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

**0**

votes

**0**answers

22 views

### Simulated Annealing and Mixed Integer Non-Linear models

Has any research or study been done so far, in which a MINLP model being solved with Simulated Annealing algorithm,without "linearzing" the proposed MINLP model at first?

**0**

votes

**0**answers

51 views

### Simulated Annealing algorithm for MINLP [on hold]

In the objective function of a mathematical programming model,we have an expression like this:
$$
\biggl(\biggl|X\biggl| \biggl) . Q
$$
in which both X and Q are continuous variables, and $||$ ...

**0**

votes

**0**answers

31 views

### How to numerically evaluate a integral whose limits are functions of x (using Gauss quadrature rule)? [on hold]

I am trying to numerically evaluate an integral $\int_q^1 \ln (\sum_i \alpha_ix_i) dq$, in which $\ln (\sum_i \alpha_i x_i)$ is related to $q$ via the following:
$z_i=(1-q)\frac{\alpha_ix_i}{\ln ...

**0**

votes

**0**answers

19 views

### Dennis More' Superlinear Convergence_refrences request

Why in the proof of superlinear convergence of restricted broyden class (for the unconstrained optimization) we need the bounded deterioration condition for the approximation of all the true hessian ...

**1**

vote

**0**answers

24 views

### Which algorithm is most efficient for a specific QP problem

I have a QP problem of the following kind:
$\min_{\alpha\in\mathbb{R}^n}\frac{1}{2}\alpha^T M \alpha - p^T\alpha$
s.t. $l\leq \alpha \leq u$
The matrix $M$ is symmetric and positive definite and of ...

**1**

vote

**0**answers

32 views

### Characterization of eigenvector

Let's say we have the following optimization problem. (All the $\Sigma_{ii}$'s are positive definite.)
$\max u^\top \Sigma_{12} v\quad$
$\text{subject to}\quad u^\top \Sigma_{11} u = 1\quad and\quad ...

**1**

vote

**1**answer

88 views

### Why polynomial $\psi^\top(t) A^{-1} \psi(t)$ attains maximum on $[-1, 1]$ at $t = \pm 1$, where $\psi_k(t) = t^k$?

Problem. Let $\psi(t) = (1, t, t^2, \ldots, t^{p-1})^\top$ - a polynomial basis.
Suppose there is a matrix
$$
A = \int_{-1}^1 \psi(t) \psi^\top(t) dt, \ \text{i.e. } \ A_{ij} = [2 \, | \, i+j] ...

**4**

votes

**1**answer

63 views

### About optimization with Renyi divergence

Can someone link me to some pedagogic example of computing the Renyi divergence between two discrete/continuous distributions? Like examples where someone has been able to obtain a neat closed form ...

**0**

votes

**0**answers

47 views

### About identifying a few diagrams

Please have a look at these beautiful seminar slides,
https://math.berkeley.edu/~bernd/coimbra1.pdf
Can someone kindly identify the algebraic description of the spectrahedron that is drawn on slide ...

**1**

vote

**0**answers

54 views

### Hessian matrix positive definiteness (concavity test) [closed]

I have a rather simple scenario based optimization problem:
Maximize
$$
Q_1{_s}(A_1{_s}-Q_1{_s}-bQ_2{_s})+ Q_2{_s}(A_2{_s}-Q_2{_s}-bQ_1{_s})-[(Q_1{_s}-K_1)^+ + (Q_2{_s}-K_2)^+]c
$$
subject to ...

**3**

votes

**0**answers

153 views

### Can the following system of equations be solved analytically/in a closed form?

From a constrained non-linear maximization problem I obtained the following system of equations:
$a_1=\frac{1+a_3-\sqrt{a_2a_3}\sqrt{v_1}}{1+\sqrt{\frac{a_3}{a_2}}\sqrt{v_1}}$
...

**2**

votes

**1**answer

105 views

### Global minimum of nonlinear least square

We have a continuous and differentiable function $f(\cdot)$ that maps from $R^n$ to $R^n$. We are trying to solve a nonlinear least square problem:
Minimize $J(x)=\Vert f(x)-z\Vert^2$
subject to box ...

**0**

votes

**0**answers

175 views

### A contractive mapping which I don't understand

Given a matrix $Y$ and a vector $c$ define the following iteration
$\hat{c} = f(c)$, where each element of $\hat{c}$ is given by
$$\hat{c}_{\ell} = \frac{\sum_k ...

**0**

votes

**0**answers

47 views

### Separable Least squares - is there a notion of conjugate directions?

I have a general question.
Suppose I have the following to optimize
$$\|Y-A(\mathbf{x})B(\mathbf{y})\|^2$$
where $Y$ is a vector, $A(\mathbf{x})$ is a matrix that depends on a vector $\mathbf{x}$ in a ...

**4**

votes

**0**answers

54 views

### What does the Von Neumann's stability analysis tell us about non-linear finite difference equations?

I've asked this question on computation science stackexchange, but it did not receive any answers so I have decided to ask it here as well.
I am reading a paper [1] where they solve the following ...

**0**

votes

**1**answer

71 views

### a sum of ratios of quadratic forms

I have the following function that I would like to optimize over the value A
$$f(A)=\sum_k \frac{\mathbf{y}_k^H\left[\begin{array}{cc} 1&0\\ 0& A \end{array} ...

**2**

votes

**0**answers

27 views

### $\min \sum_i f(w^i) +\sum_j g(w_j)$ wrt col and rows of a matrix

I've got an unconstrained optimization problem, and all function involved can be regarded as differentiable as you like.
The variable is a rectangular matrix $M$.
Target Function is $\sum_i f(w^i) ...

**2**

votes

**0**answers

60 views

### Conditions under which the dual function is self-concordant

Consider the following optimization problem
\begin{align}
\min_{x}&\quad f(x)\\
\nonumber \text{subject to } \quad&h_i(x) = 0,\,i=1,\ldots,m\\
\nonumber \quad&x\in ...

**3**

votes

**0**answers

43 views

### measure of an image under an argmax function

I am trying to find any techniques to analyze the measure of an image of a set under an argmax function.
For example, let $\Omega\subset\mathbb{R}^n$ be compact and $\phi:\Omega\to\mathbb{R}$ be ...

**2**

votes

**1**answer

100 views

### A graph assignment problem

Consider bipartite graph with vertex set $V_1\cup V_2$ where $|V_1|=\frac{n(n-1)}2$ and $|V_2|=n$. The vertices in $V_1$ all have degree $2$ and connected to two vertices in $V_2$. The vertices in ...

**0**

votes

**1**answer

98 views

### Generalized Lax-Milgram for Weak Formulation of 1D Linear Schrodinger

I am interested in the variational formulation of the 1D Schrodinger equation:
$i u_t- \beta u_{xx} = 0 $ and $u(x,0)=u_0(x)$ which upon integration by parts yields:
$i(u_t,v) + \beta (u_x,v_x) = 0$ ...

**2**

votes

**0**answers

54 views

### What's the advantage of majorization-minimization (MM) algorithm [closed]

The majorization-minimization (MM) algorithm is a framework for convex and nonconvex optimization. When applied to nonconvex optimization, the MM algorithm solves a sequence of convex problems to ...

**0**

votes

**0**answers

36 views

### Nuclear norm maximization

I am trying to solve a nuclear norm maximization problem:
$$\arg \max_{Q \in O(n)} \|WQV^T\|_*$$
where $Q$ is an $n \times n$ orthogonal matrix and $W$ and $V$ are real $d \times n$ matrices. I've ...

**-1**

votes

**2**answers

91 views

### Is finding a local minimizer of a NP-hard optimization problem is still NP-hard [closed]

I was wondering if for a NP-hard optimization problem, I only want to find its local minimizer, is it still NP-hard or NP-hard is only true when trying to find a global minimizer?

**1**

vote

**0**answers

17 views

### Deterministic global solution to find the Optimal-knot placements for continuous piecewise linear functions to fit nonlinear data

I have been searching lately for a deterministic global technique to linearize a nonlinear function with continuous piecewise linear regions.
I've a univariate nonlinear function y=f(x). where f(x) ...

**1**

vote

**1**answer

85 views

### A bound on the number of bilinear functions needed in order to obtain the minmax

For $n\in\mathbb N$, let $\Delta(n)=\{x\in\mathbb R^n:x_i\geq 0, \sum_ix_i=1\}$ be the set of probability vectors in $\mathbb R^n$.
Is there a function $m:\mathbb N\to\mathbb N$ such that for any ...

**0**

votes

**0**answers

45 views

### Question on solving an optimization problem using Variable splitting and ADMM

Tell me if I have found the right approach to the following optimization problem:
$$
min_{x} \frac{1}{2}\left \| Ax-b \right \|_2^2
\\
s.t. \ \ \Phi v=x \ , \ {x^T(1-x)}=0
$$
$A$ and $\Phi$ ...

**1**

vote

**1**answer

61 views

### Envelope theorem for second derivative

I am maximizing a function $f(x,z)$ on $x$ ($z$ is treated a parameter in the maximization). The function $f$ is strictly concave on both variables.
I know how to use the envelope theorem for the ...

**1**

vote

**0**answers

33 views

### Is this QCQP convex or nonconvex

\begin{equation}
\begin{split}
\min_{x\in \mathbb{R}^n}\:f(x)=(1/2)x^{T}Q_0x+c_0^T x
\end{split}
\end{equation}
s.t.
$$
g_i(x)=\frac{1}{2}x^T Q_ix-lmax_i\leq0,i\in\{1,...,m/2\}
$$
$$
...

**1**

vote

**1**answer

76 views

### $0/1$ programming multiple quadratic constraints

If we have an $n$-variable rank $n$-linear system it is clear we can find whether there exists a $0/1$ solution in polynomial time.
If we have an $n$-variable degree $2$ system how many constraints ...

**1**

vote

**0**answers

31 views

### solution of an infinite horizon optimization problem

Give the following formulation:
$\min_{\{x_s(t):\forall s,t\}} \sum_{s \in \mathcal{S}} \mathbf{1}\left(\lim_{T\rightarrow \infty} \frac{1}{T} \sum_{t=1}^T \frac{y_s(t)}{x_s(t)}\leq 1\right)$
$s.t. ...

**1**

vote

**0**answers

28 views

### Finding the Lagrangian dual problem for a quadratic programm [closed]

I've problems to find the Lagrangian dual problem to
\begin{align*}
\min \limits_{x \in \mathbb{R}^n} \; \frac{1}{2} x^{ T} Q x + q^{T} x \\
\text{s.t.} \quad
Ax &=b \\
x &\geq 0
...

**0**

votes

**1**answer

57 views

### Solving a nonlinear optimisation problem

I have the following nonlinear optimisation problem arising in my model.
$$\min \sum_{k=0}^{N-1} (\tau-t_k)^+\quad \text{ s.t. } {\mathbf{x}^\top\mathbf{w}\le W,\ \mathbf{x}\ge0}, t_k=t_{k-1}+x_k ...

**1**

vote

**0**answers

167 views

### Minimization of nonlinear integral operator

See also on MSE.
For non-negative self-adjoint traceclass operators $0\leq T \leq 1$ with $\mathrm{tr}T^\alpha=N$ on the Hilbert space $L^2(\mathbb{R}^3)$ s.t. $\operatorname{tr}(-\Delta ...

**2**

votes

**0**answers

86 views

### Solve non-linear Optimization Problem [closed]

I have to find $x$ that minimizes: $$ \sum_{k}(x^H\textbf A_kx - b_k)^2$$ where $A_k$ are 4 x 4 positive definite matrices ($A_1, A_2,...A_k$), $x$ is 4 x 1 vector and $b_k$ are scalars ...

**4**

votes

**0**answers

176 views

### Comparison of Constrained Optimization Methods

I am trying to solve a constrained optimization problem using filter methods and came across two papers on the topic that I am having some problems with. The original filter method paper is the ...

**5**

votes

**0**answers

78 views

### max-min optimization problem

I'm curious if there is any nice way to approach solving the following kind of optimization problem. Given a $n \times m$ matrix $A = (a_{ij})$, I want to solve
\begin{align*}
& \max_{c}\min_{1 ...

**0**

votes

**1**answer

105 views

### Convert general optimization problem to LP problem

I am trying to convert the following problem into a linear programming problem:
There are $M\times N$ matrix $T$ of real numbers between 0 and 1 and $N\times 1$ vector $w$ of real numbers between 0 ...

**2**

votes

**0**answers

51 views

### How to approximate higher-degree multivariate polynomial in space of lower-degree multivariate polynomials with some constraints?

For a polynomial $P_{1}(x)$, $x\in {\mathbb R}^n$ with a higher-degree, how to find a lower-degree polynomial $P_{2}(x)$ with determined structure or bounded degree to approximate it with the ...

**0**

votes

**0**answers

22 views

### Maximizing modular function subject to supermodular constaint

I'm trying to solve a constrained optimization problem with submodular functions and get some nice properties of the solution. Unfortunately, I think I am in a setting where Topkis' theorem does not ...

**0**

votes

**0**answers

46 views

### How to use the property of Frobenius norm in this proof?

Let $A \in \mathcal{S}^{n}$($\mathcal{S}^{n}$ is the $n \times n$ symmetric matrix space), $P,Q \in\mathbb R^{n \times n}$, and $\rho \geq 0$ be given. Show that:
$$A \succeq P^{T}ZQ+Q^{T}Z^{T}P$$
for ...

**1**

vote

**1**answer

64 views

### Quadrature formula for singular integrals over rectangular cuboids

A)Find the maximum of the following :
$$\int_{\Omega_{a,b,c}}\int_{\Omega_{a,b,c}}\frac{dV(X)dV(Y)}{\|X-Y\|^2}$$
where ${\Omega_{a,b,c}}=[0,a]\times[0,b]\times[0,c]$, given $abc=1$ with $a,b,c>0$.
...

**1**

vote

**0**answers

108 views

### Lower bound on the value $\textbf{1}^Tx$ such as $Ax\geq b$

The problem may be formulated as follows:
We are given a set of $m$ positive numbers $\{b_1,...,b_m\}$ and a set of $n$ positive numbers $\{v_1,...,v_n\}$. We have $v_j\leq K$, $j=1,...,n$, for a ...

**2**

votes

**0**answers

70 views

### Learning rule for recurrent neural network with flexible time steps

Summary: I want to train a recurrent network to output some answers, but the recurrent network is allowed to re-iterate through itself a flexible number of times for each input-output pair.
Why this ...

**2**

votes

**1**answer

85 views

### Better alternative to solve quadratic programming for large matrices

I have the following problem. Let's say we have $x_{jk}$ it is an expression value of gene $j$ in a sample $k$. It is the average of expression levels across the cell types $s_{ij}$, weighted by ...

**3**

votes

**1**answer

129 views

### Computational complexity of low rank SDP

Suppose we are given a general SDP of the form with an additinal rank requirement
\begin{array}{rl} {\displaystyle\min_{X \in \mathbb{S}^n}} & \langle C, X \rangle_{\mathbb{S}^n} \\ \text{subject ...

**2**

votes

**0**answers

112 views

### Minimize L-infinity norm with restrictions

I need to minimize the following L-infinity norm with respective to $\tau$. L-infinity norm of a matrix $A$ is defined as $\|A\| = max_{i,j}|a_{i,j}|$.
$$
min_{\tau} \| I -S(S+\tau)^{-1}\|
$$
$$
...

**1**

vote

**0**answers

73 views

### reconstructing a linear order corrupted by noise

Suppose we have a partial order (efficiently computable), $\leq$, on $\mathbb{R}^n$, a set $S \subset \mathbb{R}^n$, and let $\rho$ be the standard Euclidean metric. We want to find a set $S^\prime = ...

**2**

votes

**0**answers

75 views

### A (non-convex) minimization quadratic programming problem with d constraints

Minimize $0<\omega_{dd}<2$ subject to
$$\sum_{j=1}^{d}(\omega_{dj} - \omega_{ij})^{2} \geq 4, i=0,1,...,d-1,$$
where $-2<\omega_{ij}<2$ is known for $0 \leq i \leq d-1$ and $1 \leq j \leq ...

**5**

votes

**1**answer

157 views

### Bounding function of norms in constrained vector space

$v$ is a vector of length $n$, where $v_1 = 1$ and every element $v_i \in [0,1]$
$w = \| v \|_1^1 = \sum_i |v_i| = \sum_i v_i$
$x = \| v \|_2^2 = \sum_i |v_i|^2 = \sum_i v_i^2$
$y = \| v \|_3^3 = ...