**3**

votes

**1**answer

108 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 $x_i$. ...

**1**

vote

**2**answers

127 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 ...

**3**

votes

**1**answer

192 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 ...

**0**

votes

**0**answers

54 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: ...

**0**

votes

**0**answers

64 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 ...

**3**

votes

**1**answer

169 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
...

**1**

vote

**0**answers

84 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 ...

**2**

votes

**0**answers

326 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 $X\in\mathbf{S}_{++}^{n}$ ...

**2**

votes

**0**answers

49 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 ...

**1**

vote

**0**answers

37 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 ...

**2**

votes

**1**answer

83 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{ - ...

**0**

votes

**0**answers

54 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 ...

**1**

vote

**0**answers

44 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, ...

**4**

votes

**1**answer

183 views

### optimization problem, any solution?

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

**5**

votes

**1**answer

121 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 ...

**0**

votes

**0**answers

190 views

### 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) = ...

**2**

votes

**1**answer

131 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 ...

**1**

vote

**1**answer

73 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:}}\, & ...

**1**

vote

**1**answer

108 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 ...

**3**

votes

**2**answers

75 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 ...

**1**

vote

**0**answers

74 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[ ...

**2**

votes

**0**answers

134 views

### 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
...

**2**

votes

**0**answers

175 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 ...

**4**

votes

**0**answers

166 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 ...

**2**

votes

**0**answers

97 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 $x$ ...

**8**

votes

**3**answers

711 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 ...

**2**

votes

**2**answers

112 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 ...

**1**

vote

**0**answers

105 views

### Diagonal entries of a Cholesky factorization

Let $I$ denote an identity matrix, $E$ denote the all-one matrix of dimension $k\times k$ and $c$ some positive real number. Define $X=B(I-cE)B^T$ where $B$ is given by
$B:=\begin{pmatrix}
1 ...

**0**

votes

**0**answers

58 views

### Sufficient optimality condition for a non-smooth quasiconvex problem

The result of relaxing to an integer program is the following optimization problem:
$$\min_{\textbf{x}} \sum_{i=1}^n \alpha_i h(x_i)\quad subject \; to \quad A\textbf{x} = \textbf{0}$$
where ...

**5**

votes

**1**answer

262 views

### 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 $B$ ...

**2**

votes

**1**answer

86 views

### Convex interaction energy

Does anybody know examples of absolutely continuous probability measures $\mu_0,\mu_1$ on $\mathbb{R}^n$ with finite 2nd moments such that
$$
\frac{d^2}{dt^2}\left(\int_{\mathbb{R}^n\times ...

**1**

vote

**1**answer

182 views

### Minimize a strictly convex quadratic function subject to linearly equality and nonnegativity constraints in finite time?

I am wondering if we can minimize a strictly convex quadratic function in finite time, subject to linearly equality and nonnegativity constraints.
Thanks!

**2**

votes

**5**answers

252 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 ...

**1**

vote

**1**answer

217 views

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

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

**2**

votes

**2**answers

160 views

### How to minimize the Bregman divergence on a convex hull spanned from a set of vectors?

everyone.
It has been well known that the following minimization problem of a Bregman divergence with linear inequality
can be solved by successively projecting the current point to each constraint ...

**23**

votes

**2**answers

882 views

### An Interesting Optimization Problem

You are given n non-negative integers $a_1, a_2 ,, a_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference ...

**1**

vote

**0**answers

33 views

### Searching for the maximum of a (strictly convex) two-dimensional distribution via maximization over a series of arbitrarily specified 1D intervals

Let $f$ be a strictly convex two-dimensional distribution with a maximum $M$ at some unknown position $(x_m,y_m)$. Starting from the origin, $(x_0,y_0) = (0,0)$, we need to find $(x_m,y_m)$, however ...

**1**

vote

**1**answer

183 views

### 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

**2**answers

395 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 ...

**2**

votes

**1**answer

265 views

### Block Covariance Matrix - Positive Definite? (Quadratic Optimization) [closed]

I have a covariance matrix C. I have then formulated an quadratic optimization problem that involves the following matrix in the quadratic form:
[ C C ]
[ C C ]
However, the quadratic solver ...

**1**

vote

**0**answers

144 views

### Extreme points of a set related to semidefinite cone

Let $X \in \mathbb{R}^{n \times n}$ be symmetric matrix. Consider the following set
$$
\mathcal{C} = \{ X: X \succeq 0, \quad 0 \le X_{ij} \le 1, \forall i,j\}
$$
What are the extreme points of this ...

**3**

votes

**1**answer

267 views

### SDP formulation of noisy low rank matrix completion

Exact low rank matrix completion using nuclear norm minimization can be formulated as a semidefinite program (SDP). Following the notation in the paper, a convex problem for noisy matrix completion ...

**1**

vote

**1**answer

154 views

### No Strong Duality In Spite of Slater's Condition

I was reading some course notes here.
On Page 8, it says:
Note that strong duality holds here (Slater's condition), but the
optimal value of the last problem is not necessarily the optimal
...

**0**

votes

**1**answer

288 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 ...

**0**

votes

**1**answer

185 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 ...

**4**

votes

**2**answers

206 views

### Minimax theorem on a non convex domain

A minimax theorem is a theorem which states that under certain conditions on $\mathcal{X}$, $\mathcal{Y}$ and $f$:
$$ \inf_{x \in \mathcal{X}}{\sup_{y \in \mathcal{Y}}{f(x,y)}} = \sup_{y \in ...

**-1**

votes

**1**answer

119 views

### Uniqueness of the solution to a quadratic problem [closed]

Consider a positive definite matrix $\boldsymbol H$, the known vectors ${\boldsymbol b}$ and ${\boldsymbol a}_i$. Now the minimization problem is casted with respect to the vector ${\boldsymbol x} $ ...

**0**

votes

**1**answer

116 views

### Eigenvalues of a given parametrized matrix.

Let $\mathbf{A}$ and $\mathbf{B}$ be two complex rank-one $N\times N$ positive semi-definite matrices. Let the matrix $\mathbf{C}$ be defined as
\begin{align}
...

**1**

vote

**1**answer

260 views

### Interior of a dual cone

Let $K$ be a closed convex cone in $\mathbb{R}^n$. Its dual cone (which is also closed and convex) is defined by $K' = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\}$.
I know that the interior of ...

**1**

vote

**1**answer

165 views

### Lagrange multiplier and semidefinite programming

suppose we have a primal semidefinite programming. for finding its dual we use Lagrange multiplier $w_i$ for each semidefinite constraint. If the Lagrange multiplier be zero for one constraint what we ...