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

**-1**

votes

**1**answer

47 views

### About the critical points of quasi-convex functions

What do we know about the structure of critical points of quasi-convex functions?
I am looking for statements like "the critical points of a quasi-convex function are always either a global minima ...

**1**

vote

**0**answers

27 views

### Difference between Chebyshev first and second degree iterative methods

Consider linear equation $Au = f$.
We want to solve it with iterative method (assuming $A$ is good).
First order iterative method is:
$$
u^{k+1} = u^k - \alpha_{k+1}(Au^k - f),
$$
The second degree ...

**1**

vote

**0**answers

28 views

### About a particular definition of a Hessian of a function of tuples of matrices

Say I have a function $L : (W_1,..,W_{H+1}) \rightarrow \mathbb{R}$ i.e it takes a tuple of $n$ matrices of different dimensions and computes a number from them.
Then I see being defined a ...

**6**

votes

**0**answers

157 views

### Analytic expression for the Tsirelson bound of the I3322 inequality?

Finding Tsirelson bounds for Bell inequalities is a well-loved problem in quantum information theory. A famous case where it is still open is for the I3322 inequality. In this paper Pál and Vértesi ...

**7**

votes

**0**answers

115 views

### Solving matrix equation $X^{-1}=\sum_{i=1}^n D_i X A_i$

Does anybody know an algorithm to solve the following matrix equation?
$$X^{-1}=\sum_{i=1}^n D_i X A_i$$
where $D_i$s are diagonal and $A_i$s are symmetric matrices.
It would be great to have an ...

**0**

votes

**0**answers

39 views

### optimize a Quadratic Matrix Programming with multi-spherical constraints

I have got the following quadratic problem restricted on the Cartesian product of Euclidean spheres.
$\underset{X \in \mathbb{R}^{n\times 3}}{\text{min}}$ $Q(X) = \frac{1}{2} Tr(X^TA X) + Tr(B^T X)$
...

**0**

votes

**1**answer

30 views

### Limits of argmin ratios and sums

In my research on convergence properties of certain Bayesian methods I have encountered $\mathop{\mathrm{arg\,min}}$ limits of the forms
\begin{equation}
\lim_{n\to\infty} \mathop{\mathrm{arg\,min}}_{\...

**1**

vote

**0**answers

48 views

### Posterior consistency of non linear model

This is possibly a reference request. Let $G$ : $\mathbb{R}^p \to \mathbb{R}^q$ be a continuous injective/bijective function. Let $\mu$(we may also assume this to be a non degenerate Gaussian) be ...

**1**

vote

**0**answers

54 views

### max min of ratio of quadratic forms

Consider the optimization over two vectors $x$ and $y$
$$\max_{x,y} \min\left(\frac{x^TAx}{y^TAy},\frac{y^TBy}{x^TBx}\right)$$
for two positive definite matrices $A$ and $B$.
This problem can be ...

**0**

votes

**1**answer

53 views

### Quadratically constrained quadratic programming/optimization involving piece-wise function

I have a quadratically constrained quadratic programming/optimization problem involving kind-of piece-wise quadratic functions $f_n (x_m)=a_{n,m} (x_m-\theta_n)^2$, if $|x_m-\theta_n|<c$; $c^2a_{n....

**0**

votes

**0**answers

17 views

### Finding orthogonal basis with constraint

Is there any fast algorithm that output an orthogonal basis $e_i,i\leq n$ of $R^n$
with $e_i\in V_i$? Where $V_i,i\leq n$ are given linear subspaces of $R^n$.
And is there any condition on $V_i,i\leq ...

**0**

votes

**0**answers

43 views

### Largest instance of highly nonlinear benchmark functions (e.g. Rastrigin function)

What is the largest instance size (number of variables) ever numerically solved for highly nonlinear (continuous, not combinatorial) optimization benchmarks functions, such as Rastrigin, Schwefel or ...

**0**

votes

**0**answers

31 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

31 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

35 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

102 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

**2**answers

78 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 or ...

**0**

votes

**0**answers

51 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

58 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 $Q_1{...

**3**

votes

**0**answers

154 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:
$a_1=\frac{1+a_3-\sqrt{a_2a_3}\sqrt{v_1}}{1+\sqrt{\frac{a_3}{a_2}}\sqrt{v_1}}$
$a_2=\frac{2-a_3-\sqrt{...

**3**

votes

**1**answer

137 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 Y_{k,\ell}\frac{1}{|c_{\ell}|^2+|c_{k}|...

**0**

votes

**0**answers

52 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

130 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 non-...

**0**

votes

**1**answer

76 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} \right]\mathbf{x}_k\mathbf{x}_k^H\...

**2**

votes

**0**answers

28 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

64 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 X\subseteq\...

**3**

votes

**0**answers

49 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

121 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

109 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

63 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

45 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

99 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

21 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

104 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

50 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

87 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

42 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.
$$
g_i(x)=\frac{1}{2}x^T Q_ix-lmax_i\leq0,i\in\{1,...,m/2\}
$$
$$
g_i(x)=\frac{...

**1**

vote

**1**answer

102 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

33 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

30 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
\end{...

**0**

votes

**1**answer

58 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

172 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 T^\alpha)=\...

**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 ($b_1,b_2,......

**4**

votes

**0**answers

179 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

86 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

112 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

56 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

23 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

48 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 ...