**8**

votes

**1**answer

204 views

### Algorithm for matching in the power set lattice

Suppose that we have two probability distributions, $f$ and $g$ on the subsets of a finite set $X$, i.e. $f, g: P(X) \to [0,1]$, with
$$
\sum_{A \subseteq X} f(A) = \sum_{A \subseteq X} g(A) = 1.
$$
...

**2**

votes

**1**answer

1k 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 ...

**3**

votes

**2**answers

695 views

### Computational complexity of unconstrained convex optimisation

What is known about the relationship between unconstrained convex optimisation and computational complexity? For example, for which optimisation problems and which gradient descent algorithms is one ...

**5**

votes

**0**answers

126 views

### Minmax problem for polygons

Let $\text{Pol}_n$ be the set of all convex polygons on a plane with $n$ vertices. For $P\in \text{Pol}_n$ denote by $\text{Tr}(P)$ the set of all triangles which vertices are some vertices of $P$. I ...

**2**

votes

**1**answer

138 views

### Relations between a set of inner products of vectors

Suppose we have n normalized vectors on an arbitrarily large Hilbert space $|A_1\rangle,\dots,|A_n\rangle$, $\langle A_i|A_i\rangle=1$ for every i. And there're $\frac{n(n-1)}{2}$ inner products ...

**7**

votes

**2**answers

329 views

### Entropy conjecture for distributions over $\mathbb{Z}_n$

Suppose we have two independent random variables $X$ (with distribution $p_X$) and $Y$ (with distribution $p_Y$) which take values in the cyclic group $\mathbb{Z}_n$. Let $Z = X +Y$, where the ...

**3**

votes

**1**answer

278 views

### Stronger bound for a modified Lyapunov Equation

In regard to the stability analysis and control properties of the linear system $\dot{x}=Ax$.
Consider the solution $P$ of the continuous Lyapunov equation $AP+PA^T+Q=0$, where $A,Q,P
\in
...

**2**

votes

**1**answer

203 views

### Nearest trio of neighbours for non-intersecting ellipses

Hi,
I'm working on a problem which is to find the closest trio of neighbours for a set of arbitrarily placed non-intersecting ellipses. As a new user I'm not allowed to include image tags but I've ...

**2**

votes

**4**answers

892 views

### Fractals as solution to optimization problem?

What's the scientific reason for fractals being present in nature at such a large scale? Is it perhaps the solution of an optimization problem? For example, would the fractal based shape of certain ...

**3**

votes

**1**answer

362 views

### What is the minimum of the Frobenius norm in the intersection of positive semidefinite cones?

For scalar variables $x$, we have a simple solution for the following problem.
\begin{eqnarray}
\min_x&&\alpha(x-a)^2+\beta(x-b)^2 \\\
\mathrm{s.t. }&&x\leq a\\\
...

**3**

votes

**1**answer

188 views

### Integral of a quadratic on a polygon (variations of discrete surfaces)

This question is pretty simple to state: given two linear functions on a polygon, I'm looking for a formula for the integral of their product which depends only on the values at the (unlabelled) edges ...

**2**

votes

**3**answers

608 views

### An Optimization problem

The following unusual optimization problem came up and I don't know where to begin:
Maximize over the real variables $x_1, \dots, x_n$ the sum
$$
S = \sum_{r = 1}^n \frac{1}{x_1 + \dots + x_r}
$$
...

**3**

votes

**2**answers

264 views

### Optimal control problem with control derivative.

I faced to a bit weird control problem, that is minimize cost functional
\begin{equation}
J(u) = \int_0^Tg(t,x(t),u(t),\dot u(t))dt
\end{equation}
subject to
\begin{equation}
\dot x(t) = ...

**1**

vote

**0**answers

164 views

### Finding a curve of some approximate arc length (with uniform or zero curvature) with a specified distance to a set of points in 3-space

Imagine I define a set of $N$ points in 3-space, $P$, and I would like to define a straight-line or curve, $C$, with uniform or zero curvature, that has some desired distance, $M$, to each of these ...

**2**

votes

**1**answer

190 views

### Is the following function convex-\cap? How to maximize it?

This question got no answer or comments on Math Stackexchange:
http://math.stackexchange.com/questions/104114/is-the-following-function-convex-cap
Let $p=(p_1,\ldots,p_n)$ be a given nondegenerate ...

**4**

votes

**0**answers

87 views

### Applications of k-medians with moment constraints

Suppose I have a collection of points $p_1,\dots,p_N$ in the plane. In the Euclidean $k$-medians (or $k$-means) problems, our objective is to distribute a set of $k$ points $x_1,\dots,x_k$ in the ...

**3**

votes

**1**answer

355 views

### Solving for Hamiltonian path with constraints on allowable routes through vertices

Suppose you have a complete graph with N vertexes, with a distinguished vertex $n=1$ ("start"), and you wish to find a route traveling exactly once through each vertex so that the distance along the ...

**2**

votes

**1**answer

242 views

### Optimization of a Specific Polynomial

I have a polynomial:
$$f(x_1 \dots x_n) = \prod_{i=1}^n (c_ix_i + 1) - \frac{1}{2}c_0\sum_{i=1}^nx_i^2$$
Given some values for $c_0 \dots c_n$, I'd like to choose the maximizing values for $x_1 ...

**3**

votes

**2**answers

237 views

### Analogue of PSD matrices for permanents?

Let's begin with a few observations. Suppose we consider the set of $N\times N$ matrices and consider the matrices with positive determinant. There are several connected components in this set; let ...

**0**

votes

**0**answers

139 views

### Conic fitting with pseudoinverse technique

I am trying to fit points of an ellipse to my model using pseudoinverse technique described in this paper (it's in section 4.1). I'm sure that my understanding is wrong, please, could you give me ...

**1**

vote

**1**answer

273 views

### A positive semidefinite programming problem

Dear all,
I've got a SDP problem as follows:
$\min_{{\bf H}\succeq0}\quad trace({\bf H}) - {\bf a}^{\top}{\bf H}{\bf b}$,
where ${\bf a}$ and ${\bf b}$ are two constant vectors. May somebody tell ...

**4**

votes

**2**answers

226 views

### Lagrangian duality

Suppose we have a primal problem
$
\min_x f(x), s.t. h_i(x)=0,
$
where $h_i$ are all affine, and $f$ is convex.
Then its Lagrangian is
$\min_x \max_{z_i} f(x) + \sum_i z_i h_i(x)$
and the dual ...

**1**

vote

**1**answer

191 views

### Proving a variational problem has no solutions

Consider the following integral
$ \int_{0}^{\frac{\pi}{2}} \left( \sqrt{y(x)^2 + y'(x)^2} \left( \ln \left( \frac{\sin(x)}{1 -\cos(x)} \right) + \frac{\pi}{2} \right) + \frac{\pi}{2} y'(x) + 1 ...

**3**

votes

**0**answers

222 views

### Maximise $L^q$ norm of a vector, for fixed $L^1$ and fixed $L^p$ norms [closed]

Consider a vector $x \in \mathbb R_+^n$ and $p,q \in \mathbb R$ such that
$1 < p < q$
We fix $\sum \limits_{i=1}^{n}|x_i| = 1$ and $ \left(\sum \limits_{i=1}^{n}|x_i|^p \right)^\frac{1}{p} = ...

**1**

vote

**1**answer

257 views

### Minimizing ellipsoid over intersection of ellipsoids

Let's say I want to minimize a quadratic form $\sum_{j=1}^n c_jx_j^2$ (all $c_j$ are positive constants), which corresponds to an $n$ dimensional ellipsoid, over the outer part of the intersection of ...

**2**

votes

**0**answers

333 views

### Partial feedback linearization (Control theory)

Greetings,
I'm trying to understand a theorem about partial feedback linearization from a paper "On the largest feedback linearizable subsystem" by R.Marino (you can find it here: ...

**1**

vote

**1**answer

97 views

### Are there intuitive/classically algorithmic analogues to Semidefinite programs on networks?

Many network optimization algorithms, including shortest path, push-relabel, augmenting path, etc, actually have an interpretation in terms of linear programming.
A famous application of semidefinite ...

**2**

votes

**0**answers

81 views

### Minimum time planar paths under a bound on magnitude of acceleration

On a plane, given initial position (x1,y1), initial velocity (u1,v1), final position (x2,y2), and final velocity (u2,v2), compute the solution to x''= cos(z), y''=sin(z) that has these endpoint ...

**1**

vote

**2**answers

165 views

### Bound on expression from probability distributions

I came across this issue while trying to combine multiple probability distributions into a single distribution which approximates them all simultaneously. This boils down to maximizing this expression
...

**0**

votes

**1**answer

117 views

### Conditions under which a given scheme converges

I'm sorry in advance for how long this question is. Suppose I have a continuous function $f:\mathbb{R}^n \rightarrow \Delta_{n-1}$, where we think of the simplex $\Delta_{n-1}$ as the set
...

**2**

votes

**1**answer

656 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 ...

**1**

vote

**1**answer

163 views

### Conditions ensuring extrema are twice continuously differentiable?

For a functional $J[y]=\int_{a}^{b}F(t,y,y')dt$, are there any conditions that ensure extrema over the class of piecewise continuously differentiable functions are all in $C^2[a,b]$?

**4**

votes

**1**answer

468 views

### An optimization problem involving sum of binomial coefficients upto some value

I would like to minimize $f(s, n, \epsilon)$ with respect to $s$ where
$$f(s,n,\epsilon) = \left( 1 + \frac{n}{2^s} \right)\frac{1}{s} \sum_{k=0}^{\lfloor s\epsilon \rfloor} {s \choose k}~.$$
Note ...

**2**

votes

**1**answer

412 views

### minimizing functions over simple matrix inequalities

I'm wondering if anything is known about minimizing convex, not necessarily linear functions subject to "simple" matrix equalities. To be precise, consider the following example:
$min \Sigma x_i ln ...

**4**

votes

**0**answers

161 views

### Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l \rangle −f_1(x)−f_2(x)$ via convex duality?

I am attempting to solve the argument maximization problem
$$\arg\sup_x \{ \langle x,l \rangle − f_1(x)−f_2(x) \} \ \ \ \ \ \ \ \ \ \ (1)$$
where the functions $f_1$ and $f_2$ are concave but ...

**4**

votes

**2**answers

503 views

### Algorithm for the intersection of a vector subspace with a cone of non-negative vectors

Hi,
I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...

**1**

vote

**2**answers

735 views

### How to use DFT to solve this minimization problem?

This is a problem when I'm reading a paper.
Equation:
$min\{\sum_p(S_p-I_p)^2+\beta((\partial_xS_p-h_p)^2+(\partial_yS_p-v_p)^2) \} $
where $S,I,h,v$ are all $M*N$ matrices and p stands for every ...

**10**

votes

**1**answer

529 views

### Complexity of a weirdo two-dimensional sorting problem

Please forgive me if this is easy for some reason.
Suppose given $S$, a set of $n^2$ points in $\mathbb{R}^2$.
I want to choose a bijective map $f$ from $S$ to the set of lattice points in $\lbrace ...

**3**

votes

**1**answer

229 views

### Unique matrix satisfying a system of equations

Assume I have a $n\times n$ positive semidefinite matrix $G$ of rank $p$ satisfying a set of $np - p(p-1)/2$ equations $v^T_jGv_j = 1$, $j = 1 \ldots np - p(p-1)/2$ for some given vectors $v_j$. It is ...

**0**

votes

**0**answers

211 views

### L1-regularized Least Squares on a matrix with Toeplitz Blocks (not block-Toeplitz)

I am trying to speed up a sparse signal recovery algorithms.
My sensing matrix is a set of Toeplitz Blocks, M = [T1,T2,T3,...,Tk]
The objective is min ||Mx - b||_2^2 + ||x||1
What I'm actually ...

**4**

votes

**1**answer

733 views

### Maximize sum of largest eigenvalues

Consider the following optimization problem:
$\max_{\lambda_j(X)}\sum_{j=1}^n d_j\lambda_j(X)$ subject to $v_j^TXv_j \leq 1, X \geq 0$.
$d_j$ are such that $d_1 \geq d_2 \geq \ldots \geq d_k > ...

**6**

votes

**1**answer

271 views

### Constructing a hypersurface with given outer normals

Pick a point on each of the positive half-axes in $\mathbb{R}^n$. Put a (unit-norm?) vector at each of the n points.
(a) Is there a hypersurface in the orthant $\mathbb{R}^n_+$ going through these n ...

**1**

vote

**0**answers

120 views

### Minimizing an entropylike expression with a quadratic constraint

Let '$\{X_i\}$' be a set of n positive integers, and fix k to be a positive integer. I am interested in finding the set of solutions to the pair of inequalities:
'$\displaystyle \sum_{i=1}^n X_i ...

**3**

votes

**1**answer

318 views

### Mathematical Programming with other Algebras than Linear

Linear Programming is strongly entwined with linear algebra, as are many of its generalizations under the heading of mathematical programming / convex optimization.
What analogies are there for ...

**0**

votes

**1**answer

433 views

### Nonlinear constraint and product of variables

I have been asked to add to an existing linear programming model several constraints dealing with ratios among continuous decision variables. An example ratio constraint would be like:
$x_1*x_2 - ...

**2**

votes

**2**answers

861 views

### Quadratic problem solving with absolute value constraint

Hello,
I have been trying to solve a problem of the form :
$\max_x\quad -\tfrac{1}{2}x^TAx + b^Tx - C\sum_i |x_i|$
without the C term it is a simple quadratic problem,
but I haven't been able to ...

**2**

votes

**1**answer

259 views

### Linear and Isometric Automorphism Groups of the PSD Cone

Let $S_+$ be the cone of psd matrices ($n\times n$ real symmetric positive semidefinite matrices). This cone is a metric space induced from the inner product $\langle A,B\rangle = tr (AB)=tr(BA)$.
...

**0**

votes

**0**answers

427 views

### Decomposing max-convolution of sum of functions ?

Hello.
$R, F_1, F_2, F_3$ are random (not-convex, not-concave) 2D matrices of size 100x100.
$R$ is a linear combination of $F_1, F_2, F_3$.
Explicitly, $R = w_1 F_1 + w_2 F_2 + w_3 F_3$
where ...

**2**

votes

**1**answer

841 views

### How do I optimize over (or take derivative wrt) a square diagonal matrix?

Hello. I'd like to solve the following optimization problem.
$P_i$ is a 6x6 matrix
$X$, $Y$ is a 6xk matrix
$w_i$ is a kx1 vector
$diag(w_i)$ is a square diagonal matrix with diagonal entries equal ...

**3**

votes

**2**answers

441 views

### Is a solution of a linear system of semidefinite matrices a convex combination of rank 1 solutions?

The cone of symmetric positive semidefinite $n\times n$ matrices is the convex hull of rank $1$ matrices. That is, every symmetric positive semidefinite matrix is a convex combination of rank 1 ...