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2
votes
4answers
867 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
1answer
352 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
1answer
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
3answers
606 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
2answers
256 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
0answers
162 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
1answer
188 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
0answers
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
1answer
343 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
1answer
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
2answers
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
0answers
137 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
1answer
270 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
2answers
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
1answer
189 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
0answers
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
1answer
256 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
0answers
317 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
1answer
95 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
0answers
80 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
2answers
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
1answer
115 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
1answer
635 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
1answer
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
1answer
456 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
1answer
404 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
0answers
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
2answers
492 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
2answers
692 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
1answer
523 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
1answer
227 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
0answers
210 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
1answer
706 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
1answer
269 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
0answers
119 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
1answer
317 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
1answer
381 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
2answers
817 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
1answer
250 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
0answers
421 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
1answer
779 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
2answers
425 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 ...
13
votes
2answers
920 views

Seeking proof for linear algebra constraint problem.

Given a symmetric real matrix with a zero diagonal $M$, I am trying to find a diagonal matrix $D$, such that the matrix $M + D$ is positive definite, and $(M+D)^{-1}$ has a diagonal consisting of all ...
4
votes
3answers
671 views

Schur complement and “negative definite”!

Hi I have the following problem. Let the symmetric matrix M of the form: \begin{bmatrix} A & B \newline B^T & C \newline \end{bmatrix} We have that $C$ is positive semidefinite. Is ...
1
vote
1answer
155 views

Max Absolute Difference of Expectations under Change of Measure

Given a sample space $\Omega=\{ 1,\cdots,N \}$, a random variable $x$ defined on $\Omega$ that takes value $x_1,\cdots,x_N$, and a set of strictly positive real numbers $w_1,\cdots,w_N$. Define for ...
5
votes
1answer
326 views

Numerical linear algebra: how to compute $b^TA^{-1}b$ efficiently

What is the most efficient way to compute $b^TA^{-1}b$ for a given $A$ and $b$? Do we have to calculate $A^{-1}b$, or is this not necessary? edit: I forgot to mention that A is symmetric and ...
2
votes
3answers
700 views

Maximizing the minimum of piecewise linear functions in high dimensional space

I'd like to compute $\max_{x,t} t$ such that $\forall i$, $t < a_i + \|x - b_i\|_\infty$. where $a_i,\ldots, a_n \in \mathbb R$ and $b_1,\ldots,b_n \in [0,1]^{21}$ are fixed, $x \in [0,1]^{21}$, ...
3
votes
2answers
3k views

Linear program to maximize the minimum absolute value of linear functions ?

I'd like to compute $\max_{x,t} t$ such that $\forall i$, $t < a_i + |x - b_i|$. where $a_i,\ldots, a_n$ and $b_1,\ldots,b_n$ are fixed and $x \in [0,1]$. Can this be solved with a linear ...
2
votes
1answer
253 views

Maximum-bend TSP

I've seen minimum-bend TSP studied, has anyone looked at max-bend TSP? As a special case, I'm interested in the maximum number of turns a hamiltonian path can take in an $n \times n$ square grid. I ...
2
votes
1answer
271 views

Point-wise error estimate in polynomial regression

In our application we wish to estimate the actual path of objects. We have a set of samples of object locations $(t_i, x_i, y_i, P_i)$ where $t_i$ is the sample time, $(x_i, y_i)$ is the 2D location, ...