The convex-optimization tag has no wiki summary.

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### Homotopy with non piece-wise linear boundary

in the middle of a long proof I encounter the following problem.
Let $E$ be a closed and convex set in $\mathbb R^n$ such that for all $\vec x\in E$ it holds that $\sum_ix_i=1$. (We can understand ...

**4**

votes

**1**answer

98 views

### Algorithm to minimally connect line segments in Euclidean plane

Suppose you have finitely many line segments in the Euclidean plane. How do you "connect them to form one chain of line segments of minimal length?"
More formally and generally, what I'm looking for ...

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**1**answer

111 views

### Necessary conditions for optimality in Banach spaces

Let $X$ denote the non-negative "orthant" of the Banach space $L^2$ (or whatever you call the set of functions in $L^2$ that are non-negative), and let $C$ be a closed, convex subset of $X$. Let $f$ ...

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**1**answer

51 views

### An optimization problem in complex space

Consider the following optimization problem
$$
\min \| \textbf{Ax-B}\|
$$
$$
s.t.|x_i|=1,i=1,...,n
$$
where $\textbf{x}\in \mathbb{C}^{n}$ is the optimization varaible, $x_i$ is the $i$-th ...

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17 views

### Restricted strong convexity for biconvex functions

Recently, it has been shown (arxiv paper) that non-convex functions with the restricted strong convexity (RSC) property has the interesting property that their local minima lie within a small ball of ...

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**1**answer

95 views

### A difficult combinatorial optimization problem

Let $\mathcal{J}$ be a closed, bounded, compact, convex set in $\mathbb{R}^L$.
(Notations: vector $\mathbf{x}$ is denoted in bold letters and its $i^{th}$ co-ordinate is denoted as $x_i$. ...

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**1**answer

44 views

### Finding maximum of a function with unfixed number of variables

Can anybody solve this:
For a constant positive integer $n\geq6$
find $k$ and positive integers $a_{1},a_{2},...,a_{k}$
that maximize the expression
...

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**2**answers

297 views

### Gradient descent-like optimization on a convex landscape with noisy sampling

This is a rewrite of the original positing (below), and is crossposted to ...

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**8**answers

763 views

### When do people actually use the maximum entropy distribution?

One of the standard problems in convex optimization is the calculation of the maximum entropy distribution that satisfies some set of criteria. For example, if $\mathbf{x} \in \mathbb R^n$ is an ...

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**1**answer

130 views

### Reverse optimization of a minimum cost flow network

Given an undirected graph $(V,E)$, with $W$ as the weight of each edge, and a convex cost function $F(X)$, such as $|X|^k$ ($k>1$).
The cost to send $x$ unit of flow through edge $e_i$ is defined ...

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**1**answer

98 views

### optimization of inverse matrix with constraint on matrix elements

everyone! I have this optimization problem with constraint.
D and T are symmetric matrice, where T is known and D is the unknown parameter.
x and v are two known p-dimensional vectors.
The objective ...

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**1**answer

228 views

### Find the following transformation $G$

I asked this question 6 days ago on math.stackexchange.com (http://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here.
I'm ...

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61 views

### Matrix equation

Let $A$ be $k\times n$ matrix i.e., $A=(a_{1},\ldots, a_{n})$ where $a_{j} \in \mathbb{R}^{k}$, $rank(A)=k$ and $1\leq k \leq n$. Let $q=(q_{1},\ldots, q_{n})\in\mathbb{R}^{n}$ be such that ...

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**2**answers

164 views

### Removing constraints in convex optimization

Say I have a convex optimization problem of the form $$\min_x f(x) ~~ s.t.\\ g_1(x)\leq0,\\\vdots \\g_n(x)\leq 0$$ with all functions convex. Suppose that $x^*$ is a unique optimizer to my problem and ...

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35 views

### What is the purpose of the definition of “metric regularity”/“regularity modulus”?

A set mapping $F:X \rightrightarrows Y$ is said to be metrically regular for $\overline{x}\in X$ and $\overline{y} \in Y$ if there exists a $\kappa\in(0,\infty)$ for which
$$
d(x,F^{-1}(y))\leq ...

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85 views

### This function looks quasiconvex, can't understand why

Suppose that $\mathbf{C}$ is a given matrix with non-negative entries in $\mathbb{R}^{m\times n}$ and $d$ is a given scalar, and let $g(\mathbf{y})$ be defined by ...

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44 views

### Is there a unique tilted measure with specified marginals?

Suppose $\mathcal{A},\mathcal{B}$ are finite sets and $\mu_{A,B}(a,b)$ is a probability measure on the product set $\mathcal{A}\times \mathcal{B}$ so that $\mu_{A,B}(a,b)>0$ for each $a\in ...

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**1**answer

33 views

### Convert constraint to do convex optimization or use Lagrange multiplier method [closed]

$w_1, w_2, w_3 ... w_n$ are the weights I need to find
I have the following constraint:
$|w_1| + |w_2| + .. |w_n| <= 5$
That is the sum of the absolute values of the weights has to be less than ...

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**1**answer

94 views

### Linearly constrained eigenvalue problem

Suppose I'd like to:
\begin{align}
\mathop{\text{min}}_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\
\text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\
&& ...

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**2**answers

241 views

### Convex Sets and Nearest Neighbors

For a set $S \subseteq \mathbb{R}^n$ and a point $x \in \mathbb{R}^n$, let $c_S(x)$ be the point $s \in S$ that minimizes $\|x-s\|$ if such a point exists and is uniquely determined. It is known that ...

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**1**answer

75 views

### Can one always find sparse solutions to an $\ell^1$-minimization problem?

Consider $A\in\mathbb{R}^{m \times N}$ and $b \in \mathbb{R}^m$, with $m<N$. Is it true that the optimization problem
$$\min \|x\|_1 \quad s.t. \;\; A x = b,$$
admits an $m$-sparse solution in ...

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**2**answers

121 views

### investigating positivity/negativity of a function [closed]

I'm investigating if and how the positivity or negativity of a multivariable function can be proved. Consider $y_{1},y_{2},y_{3}\in\mathbb{R}$ and the following function
...

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**3**answers

329 views

### The distribution of the shortest path through $n$ points

In the big picture, I'd like to know: if I sample $n$ points uniformly at random in the unit square, what is the probability that the shortest path that visits each one of them is very small?
More ...

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114 views

### The set of matrices with same spectral radius

I am working on an optimization problem over the set of positive matrices (that is, matrices where all entries are positive numbers) that have the same spectral radius. My main problem is how to ...

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**1**answer

130 views

### Multiplicative gradient descent?

The normal gradient descent is additive: $w_{t+1}=w_t-\lambda_t\nabla f(w_t)$, but is there a multiplicative gradient descent that looks something like $w_{t+1}=w_t[-\lambda_t\nabla f(w_t)]$?
I know ...

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**1**answer

177 views

### optimization problem, any solution?

The objective is as follows:
$\min_{\mathbf{F}} a Tr(\mathbf{F} \mathbf{F}^H) - Re\{\mathbf{b}\mathbf{F}^H \mathbf{C} \mathbf{F} \mathbf{d}\}$
$s.t.\ \ \ Tr(\Sigma \mathbf{F} \mathbf{F}^H)<p$
...

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34 views

### Convergence of Coordinate Descent / Alternating directions

My question regards this method
http://en.wikipedia.org/wiki/Coordinate_descent,
where at each step a function $f$ is minimized along one coordinate axis (or block of coordinates).
Assume that $f: ...

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59 views

### numerical and functional mixed optimization problem $\max f - \min f +\int_{-1}^1 (f'(x)-x)^2dx$

Given a function $g(x)$ and its domain, we want to get another function $f(x)$ whose derivative is approximately $g(x)$, but so that $f(x)$ itself has small variation. For example, for ...

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**1**answer

165 views

### Linear dependency of real numbers with integer coefficients adding up to zero [closed]

Let $x = (x_1, \dots, x_n)$ be a vector of real number. I was asking myself if there was an efficient way of telling whether there exists a non-zero vector of integers $z \in \mathbb Z$ such that both
...

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307 views

### On increasing the penalty term in convex optimization with regularization

Given the two strictly convex (unique solution) optimization problems as:
$$Problem\:1:\min_{X} f(X)+\|X\|_{F}^2 \hspace{2cm}Problem \:2:\min_{X}f(X)+n\|X\|_2^2$$
where $X\in\mathbf{S}_{++}^{n}$ ...

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75 views

### Convex Optimization related problem

Suppose two non-negative convex functions $f$ and $g$ be given.
We want to solve the following optimization
$$\max_{g\leq\epsilon}f.$$
Now suppose that both $f$ and $g$ can be upper-bounded by a ...

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37 views

### Reference request: Edmond's Algorithm for integer hull

I'm looking for a good reference for the algorithm (supposedly by Edmonds) to compute the integer hull of a polytope, not by cutting plane methods but by starting with a set of integer points and then ...

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66 views

### dual problem of SDP [closed]

suppose we have the following optimization problem:
\begin{array}{l}
\mathop {\min }\limits_{{\bf{X}},{\bf{x}}} \,\,Tr\left( {{\bf{XA}}} \right) + 2{{\bf{a}}^H}{\bf{x}} + b\\
s.t:\,\,\,\,\left[ ...

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**1**answer

75 views

### Is first term of my cost function convex?

I have an optimization problem in the form of
[\begin{array}{l}
\mathop {{\rm{Minimize}}}\limits_{\bf{X}} \,\,\,2\left| \delta \right|\sqrt {{\rm{Tr}}\left( {{\bf{A}}{{\bf{X}}^2}} \right)} {\rm{ - ...

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31 views

### Changing a nonlinear equality constraint into some conic inequality plus rank constraint

If we have a constraint optimization problem in which one of our constraint is $\prod\limits_{k = 1}^N {\left( {x - {a_k}} \right) = 0} $ . How could this nonlinear equality condition be changed into ...

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48 views

### Finding gradient of an optimization

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me?
Assume that we have an optimization ...

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33 views

### Characterization of the optimal solution in relative entropy minimization

The following optimization problem is related to relative entropy and to the limit of the iterative proportional fitting procedure.
For $1 \leq i,j \leq n$ and fixed $w_{ij} \geq 0$, and fixed $a_i, ...

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**1**answer

112 views

### Reference request: Continuity of unique maximizer of linear functional on convex set

Does anyone know reference for a theorem of the following sort:
Proposition: Let $K \subset\mathbb {R}^n$ be a compact convex set, and assume that
$$f(w):=\operatorname{argmax}_{x\in K}w(x) $$ is ...

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**1**answer

69 views

### mixed semi definite and second order programming complexity order

Consider the following mixed semi definite and second order programming:
$\begin{array}{l}
\mathop {{\rm{min}}}\limits_{\bf{X}} \,{\rm{Tr}}\left( {{\bf{XA}}} \right)\\
{\rm{s}}{\rm{.t:}}\, & ...

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**1**answer

123 views

### Projection onto $\ell^{2,1}$ ball

Does anyone have an idea how to project onto the $\ell^{2,1}$ ball efficiently, i.e. how to solve
$$ u = \arg \min_u \|u-f \|^2 \text{such that } \left(\sum_i \big(\sum_j |u_{i,j}|\big)^2 ...

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### Proximal operator of modified L1 matrix norm

In literature proximal operator $prox_{\lambda f} : R^n \rightarrow R^n$ of $f$ is defined as:
$prox_{\lambda f}(V) = argmin(X) (f(X) + (1/2 \lambda)||X-V||^2_2)$
Consider now $g(X) = ...

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**1**answer

50 views

### Convergence rate of stochastic gradient decent with projections

Given a strong (not only strict) convex function $f: \mathbb{R}^n\to\mathbb{R}$. On such problems, stochastic gradient decent (SGD) has a convergence rate of $O(1/T)$, where $T$ is the number of ...

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**2**answers

74 views

### Is the prox-residual monotone?

$\newcommand{\scp}[2]{\langle #1,#2\rangle}\newcommand{\id}{\mathrm{Id}}$
Let $f$ and $g$ be two proper, convex and lower semi-continuous functions (on a Hilbert space $X$ or $X=\mathbb{R}^n$) and let ...

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### proving quasi convexity of multivariable function

Given
an arbitrary $(N \times N)$ square matrix ${\bf X}$
a positive definite $(M\times M)$ matrix ${\bf T}$
a $(Q\times MN), Q< MN$ matrix ${\bf Z}$ consisting of only 1s and 0s where there is
...

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149 views

### Continuity of minimizers to distance function from point to convex set

Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$):
$\min_{x\in U}\|x-y\|$.
I believe the minimizer $x_{U}^{*}$ is ...

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156 views

### When is the sum of a weak-$*$ closed convex cone and a subspace also weak-$*$ closed?

Let $X$ be a Banach space. Suppose $C \subset X^*$ is a convex cone and $V \subset X^*$ is a subspace, and suppose both $C$ and $V$ are closed in the weak-$*$ topology. Are there any general ...

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80 views

### Optimality condition for non-differentiable constrained convex optimization problem

(EDIT: see proof at the end) Consider the problem
$$
\min f(x) \; \text{s.t.} \; x\in D
$$
where $f(x)$ is convex but not differentiable, and $D$ is convex.
For differentiable $f$, we know that $x$ ...

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votes

**3**answers

663 views

### Is group theory useful in any way to optimization?

For what I have seen, optimization uses a lot of linear algebra and convex analysis, but I have not seen any group theory being used, so I was curious about it.
Is group theory useful in any way to ...

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**2**answers

103 views

### sensitivity analysis in conic optimization

I have a conic optimization of the form:
$\min_x \langle c, x \rangle$, s.t. $Ax = b$, $x \in K$.
Where $x \in \mathbb{R}^{n}$, $A$ is an $m \times n$ matrix, $b \in \mathbb{R}^m$, $K$ is a self ...

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**1**answer

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### What it is the volume of the unit ball section of the cone of positive definite matrices?

Let $PD_{n}$ be the cone of positive definite $n \times n$ real matrices and let $B$ be the unit sphere in $n \times n$ dimensions. What is the volume of $PD_{n} \cap B$?
EDIT: Let's assume that $B$ ...