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

**2**

votes

**5**answers

242 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 ...

**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 $y=x^2$, ...

**2**

votes

**2**answers

142 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 ...

**6**

votes

**0**answers

59 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

294 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 $g$, ...

**0**

votes

**0**answers

120 views

### No Strong Duality In Spite of Slater's Condition

I was reading some course notes here.
On Page 8, it says:
Note that strong duality holds here (Slater's condition), but the
optimal value of the last problem is not necessarily the optimal
...

**2**

votes

**0**answers

82 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}
...

**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

**1**answer

192 views

### How to find the necessary and sufficient conditions for a non-convex function to be locally convex?

Let $f(X)\geq 0$ be a nonconvex $C^\infty$ function: $\mathbb R^3\to \mathbb R$.
Give any fixed $X_0$ such that $f(X_0)=\epsilon^+$, and the level set:
${L}=\{X\in \mathbb R^3:f(X)\leq \epsilon^+\}$ ...

**1**

vote

**1**answer

149 views

### Lagrange multiplier and semidefinite programming

suppose we have a primal semidefinite programming. for finding its dual we use Lagrange multiplier $w_i$ for each semidefinite constraint. If the Lagrange multiplier be zero for one constraint what we ...

**2**

votes

**0**answers

225 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 ...

**3**

votes

**2**answers

240 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 ...

**3**

votes

**1**answer

121 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 ...

**1**

vote

**0**answers

137 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 ...

**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, ...

**3**

votes

**2**answers

959 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 ...

**4**

votes

**1**answer

267 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}) = ...

**3**

votes

**1**answer

194 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 ...

**2**

votes

**0**answers

186 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

**1**answer

238 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

367 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 ...

**0**

votes

**1**answer

354 views

### Nonconvex optimization problem

I have a nonconvex optimization problem. It is actually optimizing a linear objective function over a set of linear constraints and a set of nonlinear, non convex constraints.
Is this problem ...

**1**

vote

**1**answer

129 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, ...

**1**

vote

**0**answers

129 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 ...

**0**

votes

**1**answer

248 views

### solve non-convex quadratic constrained quadratic programming

$\min_{\beta}\beta^{T} A \beta$
$s.t. \ \beta^{T} C \beta=1\ and\ \beta\geqslant 0$
Here $A,C\in \mathbb{R}^{M\times M}$, $\beta \in \mathbb{R}^{M}$
I saw in one paper saying that it could be ...

**3**

votes

**1**answer

207 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 ...

**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 ...

**3**

votes

**1**answer

75 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 ...

**2**

votes

**0**answers

146 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 ...

**2**

votes

**3**answers

154 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 ...

**3**

votes

**1**answer

486 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 ...

**2**

votes

**1**answer

410 views

### Maximizing supermodular functions

I have a real supermodular objective function which I want to maximize with constraint. The constraint is on the size, like |A|=k .
I am wondering if anyone can give me more information about a ...

**0**

votes

**1**answer

242 views

### minimization of a function when the feasible set is an unbounded cone

I have the following semi-infinite programming problem: I need to minimize a strictly convex real-valued function $f:\mathbb R^n\to\mathbb R$ subject to infinite linear constraints. I know in advance ...

**2**

votes

**1**answer

315 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 ...

**0**

votes

**0**answers

57 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 ...

**6**

votes

**1**answer

243 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 ...

**1**

vote

**1**answer

134 views

### What kind is this optimization problem

I come across a problem like
$\max {\frac{1+v}{1-u}}$
$s.t.~$ $ux^2+vy^2-xy\ge0$ $\forall x,y\in\mathbb{R}$
I do not know much of optimization.
What I have done is that $ux^2+vy^2\ge ...

**0**

votes

**0**answers

54 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

**1**answer

155 views

### Can one always Decide whether a Systems of Nonlinear Equations with Bilinear terms is Feasible?

I have come to a point in my PhD research were i need to prove that a particular decision procedure is decidable or not. And if i can solve the sub-problem described below, i shall have proved it. The ...

**1**

vote

**2**answers

177 views

### Convex optimization problem to QPP

Briefly, have the following problem:
\begin{equation}
\sum_{i = 0}^n a_i \ (max [ F_i( \bar x ), 0 ] )^2 \rightarrow min, \\\\
s.t.\\\\
A \bar x \leq b
\end{equation}
where $ F( \bar x ) $ is a ...

**5**

votes

**1**answer

308 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 ...

**0**

votes

**1**answer

174 views

### A certain type of quadratic problem.

I am interested in solving the following equality constrained quadratic (?) problem.
\begin{align}
\min_{u^{H}u=1}~(u^{H}A_1u) \\\
s.t.~ u^{H}A_2u=0
\end{align}
$A_1$ and $A_2$ are $N\times N$ ...

**11**

votes

**2**answers

557 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 ...

**5**

votes

**2**answers

267 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 ...

**0**

votes

**1**answer

128 views

### Nonlinear matrix equation 2

Solve the following nonlinear equations for $v$ and $w$
$Avv^TAw+Bvv^TBw=\lambda_1v+\lambda_2w$
$Aww^TAv+Bww^TBv=\lambda_1w+\lambda_2v$
$v^Tw=w^Tv=0$
$v^Tv=w^Tw=1$
where $\lambda_1, \lambda_2, ...

**1**

vote

**1**answer

311 views

### Can you maximize the spectral norm of a matrix in a semidefinite program?

Consider the following optimization problem: Maximize $||X||$, subject to $X$ being Hermitian (or symmetric, if you prefer) and a bunch of semidefinite constraints on $X$. I want to know if this can ...

**1**

vote

**1**answer

777 views

### linear objective function, non-linear constraints involving square-root of variables

i am trying to solve a general linear objective function with non-linear constraints. Can someone help me solve this.
Here is an example of one problem i am trying to solve,
minimize $b$
subject ...

**2**

votes

**1**answer

215 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 ...

**2**

votes

**2**answers

190 views

### product of variables in objective function

Hi there,
I'm looking for a solver that allows me to solve an optimization problem of the form min x1*x2*x3,...,xn subject to some linear constraints. I've used gurobi before, however I couldn't find ...

**1**

vote

**2**answers

189 views

### what method can I employ to solve this optimization problem which involves \min?

The optimization problem is:
maximize $$\min(\sum\limits_{i=1}^N \log\left(a_{1,i}+\frac{b_{1,i}}{c_{1,i}+d_{1,i}x_i}\right),\sum\limits_{i=1}^N ...