Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.

**0**

votes

**0**answers

32 views

### Differentiability and the Maximum Theorem [on hold]

Suppose that in Berge's Maximum Theorem the argmax correspondence happens to be single-valued. Are there assumptions that will guarantee that this argmax function will be smooth (e.g., continuously ...

**0**

votes

**0**answers

14 views

### Complexity of optimizing a bi-objective function with integer constraints

I have two different objective functions, each of which can be solved optimally in polynomial time. Does this mean, I can optimize a linear combination of these objective functions in polynomial time ...

**0**

votes

**0**answers

83 views

### How to decide a value of learning rate for Stochastic Gradient Descent?

I'd like to know how to decide a value of learning rate for Stochastic Gradient Descent (SGD), such as $\eta$ on the following parameter update iteration equation,
$w_{i+1} = w_i + -\eta \nabla ...

**4**

votes

**2**answers

97 views

### Convexity of a (non-symmetric) function of matrices

Let $f : H_{n\times n} (\mathbb{C}) \rightarrow \mathbb{R}$ be the function on Hermitian, positive semidefinite matrices $f(A) = \frac{M_i (A)}{\det(A)}$ where $M_i(A)$ is the determinant of the the ...

**3**

votes

**1**answer

55 views

### Analysis of first-order methods for constrained convex optimization with approximate oracles

In many first-order optimization methods an oracle is needed whose action enforces the constraint/regularizations. For example, in projected gradient descent, conditional gradient method, and proximal ...

**2**

votes

**1**answer

101 views

### Optimization over symmetric polynomials

Consider the following constraint satisfaction problem: Let $\alpha_1 , \ldots, \alpha_k \in \mathbb{R}$ be given as well as an error parameter $\epsilon$. Find $p_1, \ldots, p_n$ such that
(i) $0 ...

**5**

votes

**0**answers

144 views

### Characterizing matrices with rank constraint

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M,b]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad ...

**6**

votes

**1**answer

455 views

### Bound on the sum of arguments

Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has
$$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)}
+\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi,
$$
where $f$ is the ...

**1**

vote

**0**answers

33 views

### Likelihood convexification

I am doing constrained vector optimization using a non-convex non-linear likelihood function. My problem is of the following form:
$$\begin{align*}\hat Q &= \underset{\vec Q}{\arg\min} -\log ...

**2**

votes

**0**answers

60 views

### How to solve the following generalized quadratic programming problem [closed]

I want to solve a generalized form of a quadratic programming problem
$$\min_x \left(\sqrt{x^TPx}+\sqrt{x^TQx}\right)^2+c^Tx$$,
$$\textrm{ s.t. } Ax\le b.$$ Here, $P$ and $Q$ are both positive ...

**0**

votes

**0**answers

56 views

### Specific optimization problem solution procedures

Is there a standard procedure to solve following two optimization problems?
$$\mathsf{Problem\mbox{ }I}:\mbox{ }\min_{A\in\{0,1\}^{n\times n}:rk(A)=r}\mbox{ }\max_{R,S\in\Bbb R^{n\times ...

**4**

votes

**2**answers

205 views

### Convexity of a function of matrices

Let $A$ be an $n\times n$ positive-definite matrix. Let $0<\lambda _1 \leq \lambda_2 \leq \lambda _3 \ldots \leq \lambda _n$ be the eigenvalues of $A$. Let $n\geq k\geq 1$. Is the function $f(A) = ...

**1**

vote

**0**answers

37 views

### modern exposition of exact ground state of classical XY model or Ising model

What is the state of art technique in solving exact ground state of Heisenberg model, meaning minimization of the H terms (hamiltonian) provided infinite spin space?
...

**5**

votes

**1**answer

153 views

### Find subset of collection of sets whose intersection has minimum average value

Let $a_1,\ldots,a_n>0$, and let $S_1,\ldots,S_d\subset\{1,\ldots,n\}$ (all non-empty).
For any $I\subseteq\{1,\ldots,d\}$, define $S(I)=\bigcap_{i\in I} S_i$.
Given some $1\leq s < d$, consider ...

**1**

vote

**0**answers

30 views

### Optimization of a multilinear function over a product of hypersimplices

Let $P = \Delta_1 \times \cdots \times \Delta_N$ be the Cartesian product of $N$ hypersimplices. Let $f : P \to \mathbb{R}$ be a multilinear function of $N$ variables, ie $x_i \mapsto f(x_1, \ldots, ...

**0**

votes

**0**answers

25 views

### non-coherent estimation problem

I have the following signals
$$\left[\begin{array}{c} y_{mn} \\ y_{nm}\end{array}\right] =\left[\begin{array}{c} x_{n} \\ x_{m}\end{array}\right]h_{nm} +\left[\begin{array}{c} e_{mn} \\ ...

**2**

votes

**0**answers

96 views

### Solution of a linearly constrained quadratic programming problem [closed]

What is the solution of the following optimization problem:
\begin{align}
&\min{\mathbf{p}^\mathrm{T} \mathbf{B} \mathbf{p}}\\
&\text{subject to}: \mathbf{0}\leq{\mathbf{p}}\leq \mathbf{1}.
...

**2**

votes

**0**answers

45 views

### Sequence transformations that are entropy invariant

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Define entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$
by ...

**5**

votes

**1**answer

156 views

### Finding sparsest solution of a linear system

I want to find the solution with most zero-components for the following problem:
$Ax=b$ for $A\in \mathbb{R}^{k\times n}, b \in \mathbb{R}^{k},k<n$, where $x$ is real and has no additional ...

**1**

vote

**0**answers

42 views

### Limiting Entropy of deterministic sequences - 2

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_a$ be limiting distribution ...

**1**

vote

**1**answer

104 views

### Limiting Entropy of deterministic sequences - 1

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$.
Consider distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_{a,m}$ be distribution at ...

**1**

vote

**1**answer

77 views

### Entropy difference dominance of sequences

Consider a collection of positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$.
Similarly for the collection $\{a_i\}_{i=1}^{m+1}$ form the distribution ...

**1**

vote

**1**answer

56 views

### Entropy dominance of certain restricted sequenes

Say you have positive $\{a_i\}_{i=1}^n$ and you have $p_i=\frac{a_i}{\sum_{i=1}^na_i}$, then assume you have a $C$ such that $C<2a_n\ll\sum_{i=1}^na_i$ (that is $C$ is not very large), then define ...

**2**

votes

**1**answer

65 views

### Entropy dominance

Let $0<a<b<c$ be distinct positive reals.
Define four different probability distributions:
$$\mathcal{P}_{ab}:P_{a,ab}=\frac{a}{a+b}=1-P_{b,ab}$$
...

**1**

vote

**1**answer

49 views

### Maximizing a certain concave function over a non-convex set

I have been working on a problem which involves maximizing a concave function over a non convex constraint set. The problem is the following $$max. \frac{1}{2}x^TBx\\ s.t.\ x_i(1-x_i)=0,\ \ ...

**1**

vote

**1**answer

72 views

### Smooth unit vector field on a tetrahedron to interpolate vertex constraints

For a tetrahedron $T\subset \mathbb{R}^3$ with vertices $r_i\in \mathbb{R}^3$ , $i=1,\ldots,4$, and unit vectors $u_i\in \mathbb{S}^2$ at each vertex $i=1,\ldots,4$
consider the (energy) functional
...

**11**

votes

**2**answers

566 views

### Conjecture on maximum of symmetric combinatoric function

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture).
(The question was first asked at math.SE, ...

**0**

votes

**1**answer

59 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
...

**0**

votes

**0**answers

57 views

### Best s-term approximation and unit balls in weak $\ell^p$ norm

In the book "A Mathematical Introduction to Compressive Sensing" by Foucart and Rauhut there is the following asymptotic estimate at page 332, equation (11.1):
$$\sup_{\mathbf{x}\in B^N_{r,\infty}} ...

**1**

vote

**1**answer

79 views

### Nonlinear system of equations whereas most of the equations are linear. How to minimise operation?

Let us say we have a n * n system of equations like KU=F where K is a n*n matrix and U and F are n*1 vectors. K and F are defined and the final goal is to find U values.
K is a sparse banded matrix ...

**0**

votes

**0**answers

33 views

### How can be a conservative field constraint be efficiently implemented in a continuous optimization problem?

Suppose we have the following continuous optimization problem:
$$
\underset{x}{\mathrm{minimize}}f\left(x\right)
$$
subject to
$$
\exists X:\nabla X=Jac\left(X\right)=x
$$
where $f$ is a function ...

**-3**

votes

**1**answer

52 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 ...

**4**

votes

**2**answers

265 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 ...

**3**

votes

**0**answers

81 views

### Containing a “fuzzy” ellipsoid within an ordinary ellipsoid

Consider the ellipsoid described by the inequality $(x - x_c)^T P^{-1} (x - x_c) \leq 1$, where the vector $x_c \in \mathbb{R}^n$ denotes the center of the ellipsoid and the symmetric positive ...

**0**

votes

**0**answers

27 views

### Optimization problem involving an entrywise function

Let $X$ a $n\times p$ real-valued matrix and $Y$ a $p\times q$ real-valued matrix. Let $\phi:\mathbb{R} \to \mathbb{R}$ a function. What is the appropriate way to deal with the following optimization ...

**0**

votes

**0**answers

50 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: ...

**1**

vote

**0**answers

71 views

### Maximizing the “uniformity” of a probability measure, with constraints, via path length minimization

Background
I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below:
Definition: Maximally Uniform ...

**0**

votes

**0**answers

44 views

### minimizing concave function

I am interested in using the algorithm of Harold Benson described in his 1991 paper: "A Branch and Bound-Outer Approximation Algorithm for Concave Minimization over a Convex Set". In the paper, he ...

**0**

votes

**0**answers

24 views

### Recursively calculate Tikhonov regularizer in b-spline objective function

I'm trying to write a program to calculate cubic b-spline based on set of inputs. But I can't figure out how to calculate value of Tikhonov regularizer.
My b-spline function is this:
I have ...

**7**

votes

**2**answers

427 views

### How to solve such an optimization problem

I encounter the following optimization problem, but I can't solve it.
Given $N$ variables satisfying $0 \leq x_1 \leq x_2 \leq x_3 \leq ... \leq x_N \leq 1$ and an integer $K$ no large than $N$, find ...

**1**

vote

**1**answer

225 views

### Levenberg's original article “A method for the solution of certain problems in least squares”

Does there exist any digital copy of the original article (or a transcript) K. Levenberg, A method for the solution of certain problems in least-squares, Quart. Appl. Math. 2 (1944): 164-168?
It is ...

**2**

votes

**2**answers

195 views

### How to fit the parameters of differential equations with known data?

I have the following data from chemical kinetics research to fit the parameters of ordinary differential equations:
$$
\left[
\begin{array}{ccccccc}
\text{No.}& t & y_1(t)&y_2(t) & ...

**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 ...

**0**

votes

**0**answers

52 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 ...

**4**

votes

**1**answer

182 views

### optimization problem, any solution?

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

**0**

votes

**0**answers

51 views

### Solution of a nonlinear system of two equations

Given the matrix $A_{M,N}$ with $N\gt M$, the vector $y$, I have to find the vectors $x$ and $u$, satisfying the following equations:
$$D(x)x=A^Tu$$
$$y=Ax$$
where: $$D(x) = \left| \begin{array}{ccc}
...

**0**

votes

**0**answers

42 views

### A Optimization problem using co-ordinates of joint numerical range.

Let $\mathbf{A}_1,\dots,\mathbf{A}_L$ be $N\times N$ hermitian matrices. Define the mapping from the $N-$dimensional unit sphere to $\mathbb{R}^L$ as
\begin{align}
...

**0**

votes

**2**answers

118 views

### Approximate solution to large mixed integer programming problem

What are the available approaches to find an approximate solution to a large mixed integer programming problem?
I ran my problem in the Gurobi MIP solver.
It can find a feasible solution in ...

**2**

votes

**3**answers

335 views

### Solving a quadratic matrix equation with non-squared matrix

I was trying to solve the problem of finding the value of a non-squared matrix $T$ ($n \times m$) which solves
$$ T^T T = X$$
where $X$ is a symmetric and positive semidefinite $m \times m$ matrix, ...

**4**

votes

**1**answer

198 views

### Finding a point maximizing the minimal distance to a set of points

Given a set of of $N$ points $\{\mathbf x_i \in \mathcal{S}^d\}_{i = 1, \ldots, N}$, where $\mathcal{S}$ is a set of possible values, how can I find the point $\mathbf x^*$ that maximizes the minimum ...