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

**11**

votes

**2**answers

577 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, ...

**11**

votes

**2**answers

580 views

### Quadratic Farkas' Lemma?

The Farkas Lemma says that if a system of linear inequalities implies
yet another linear inequality, then this last inequality can be obtained by
taking a positive linear combination of the ...

**7**

votes

**2**answers

436 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 ...

**6**

votes

**1**answer

246 views

### Generalization of the equilateral triangle?

I consider points in the two-dimensional plane.
An equilateral triangle is a set of three points in the plane which are equidistant.
Suppose now I have $n$ points $x_1,...,x_n$. What is the ...

**6**

votes

**1**answer

464 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 ...

**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

**2**answers

494 views

### Find minimum-area ellipse which encloses two ellipses

I need an efficient algorithm to find the ellipse with the smallest possible area which encloses two given ellipses. The given ellipses are constrained to have coincident centers at the origin but can ...

**5**

votes

**2**answers

301 views

### Can we decompose a polynomial into difference of convex polynomials?

Given a multivariate polynomial $p(x_1, ..., x_n)$ on $\mathbb{R}^n$, can we always decompose it into the difference of two convex polynomials? i.e., is there a pair of convex polynomials $f$ and $g$, ...

**5**

votes

**1**answer

312 views

### Solve equation with matrix variable

I want to solve a matrix $\Omega$ from a equation $\sum_k (\Omega + \Theta_k)^{-1} = Q$. The $Q$ and $\Theta, \forall k=1...K$ are known, and are positive definite matrices. $\Omega$ also has to be ...

**5**

votes

**2**answers

270 views

### Simultaneous maximization of two Generalized Rayleigh Ritz Ratios

Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios ...

**5**

votes

**1**answer

163 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 ...

**5**

votes

**0**answers

163 views

### Characterizing matrices with rank constraint

Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M]=\{Q\in\Bbb\{0,1\}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb\{0,1\}^{n\times ...

**5**

votes

**0**answers

183 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 ...

**4**

votes

**2**answers

275 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 ...

**4**

votes

**2**answers

217 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) = ...

**4**

votes

**2**answers

106 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 ...

**4**

votes

**1**answer

241 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 ...

**4**

votes

**1**answer

183 views

### optimization problem, any solution?

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

**4**

votes

**1**answer

278 views

### Finding all local maximum points of a function?

Let ${\boldsymbol \theta}=(\theta_1,\theta_2,\ldots,\theta_n) \in{\mathbb T}^n$ and $P:{\mathbb T}^n\rightarrow {\mathbb R}$ be a function defined on $n$-torus as
$$
P({\boldsymbol \theta}) = ...

**4**

votes

**0**answers

151 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

**2**answers

1k views

### Sparse approximation of the inverse of a sparse matrix

Is it possible to approximate an inverse of a sparse matrix with a sparse matrix?
The problem comes up in numerical non-linear quasi-Newton optimization: given a sparse Hessian a good starting point ...

**3**

votes

**1**answer

525 views

### The average number of people that can sit on a bench of a given length.

Let me explain what I mean:
The width of the average person varies, perhaps with a normal distribution.
Given a specific variance, how many people (on average) can sit side-by-side on a bench of a ...

**3**

votes

**2**answers

139 views

### QR-Decomposition of matrix valued function

Suppose I have a matrix valued function
$$
F:\mathbb{R}\rightarrow\mathbb{R}^{m\times n},\qquad F(x)=\tilde Q\tilde R+xu_1v_1^T+xu_2v_2^T
$$
where $\tilde Q\in\mathbb{R}^{m\times m}$ is orthogonal, ...

**3**

votes

**1**answer

150 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 $s_1$ ...

**3**

votes

**1**answer

59 views

### Constrained vs Unconstrained Optimization

I'm currently working on an optimization problem with a linear objective with linear and nonlinear constraints, i'm facing difficulties reaching a good solution, so i was advised to move the nonlinear ...

**3**

votes

**1**answer

65 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 ...

**3**

votes

**1**answer

132 views

### Decomposition of a semi-definite matrix into sums of sparse semi-definite matrices

I'll first provide the background.
Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables
$x^{(1)},\ldots,x^{(n)}$.
We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...

**3**

votes

**1**answer

196 views

### Intuition on a certain class of quadratic optimization problems

Let $\mathcal{X} = \{\mathbf{X}\in\mathbb{C}^{d\times d}:\|\mathbf{X}\|\leq 1\}$, where $\|\cdot\|$ is the Frobenius norm. Let $\mathbf{y}\in\mathbb{C}^{d\times 1}$. We are familiar with the following ...

**3**

votes

**1**answer

216 views

### Optimization problem - maximizing number of satisfied linear inequalities subject to a quadratic constraint

I am wondering what is known about optimization problems of the following type.
Our control x is a unit vector in $\mathbb{R}^n$. We are given a finite number of linear inequalities
$$Az≥b,$$
and we ...

**3**

votes

**1**answer

77 views

### Conjugate gradient algorithm where first search direction is not equal to residual

In usual formulation of conjugate gradient algorithm initial search direction is taken to be the residual (so residual and search direction spans Krylov subspace). However, in cases where inexact ...

**3**

votes

**0**answers

105 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 ...

**3**

votes

**0**answers

273 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

**5**answers

252 views

### Distance between two sets

Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem.
$$ \min ...

**2**

votes

**3**answers

155 views

### Constraint optimization problem for any dimensionality $n>1$.

I am going to post a particular example for the sake of clarity.
One needs to maximize a real function
$F = a_1a_2 + a_2a_3 + \cdots + a_{n -
> 1}a_n + a_na_1;$
with active ...

**2**

votes

**2**answers

261 views

### Why eigenvectors optimize this orthogonally constrained nonlinear minimization problem?

Given a $p \times p$ positive definite matrix $\Sigma$, why eigenvectors of $\Sigma$, stacked as columns of a matrix $R \equiv [r_1 \, r_2 \, \ldots \, r_p]$, optimize the following orthogonally ...

**2**

votes

**1**answer

106 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 ...

**2**

votes

**2**answers

256 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) & ...

**2**

votes

**1**answer

236 views

### How to minimize $-\sum p_b \ln{p_b}$?

Consider multisets of the form $A = \{a_1,\dots,a_n\}$ of integers. Let $q = P(a_i = a_j)$ when $i$ and $j$ are chosen independently and uniformly from $\{1,\dots, n\}$. Let $B$ be the set of ...

**2**

votes

**1**answer

145 views

### Is the feasibility of a system of nonlinear, non-convex equations (inequalities) decidable?

I would like to know whether the following problem is decidable.
Is the system
$x^T Q_i x + r_i = 0 \mbox{ for } i = 1, ..., k$
$x^T Q_j x + r_j \neq 0 \mbox{ for } j = k+1, ..., t$
feasible, ...

**2**

votes

**1**answer

357 views

### lipschitz constant of a multivariate function

I have a function $f:\mathbb{R}^{50} \rightarrow \mathbb{R}$ and I need to compute the Lipschitz constant of $f$ to solve an optimization problem using a specific algorithm. Does any one have ...

**2**

votes

**2**answers

1k views

### Solving a system of equations/inequalities that have trigonometric functions on the left-hand side

Is there any known (symbolic) method that solves a system of equations/inequalities that have trigonometric functions on the left-hand side of the system?
Ex) Find $x,y,\theta \in \mathbb{R}$ that ...

**2**

votes

**1**answer

56 views

### Rate of convergence for cyclic gradient descent

I'm trying to solve the optimization problem
$\min_x \frac{1}{n} \sum_{i=1}^n f_i(x)$
where $f_i$ are (strongly) convex, smooth, lower semi-continuous, etc.
However, I am not able to do conventional ...

**2**

votes

**1**answer

46 views

### When is a convex program continuous in its constraint vectors?

Consider $$F(z)=\min ae^{-x}+b e^{-y} s.t. x\ge 0, y\ge 0\text{ and } x+y=z$$
I checked if this function is continuous, but it is not at $z=0$. $F(z)=2\sqrt{ab}e^{-z/2}$ when $z\ne 0$, and ...

**2**

votes

**1**answer

68 views

### Entropy dominance

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

**2**

votes

**3**answers

349 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, ...

**2**

votes

**2**answers

158 views

### How to minimize the Bregman divergence on a convex hull spanned from a set of vectors?

everyone.
It has been well known that the following minimization problem of a Bregman divergence with linear inequality
can be solved by successively projecting the current point to each constraint ...

**2**

votes

**1**answer

242 views

### Non-linear 1st order difference equation

I have been trying to solve the following difference equation for some time now : $$u^3(n+1) = a - b\cdot u^2(n) + u^3(n), \qquad a \ne 0 \ne b$$
I have tried various substitutions, simplifications ...

**2**

votes

**1**answer

401 views

### Finding zeros of a multi-variable nonlinear trigonometric function

I am trying to calculate analytic solution (or locus) of zeros of a very large multi-variable function which is consisted of thousands of nonlinear trigonometric terms. All the variables are real ...

**2**

votes

**1**answer

223 views

### Does Quadratic Programming get easier when it's described by a diagonal matrix?

Generally, Quadratic Programming solves the problem
$$\text{Given }Q, c, A, b,\text{ choose }x \text{ to maximize } x^TQx + c^Tx \text{ subject to } Ax \le b$$
In this form, Quadratic Programming is ...