**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

236 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

129 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

263 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

185 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

219 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

251 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

306 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

93 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

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

**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

114 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

619 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

439 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

395 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

160 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

484 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

685 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

520 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

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

**0**answers

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

**1**answer

676 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

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

**0**answers

117 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

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

**1**answer

327 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 - ...

**1**

vote

**2**answers

754 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

244 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

412 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

713 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

410 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

**2**answers

914 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

**3**answers

652 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

**1**answer

154 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

**1**answer

325 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

**3**answers

673 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

**2**answers

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

**1**answer

251 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

**1**answer

270 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, ...

**6**

votes

**1**answer

418 views

### Eigenspace of Euclidean distance matrix.

What is the necessary and sufficient condition (if there is any) that $n$ orthonormal vectors $v_1,v_2,\cdots,v_n$ are eigenvectors of a Euclidean distance matrix. When $n=2$, the orthonormal ...

**0**

votes

**1**answer

240 views

### Is this a Karusch-Kuhn-Tucker method or something else? [closed]

This is a question that originated with World of Warcraft. I have a solution, but I don't know where to look up other problems of the same kind for a better explanation.
There is a plane with p axis ...

**0**

votes

**1**answer

276 views

### Infinite dimensional optimization

Assume that we optimize a convex problem (convex objective and linear constraint) over a set of functions (say $L2$). Consider now the same optimization problem (same objective and same linear ...

**3**

votes

**2**answers

486 views

### Cubic graphs which are “difficult to navigate”

Suppose I am inside a finite, weighted cubic graph without loops, with no information regarding its layout, including the number of vertices or distances to the adjacent vertices. I want to reach a ...

**2**

votes

**2**answers

430 views

### How can I simplify this quadratic optimization?

I have no experience in the field of optimization, so I have no idea how hard or naive it is. I got no response on math.stackexchange so I am posting it here, though I doubt it is research-level.
...

**1**

vote

**1**answer

232 views

### How to get constraint qualifications conditions for optimization on banach spaces

From the book of bertesekas(1999),borwein(2006),
we learned some constraint qualifications on R^n spaces,such as:
Linear independence constraint qualification(1951)
Mangasarian–Fromovitz constraint ...

**5**

votes

**2**answers

501 views

### Is there a name for this type of matrix? (Reference Request)

I am working on a problem were I encounter matrices of the form
$X = \begin{bmatrix}\frac{1}{1 - a_ib_j}\end{bmatrix}_{ij}$
I am aware of Cauchy matrices, which have the form
$X = ...

**0**

votes

**0**answers

387 views

### Combining Convex and Concave in CVX

Hello,
I have an unconstrained optimization in which I need to minimize a sum of convex functions and maximize a concave function together. I combined both the problem by adding a minus sign to the ...

**5**

votes

**2**answers

416 views

### Maximize the intersection of a n-dimensional sphere and an ellipsoid.

I have the conjecture that the volume of the intersection between an $n$-dim sphere (of radius $r$) and an ellipsoid (with one semi-axis larger than $r$) is maximized when the two are concentric, but ...