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

**2**

votes

**1**answer

105 views

### Find a line such that sum of perpendicular distances of points to the line is minimized

Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...

**6**

votes

**0**answers

61 views

### Bounding volume of cell in complement of zero set

I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...

**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 ...

**4**

votes

**0**answers

150 views

### Optimization problem involving Multivariate Normal

I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all
$n\geq3$, the function:
...

**3**

votes

**0**answers

75 views

### existence of optimal control

I'm looking for an existential result in optimal control for the following class of problems:
Given $T > 0$, $\bar x, \hat x\in\mathbb R^d$, an instantaneous cost function $c:\mathbb R^d\times ...

**3**

votes

**0**answers

82 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 ...

**2**

votes

**0**answers

46 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 ...

**2**

votes

**0**answers

83 views

### Techniques for proving that a set of constraints over the integers are inconsistent

I have a problem which boils down to showing that a set of constraints has no solutions. A simplified version of this constraint system would be the following system:
$$
\left\{
\begin{array}{l}
...

**2**

votes

**0**answers

201 views

### An intuition for three different types of subgradients (proximal, regular, limiting)

I'm having a bit of difficulty getting my head around the different types of subgradients we're currently covering in a nonsmooth optimisation class I'm taking.
These subgradients are (assume $x \in$ ...

**2**

votes

**0**answers

150 views

### Quadratic optimization with parameter in constraint

Disclaimer: I posted the same question on math.stackexchange. However, the FAQ suggests to post research-level questions in this forum.
Question: Given a function $q: \mathbb R^{N\times N}\mapsto ...

**1**

vote

**0**answers

24 views

### Minimizing sum of functions, while keeping their values non-negative

Suppose we have data $\mathbf{x}_i$, $i\in \{1, \ldots, K\}$, and we're trying to find parameters $\hat{\mathbf{\theta}}$ such that
$$\hat{\mathbf{\theta}} = \underset{\mathbf{\theta}} ...

**1**

vote

**0**answers

62 views

### Minimize Product of Sums of Squared Distances

The Question
Given two sets of vectors $S_1$ and $S_2$，we want to find a unit vector $s$ such that
$$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\}
\cdot
\{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...

**1**

vote

**0**answers

34 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 ...

**1**

vote

**0**answers

40 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?
...

**1**

vote

**0**answers

31 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, ...

**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

**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 ...

**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 ...

**1**

vote

**0**answers

65 views

### Estimation of part of parameters from an ODE

Suppose, we have an ODE
$$ \frac{dy}{dt}= f(t,y;p',a)$$
or alternatively
$$ \frac{dy}{dt}= f(t,y;p)$$
where the set of all parameters $p = (p',a)$. We only need to estimate part of parameter set ...

**1**

vote

**0**answers

257 views

### Subtour Elimination in Travelling Salesman Problem using MTZ

I am trying to formulation a problem similar to a Traveling Salesman with Time Window constraints.
To eliminate subtours, I need to use some constraint similar to a generalization of MTZ constraints ...

**1**

vote

**0**answers

143 views

### Constructing an $\epsilon$-net for a Lipschitz subspace of $L^2$

Let $X$ be a subset of $L^2([0,1])$ which contains only Lipschitz function.
Also, the member of $X$ are uniformly bounded
$$
|x(t)| < M, \text{ for all $x \in X$ and $t \in [0,1]$}.
$$
Let $F: X ...

**1**

vote

**0**answers

130 views

### Recovering a partition from spectral properties of the graph Laplacian

Let $G$ be a weighted graph with vertices $V$. Let $W$ be its real-valued, non-negative, $|V|\times|V|$ adjacency/affinity matrix. Let $L = \mathrm{diag}(W\mathbf1)-W$ be the (unnormalized) graph ...

**1**

vote

**0**answers

432 views

### Solving a system of complex non-linear equations

I have a set of five equations which can be described as follows:
$m_{i}=\frac{k_{1}}{(x+a)^{i}} + \frac{k_{2}}{(b+d)^{i}}+ \frac{k_{3}}{c^{i}}$
for i=1 to 5 where
$$\eqalign{
...

**1**

vote

**0**answers

227 views

### Multiobjective semidefinite programming

Let $C$ be size $n \times n^{2}$.
Let $B$ be size $2^{g(n)} \times n^{2}$ where $g(n) > n$.
There is only one $\mathcal{1}$ per row of $C$ and remaining entries of $C$ are $\mathcal{0}$.
$B$ is ...

**0**

votes

**0**answers

15 views

### Global and local maxima in a weighted sum of logarithms of linear functionals?

Initially posted on math.stackexchange, was recommended that this is a more relevant forum:
Is is possible to describe, and locate efficiently, the maxima of the function below in the parameters ...

**0**

votes

**0**answers

55 views

### Can this equation have an explicit solution?

Given $n > 0$, $0 \leq i \leq n$ is an integer, $D = diag(d_1, \dots, d_n)$ is
positive definite, $e_i$ is the $i$th column of a $n \times n$ identity matrix, $u \in R^n$ such that $B = D + u * ...

**0**

votes

**0**answers

14 views

### Is it possible to design an algorithm an optimization problem with non-convex and coupled constraints, using decomposition methods?

Recently, I am working on an optimization problem in wireless networks. I formulate the problem to be an optimization problem with non-convex and coupled non-linear constraints. And I want to design a ...

**0**

votes

**0**answers

17 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

89 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 ...

**0**

votes

**0**answers

57 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 ...

**0**

votes

**0**answers

27 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} \\ ...

**0**

votes

**0**answers

60 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}} ...

**0**

votes

**0**answers

34 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 ...

**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

52 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

45 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 ...

**0**

votes

**0**answers

53 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 ...

**0**

votes

**0**answers

52 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

43 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

**0**answers

77 views

### Complexity of turning a d-degree polynomial to 2-degree polynomial

For a very simple example,
$(1+x)^4=x^4+4x^3+6x^2+4x+1$ is a 4 degree polynomial, and I want to change it to a 2-degree polynomial by add more variables, for this example, we can simply let $y=x^2$, ...

**0**

votes

**0**answers

103 views

### Modifying a QP to incorporate more constraints

Consider the following problem:
$$\min \sum_{i=1}^n (Y_i - Z^{(i)})^2 \\
\text{subjected to}~ \epsilon_k^{\top}(X_j-X_k) \leq Z^{(j)}-Z^{(k)} ~ \forall k,j = 1 \ldots n. $$
where $\epsilon_1, ...

**0**

votes

**0**answers

70 views

### Big eigenvalues of a special stochastic matrix

Given a matrix $M$ of size $n\times n,$ we write its different eigenvalues by $x_1,x_2,\ldots,x_m$ with $m\leq n$ such that $|x_1|>|x_2|>|x_3|>\cdots|x_m|,$ and call $x_2\doteq ...

**0**

votes

**0**answers

58 views

### functional maximization

Define a functional space of functions of the form $F(t)=p_1 exp^{-\mu_1(\delta-t)}+p'_1 (1-exp^{-\mu_1(\delta-t)}))$. $p_1,p'_1,\delta,\mu$ are parameters in [0,1] and trivially, variation of these ...

**0**

votes

**0**answers

55 views

### Minimizing inside a spherical uncertainty region

I am trying to figure out how to solve:
$\min_U r_{p}$
where $r_{p}=\alpha^\intercal\omega$ and $U$ is a sphere centered at $\alpha$ with radius equal to $\chi|\alpha|$ . ( $\omega$ is a vector or ...

**0**

votes

**0**answers

119 views

### Question on non-linear parametric mixed integer program

I am trying to solve a mixed integer minimization problem, where there are a number of parameters, and there are products of parameters with variables appearing in the objective function. I assume ...