The nonlinear-optimization tag has no wiki summary.
2
votes
1answer
81 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 ...
0
votes
0answers
38 views
A minimization problem [migrated]
Define $$L(w,u)=\frac{1}{2}\|w-u\|^2+\beta \|\frac{w}{x}\|,~w,u\in R^n$$
where $$\frac{w}{x}=(\frac{w_1}{x_1},\dots, \frac{w_n}{x_n})$$
$$\|x\|=\sqrt{x_1^2+\cdots+x_n^2}$$
Given $u$, $x$ and $\beta$, ...
0
votes
0answers
27 views
An example of the Sequential Quadratic Programming (SQP) [migrated]
Given an objective function $f(x,y)=(x+6)^2+(y-10)^2$, we want to minimize the function $f$ under a constrained condition $xy-5\leq0$. Obviously it is a constrained optimization, a good algorithm to ...
0
votes
0answers
31 views
Optimization with differential inequality constraint
Consider the closed set $[t_1,t_2]⊂R_{>0}$ and $V(t):[t1,t2]→R_{>0}$ being a continuous and piecewise continuously differentiable function. We want to find a continuously differentiable function ...
0
votes
1answer
51 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 ...
0
votes
0answers
19 views
Minimum of a real-valued function of multiple variables
Suppose we are given two functions $$\phi(x) = \frac{\sigma^2}{2\mu^2} \left(1 - e^{-(2\mu/\sigma^2)x}\right) - \frac{x}{\mu}，x\in \mathbb R$$ and $$g_0(x) = \frac{\sigma^6}{4\mu^4} ...
1
vote
0answers
104 views
Recovering a partition from spectral properties of the graph Laplacian
Let $G$ be a weighted graph with vertices $V$. Let $W$ be its real-valued, non-negative, $|V|\times|V|$ adjacency/affinity matrix. Let $L = \mathrm{diag}(W\mathbf1)-W$ be the (unnormalized) graph ...
2
votes
1answer
118 views
Optimization problem - maximizing number of satisfied linear inequalities subject to a quadratic constraint
I am wondering what is known about optimization problems of the following type.
Our control x is a unit vector in $\mathbb{R}^n$. We are given a finite number of linear inequalities
$$Az≥b,$$
and we ...
0
votes
0answers
49 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 ...
2
votes
1answer
49 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
0answers
92 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 ...
3
votes
3answers
134 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
1answer
366 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
1answer
122 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
0answers
63 views
Optimal instance of quadratically constrained program
Consider the following optimization problem. Let $n, m \in \mathbb N$ and $0 < p_1 \leq \ldots \leq p_n ~ (p_i \in \mathbb R)$ be constant. The feasible region is described by a partition $T_1, ...
0
votes
0answers
106 views
minimum of some functions
Denote $U=\{(x_1,x_2,...,x_n):x_j\in(0,1) (1\le j\le n),\sum_{j=1}^nx_j=1\}$.
Let $f_i=f_i(x_1,x_2,...,x_n)$ ($1\leq i\leq n-1$) be $n-1$ real functions which satisfy:
...
0
votes
1answer
58 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 ...
1
vote
1answer
121 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
0answers
37 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 ...
1
vote
1answer
119 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
0answers
48 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 ...
1
vote
2answers
148 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
1answer
234 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
1answer
143 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
2answers
422 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 ...
4
votes
1answer
167 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
1answer
109 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, ...
2
votes
1answer
139 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
2answers
98 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 ...
0
votes
0answers
96 views
Non-linear optimzation with multipe objectives and multiple inequality constraints
I posted an earlier version of this question on math.stackexchange almost a week ago but have not had any replies. I'm recasting the problem in light of new understanding. I'd like to know if the the ...
