave functions is not. Yet if $f$ is log-concave, $\ln(k f) = \ln(k)+\ln(f)$, with $k \in \mathbb{R}^+$ constant, is ...
3
votes
1answer
155 views

Characterization of a set in $\mathbb{R}^d$

Let $X= (X_1,\dots, X_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set. \begin{equation}\label{main12} C= \{x\in \mathbb{R}^d ~|~ ...
3
votes
1answer
348 views

Quantitative Version of Jensen's Inequality?

Hi, I've been looking at situations where Jensen's inequality is almost tight, and found myself proving a lemma that I'm nearly certain exists somewhere in the literature. The specifics are as ...
3
votes
0answers
143 views

Ways to establish equality of measures on locally compact spaces

Let $M$ be a locally compact space and $\mu$ be some probability measure on $M$. Let $y^\ast \in M$, $f(x,y)$ be a real continuous bounded function $M \times M \to \mathbb{R}$. Consider an equality $$ ...
2
votes
2answers
348 views

Does the minima of a sequence of convex convergent functions converge?

Suppose $f_1,f_2,\ldots $ is a sequence of convex functions that converges to a continuous convex $f$. Let $x_1^*,x_2^*$ be their respective (not necessarily unique) minima, and let y be a minima of ...
2
votes
5answers
187 views

Distance between two sets

Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem. $$ \min ...
2
votes
2answers
84 views

Projection onto rotated box

Does anyone know if there is an efficient way to find the projection of an arbitrary point $z$ onto a rotated box, i.e. onto the set $\Omega=\{x \mid a \leq Ux \leq b\}$ where $U$ is a unitary matrix? ...
2
votes
1answer
583 views

Projection exists => Uniformly convex?

Hello, I know that: Let X be a uniformly convex Banach-Space, $a\in X$ and $C\subset X$ closed and convex, then there is a unique $b\in C$ with $\left\Vert a-b\right\Vert=\inf_{x\in C}\left\Vert a-x ...
2
votes
2answers
126 views

Perimeter of a 'trapped' convex set

Consider the following setup: three bounded, 'nice' convex sets $A \subseteq B \subsetneq C \subset \mathbb{R}^2$, and three points $x,y,z\in \partial A\cap \partial B\cap \partial C$ (see edit ...
2
votes
1answer
166 views

How to examine the convexity of a complex function numerically?

I have a function which does not have a closed form . Large numerical effort should be done to evaluate the function for even a single point. How can I examine the convexity of my function over the ...
2
votes
1answer
378 views

Integral inequality for convex function

Let $u(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Suppose that for some real numbers $a,b$ with $a < b$ the following equality is true: \begin{equation} \frac{1}{b-a} \int_a^b ...
2
votes
2answers
223 views

A certain type of Quadratic Constrained Quadratic Programming Problem (QCQP)

Let $P_1$, $P_2$ be two hermitian matrices. Can anyone comment about the following (QCQP) \begin{equation} \min_{z}~z^{H}z \\\ ~~subject~to ~z^{H}P_1z+1\leq 0,~z^{H}P_2z+1\leq 0 \end{equation} I am ...
2
votes
1answer
967 views

Sufficient conditions for gradient descent convergence

I have an unconstrained optimisation problem with convex objective function $f(x)$. Suppose I have access only to some function of the gradient $\hat{\nabla}= g(\nabla f)$, and I take gradient steps ...
2
votes
1answer
311 views

existence of a minimum for a convex functional on a non-reflexive space

Let $X$ be a Banach space; $K\subset X$ nonempty, closed and convex; and $f:K\to \mathbb R$ lower semicontinuous, convex functional. Let also $f$ be coercive, i.e., $f(x)\to +\infty$ as $\|x\|\to ...
2
votes
1answer
65 views

Known Results on Convexity of Numerical Range

Let $A_1,A_2,\dots,A_M$ be given $N\times N$ hermitian matrices. The numerical range is defined as the set \begin{align} \mathbb{S}=\{(u^HA_1u,\dots,u^HA_Mu)\in \mathbb{R}^M\mid u^Hu=1\} \end{align} ...
2
votes
1answer
275 views

Proving that a specific function is quasiconvex

Hello all, Assume we have a sequence of quasiconcave functions (in $X$) denoted by $f_{i,j}(X)$ for $i,j = 1,\ldots,n$. Denote by $F(X)$ the $n\times n$ matrix whose $(i,j)$ entry is the function ...
2
votes
1answer
110 views

The epigraph of a semi-convex function has positive reach

I've been trying to prove the following theorem for several hours with no result so far. Claim. Let $f:\mathbb{R} \to \mathbb{R}$ be a semi-convex function, i.e. there exists a constant $C > 0$ ...
2
votes
0answers
113 views

An intuition for three different types of subgradients (proximal, regular, limiting)

I'm having a bit of difficulty getting my head around the different types of subgradients we're currently covering in a nonsmooth optimisation class I'm taking. These subgradients are (assume $x \in$ ...
2
votes
0answers
83 views

How to prove convexity for a complex integral including the variable in both limits and integrand?

During my research [on inventory management policies, i.e., something really applied ;-) ] I stumbled on integrals of the following type and I'm curious under which circumstances there are convex. ...
1
vote
1answer
158 views

Are all points x of the boundary of a convex set C of a Hilbert space H projections onto C of a point different than x?

It appears that this is not true if H is of infinite dimension. My question is therefore the following: does anyone have a counter-example? Is there a caracterisation for the points of the boundary ...
1
vote
1answer
187 views

positive semigroups and convex operator

Let $\{Z(t)\}_{t\geq 0}$ be a strongly continuous positive semigroup on a Banach lattice V. (endowed with ordering $\leq$). $\phi:V\rightarrow V$ is a convex operator. I want to prove that ...
1
vote
1answer
218 views

When does the finite union of convex sets have a hole in it?

Let $f_1, \dots, f_j$ be convex functions from $\mathbb{R}^n \to \mathbb{R}$. I am trying to develop a test that decides whether or not the set $\{x | f_1(x) \le k_1\} \cup \dots \cup \{x | f_n(x) ...
1
vote
2answers
130 views

determining a convex set by mixed volumes

For a convex set $K \subset \mathbb{R}^2$ let $\phi_K:$ convexsets in $\mathbb{R}^2 \rightarrow [0,\infty), A \mapsto MV(A,K)$. Where by $MV(A,K)$ I mean the mixed volume of $A$ and $K$ in the ...
1
vote
1answer
97 views

Relative interior and dense subsets

(This is a cross-post from here.) Let $A,B\subseteq \mathbb R^d$ be non-empty, such that $B\subseteq \overline A.$ For $S\subseteq\mathbb R^d$ define the relative interior of $S$ by ...
1
vote
2answers
364 views

Hessian of function of covariance matrices

Suppose we have a typical logdet function $\mathcal{L}$ with respect to a covariance matrix $\mathbf{A}$, $$ \mathcal{L}(\mathbf{A}) = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - ...
1
vote
1answer
230 views

How does the complex convex set look like?

The usual convex set is the real linear convex set, if we change the real linear map into complex linear map, we can get the complex convex set. A system way to do this is in the several complex ...
1
vote
2answers
143 views

What is the dual of an semidefinitely representable (SDR) cone?

The Question Let $V\simeq \mathbb{R}^r$ be an $r$-dimensional vector space with the usual Euclidean inner product. Let $\mathcal K\subset V$ be a cone defined as $$ \mathcal K=\Big\{x\in ...
1
vote
1answer
228 views

Question regard checking convexity by “restriction to any line that intersects the function domain”

Hello all, I have a question (probably stupid one) about the fact that " A function is convex if and only if it is convex when restricted to any line that intersects its domain". In Stephen Boyd and ...
1
vote
0answers
45 views

Directional derivates and unique subgradients

I have a question about the fine structure of convex functions. Convex functions behave very regular in the interior of their domain of definition (e.g. they are locally Lipschitz continuous there) ...
1
vote
0answers
57 views

monotone parabolic systems, convex variational structure and Legendre transform

The context: for my research I am currently looking at parabolic systems of the type $$ \left\{ \begin{array}{ll} \partial_t b(u)-\Delta u=0 \qquad & (t,x)\in \mathbb{R}^+\times\Omega\\ u=0 & ...
1
vote
0answers
91 views

Lipschitz continuity of solution set mapping of a parametric convex optimization problem

I have a parametric convex optimization problem: \begin{array}{cl} \underset{x}{\text{minimize}} & f\left(x,z\right)\\ \text{subject to} & g\left(x\right)\leq0 \end{array} where $x$ is the ...
1
vote
1answer
96 views

Does John's Ellipsoid preserve subset ordering? [duplicate]

Let $K \subset \mathbb{R}^d$ be a convex body, symmetric about the origin and with nonempty interior. Then John's theorem asserts that there exists a unique ellipsoid $E$ of minimal volume such that ...
1
vote
0answers
86 views

Linearization of cones

Suppose that $K$ is a closed convex cone in $R^{n}$. Is there a "nice" function $f:R^{n} \rightarrow R^{m}$ so that $f(K)$ is a subspace? What about an approximate subspace?
1
vote
0answers
83 views

Concavity of a ratio of Kullback-Leibler divergences

For $0\leq r\leq 1$ and $0 < p < 1$, define the Kullback-Leibler divergence between the Bernoulli(r) and Bernoulli(p) distributions as: $D(r||p) := r\log\frac{r}{p} + ...
1
vote
0answers
109 views

Optimize a convex hull on a 2D histogram so the selected points match a target shape

I have an image (can be 2D or 3D), and compute a 2D histogram of the image (for example, the pixel intensity and gradient along certain direction). There is a known target region $R^*$ in the image. I ...
1
vote
1answer
106 views

Subgradient of Minimum Eigenvalue

Consider three $N \times N$ Hermitian matrices $A_0$, $A_1$, $A_2$. Consider the function \begin{align} f(t_1,t_2)=\lambda_{\text{min}}(A_0+t_1A_1+t_2A_2) \end{align} where $\lambda_{\text{min}}$ ...
1
vote
0answers
340 views

Definition and Convergence of Iteratively Reweighted Least Squares

I've been using iteratively reweighted least squares (IRLS) to minimize functions of the following form, $J(m) = \sum_{i=1}^{N} \rho \left(\left| x_i - m \right|\right)$ where $N$ is the number of ...
0
votes
1answer
160 views

A certain type of quadratic problem.

I am interested in solving the following equality constrained quadratic (?) problem. \begin{align} \min_{u^{H}u=1}~(u^{H}A_1u) \\\ s.t.~ u^{H}A_2u=0 \end{align} $A_1$ and $A_2$ are $N\times N$ ...
0
votes
1answer
128 views

Exponential Convexity

$\textbf{Definition:}$ 1. A function $h : (a,b)\rightarrow\mathbb{R}$ is exponentially convex if it is continuous and $$\sum _{i, j=1}^n\xi_i\xi_jh(x_i+x_j)\geq 0,$$ for all $n\in\mathbb{N}$ and all ...
0
votes
2answers
87 views

Feasibility of a given set of homogenuous nonconvex quadratic inequality constraints

Let $C_1$,$C_2$,...$C_N$ be $M \times M$ indefinite hermitian matrices. What can we say about the following quadratic constriants \begin{align} w^{H}C_1w>0 \\\ w^{H}C_2w>0 \\\ ...~~~~~~~~~~ \\\ ...
0
votes
1answer
121 views

Minimal point of the intersection of convex sets.

I am trying to find out if there is any known result in convex optimization that implies the following statement: "A minimal point of the intersection of $N > 2$ convex sets in $\mathbb{R}^2$ is ...
0
votes
1answer
98 views

Is these two optimization problems share the same solution?

Hello all, I am dealing with some SDP optimization, and I come across the following problem. The optimization problem is given by \begin{aligned} &\operatorname*{min}_{t_1,\ldots,t_m,X}\ \sum ...
0
votes
1answer
619 views

Is a jointly convex function of x and y convex as a function of x when y=z(x)?

Hi, Suppose that $x \in R^m, y \in R^n, z(x) \in R^n$, and $f(x,y)$ is convex in $(x,y)$. Is $f(x,z(x))$ a convex function in $x$ for arbitrary continuous functions $z(x)$? Thanks!
0
votes
1answer
123 views

Convex functions: bounding the difference

Suppose you are given a convex function $f: R^d \rightarrow R$. Let us say you are given $x,x' \in R^d$ and $x_1, x_2, \ldots, x_n \in R^d$ such that $$\sum_{i=1}^n (x_i - x') = x - x'.$$ Is it ...
0
votes
0answers
37 views

additive support functions of a convex set

Let $K \subset \mathbb{R}^d$ be a compact, convex set. It could be uniquely determined by its support function (for $u$ on the unit-sphere $S^{d-1}$), given by $$h_K(u) = \sup \{ \sum_{i=1}^d x_i ...