The nonlinear-optimization tag has no wiki summary.
2
votes
2answers
687 views
Using KKT conditions to maximize function
The goal is to maximize the following function:
\begin{align}
K_p(q) = q\log \frac{q}{p} + (1-q)\log \frac{1-q}{1-p}
\end{align}
where
\begin{align}
0 \leq q \leq 1
\end{align}
and $p \in (0,0.5)$ and ...
2
votes
0answers
19 views
Solver for sparse linearly-constrained non-linear least-squares
Reposted from stackoverflow on the advice of Nick Rosencrantz:
Are there any algorithms or solvers for solving non-linear least-squares problems where the jacobian is known to always be sparse, and ...
2
votes
0answers
18 views
System of many non-linear (quadratic) first order O.D.E. (numerical strategy or simplification)
I have a large system (N>100) of equations
$\frac{d\vec{P}}{dt}= A(t) + B(t) \vec{P} + \vec{P}^T C(t) \vec{P}$
where $\vec{P}$ is a vector of N functions of the variable t.
What is the correct ...
2
votes
0answers
44 views
Nonlinear optimization of constraint parameter - subdifferential?
Disclaimer: I discovered that the FAQ suggests to post research-level to mathoverflow instead of math.stackexchange. I "moved" the question accordingly, cp. post at mathoverflow. Sorry for the ...
2
votes
1answer
58 views
Is this linear programming?
I have the following problem and I'd like to know if it is formalizable as a LP program.
(or, more generally, if it is solvable in polynomial time).
Let us fix some terminology first which will ...
2
votes
0answers
103 views
SDP relaxation of non-convex QCQP and duality gap
Short version
Is there a duality gap between a QCQP problem and the SDP problem obtained through lagrangian relaxation?
A paper I'm studying is using this fact, but I cannot achieve the authors' ...
2
votes
1answer
143 views
Python numerical solution for a nonlinear second order ODE with two boundary conditions
I want to solve numerical the next equation, in Python
$$u''(x) = \left( a - \Big(b\big(u(x)^{2}\big)\Big) \right) \big(u'(x)\big)^{3}$$
it is a nonlinear second order $ODE$ with two $B.C$. ...
2
votes
0answers
49 views
Linearization of a dynamic system, $\small 10 \frac{d^2 y}{dt^2} + \frac{d y}{dt} = u^2(t) $
I have a dynamic system, with in signal u and out signal y. The system is described with the following differential equation
$$
10 \frac{d^2 y}{dt^2} + \frac{d y}{dt} = u^2(t)
$$
The task is to ...
2
votes
2answers
75 views
A Quadratic Problem (which looks very simple)
This arises as a part of my work.
\begin{align}
\min_{x^{H}x=1}~&x^{H}A_1x \\
subject~to~&x^{H}A_2x=0
\end{align}
$A_1$ and $A_2$ are $N\times N$ hermitian matrices and $x$ is a unit norm ...
2
votes
0answers
179 views
Minimizing with Lagrange multipliers and Newton-Raphson
I am writing a program minimizing a real-valued non-linear function of around 90 real variables subject to around 30 non-linear constraints. I found handy explanation in CERN's Data Analysis ...
2
votes
0answers
324 views
Convex minimization over the Unit Simplex
I have a simple (few variables), continuous, twice differentiable convex function that I wish to minimize over the unit simplex. In other words, $\min. f(\mathbf{x})$, $\text{s.t. } \mathbf{0} \preceq ...
2
votes
4answers
462 views
Does Slater's condition hold for the following problem?
Does Slater condition hold trivially (because there are no inequality constraints) for the problem:
$$\min_{x,y} \:\: cx+dy$$
s.t.
$$e^x + e^y = 1.$$
Can I conclude there is a zero duality gap ...
2
votes
1answer
29 views
Smooth Reformulation of NonSmooth Constraints
If I have something like :
\begin{align}
\min_x \max_i f_i(x)
\end{align}
I can reformulate this nonsmooth formulation as:
$$\min_x z$$
$$z\geq f_i(x)$$
and I have a smooth formulation of the problem.
...
1
vote
4answers
66 views
A non-linear maximisation
We know that $x+y=3$ where x and y are positive real numbers. How can one find the maximum value of $x^2y$? Is it $4,3\sqrt{2}, 9/4$ or $2$?
1
vote
3answers
67 views
What is the dual of this optimization problem?
Consider the points $x_1, \ldots, x_N \in \mathbb{R}^n$, and a (locally bounded, convex) function $f: \mathbb{R}^n \rightarrow \mathbb{R}$.
I am looking for the dual of the following optimization ...
1
vote
3answers
184 views
Summary of Optimization Methods.
Context: So in a lot of my self-studies, I come across ways to solve problems that involve optimization of some objective function. (I am coming from signal processing background).
Anyway, I seem to ...
1
vote
2answers
541 views
lagrange multiplier for more than 2 equality constraints
i couldn't do the following question for hours
minimize $\sum_{i=1}^{n}x_{i}^{3}$
s.t. $\sum_{i=1}^{n}x_{i}=0$
and $\sum_{i=1}^{n}x_{i}^{2}=n$.
by Lagrange multiplier rule
?
1
vote
5answers
124 views
Independence of Rotation Matrix Definitions
I am trying to solve a system of non-linear equations. I know that 9 of my variables put together form a 3x3 rotation matrix
$$
A = \left(
\begin{matrix}
a_{11}& a_{12}& a_{13}\\
a_{21}& ...
1
vote
1answer
52 views
Find a decoupled explicit formula for a minimizer
Consider the energy
$F(u,v) = \int^1_0((\frac{1}{4}(u')^2+(v')^2 +\frac{1}{2}(u-v+1)^2)dx$
for $C^1$ functions u and v on the interval (0,1) that satisfy the boundary conditions ...
1
vote
3answers
248 views
Constrained optimization: equality constraint
I have this very general problem (for $n>2$):
$$
\begin{align}
& \max Z = f(x_1,\ldots ,x_n) \\[10pt]
\text{s.t. } & \sum_{i=1}^{n} x_i = B \\[10pt]
& x_i \geq 0
\end{align}
$$
Assume ...
1
vote
1answer
200 views
LP relaxation for ILP\IP (integer linear programming)
I am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a ...
1
vote
2answers
146 views
Find the range of $x$, given $y_{min} \leq y(x) \leq y_{max}$, where $y(x) $ can be any function ( Updated)
I have a series of inequalities:
$$y_{1min} \leq y_{1}(x) \leq y_{1max}$$
$$y_{2min} \leq y_{2}(x) \leq y_{2max}$$
$$..$$
$$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$
Note that $x\in\mathbb{R}$
The ...
1
vote
1answer
69 views
Cost minimization problem
The problem is as follows:
A firm uses $k$ units of capital and $l$ units of labor to produce
$(k^{\alpha}l^{1-\alpha})^{1/\beta}$
units of output, where $\alpha$ and $\beta$ satisfying $0 < ...
1
vote
2answers
95 views
Maximize the product of linear functions
Suppose $f(x,y) = \prod_{i=1}^n (a_ix+b_iy)$
where $n$ is a constant larger than 500, and $a_i>0$, $b_i>0$ are known coefficient. There is only one global maximum.
What's the most efficient ...
1
vote
1answer
45 views
Reformulation of BQP to SDP
I run into the following reading some optimization papars:
$$\min_x x^TAx $$ where $x\in\{-1,1\}^n$ and $A\in S_n$,
Is equivalent to
$$ \min <X,A>$$
s.t $diag(X) = (1,1,...,1)\;\; rank(X) = 1$.
...
1
vote
1answer
49 views
(system of) nonlinear equations and instability
I heard that a system of nonlinear equations is unstable.
I am curious of how "instability" is defined, and why do nonlinear equations show instability?
Edit:
OK, so what about contexts in matrices ...
1
vote
1answer
151 views
Comparison of nonlinear system solvers?
I am dealing with nonlinear systems of equations that I am trying to solve numerically. These sets of equations derive from structural mechanics involving strong nonlinearities, like contact. The size ...
1
vote
1answer
104 views
Does a local minimum of a function always satify the Armijo rule
Does a local minimum of a function always satify the Armijo rule?
1
vote
1answer
138 views
Combinatorial Optimization Problem (can I/how do I solve this with integer programming?)
Inputs:
1) A set of M x N matrices, {A,B,C...N} containing only integers.
2) A single 1 x N matrix of floats, W (weights).
I need to pull one row from each input matrix and sum values for each ...
1
vote
2answers
37 views
Solve the Lagrangian dual problem
Consider the (non-linear) optimization problem ($P$)
$$max \quad3x_1 + 4x_2$$
$$s.t. \quad x_1^2 + x_2^2 \leq 25$$
$$ \quad x_1,x_2 \geq 0$$
Solve the Lagrangian dual problem.
I ...
1
vote
2answers
87 views
Classifying local behavior of fixed points using eigenvalues from linear stability analysis of 3D system
I've learned about classification of fixed points of 2D systems using linear stability analysis and I am wondering how if at all I can apply the same process to analyzing local behavior of fixed ...
1
vote
1answer
36 views
Quadratic Forms in Non-Linear Optimization
This is a rather trivial question but I am having a great deal of trouble:
Let $f(x) = (1/2)xQx-xb$
and $E(x) = (1/2)(x-x^*)Q(x-x^*)$
then $E(x) = f(x) + (1/2)x^*Qx^*$
where $x,x^*,b$ are vectors ...
1
vote
1answer
39 views
Distance between a point to a $2d$ ellipse in $3d$ ambient space
Suppose we are working in the 3D Euclidean space. We are given an arbitrary point $p$ and a 2d ellipse:
$$E=\{x:x^TQx\leq1,x^Tq=0\},$$
where $Q$ is a positive definite matrix and $q$ is an ...
1
vote
1answer
53 views
Numerical/artifical damping in forward Euler?
I'm testing a code to find periodic solutions of nonlinear structural vibrating systems by solving a global time-discretized periodic system of equations. I am using a forward Euler (first order ...
1
vote
1answer
68 views
Check Kuhn-Tucker conditions
How to check if $(0,1)$ point is the solution of this optimization problem using Kuhn-Tucker Theorem.
Find the min of $e^{x_1-x_2}-x_1-x_2$ where $x_1+x_2\le1,\ x_1\ge 0,\ x_2\ge0$
I am thinking ...
1
vote
1answer
101 views
Max function on a closed compact convex set.
Consider a closed convex compact subset $\mathbb{S}$ of $\mathbb{R}^N$ while we denote any of its point by $x=[x_1,x_2,\ldots,x_N]^T$. Define the function
\begin{align}
f(x)=max(x_1,x_2,\ldots,x_N)
...
1
vote
1answer
136 views
A sufficient condition for a unique maximum of the product of two concave functions
Given two concave functions $f(x)$ and $g(x)$, what conditions in terms of these functions can ensure that
$h(x)=f(x)g(x)$
have a unique maximizer on an interval $[a,b]$ for $a<b$?
1
vote
1answer
132 views
Optimization of multivariate non-linear function with linear constraint
I have this huge and ugly function.
$$
f\left(x,y\right)=
$$
$$
...
1
vote
1answer
84 views
Is nonlinear conjugate gradient a quasi-newton optimization technique?
Can the non-linear conjugate gradient optimization method with Polak-Ribier line-search choice, be named as a quasi-Newton optimization technique?
If not, why?
1
vote
1answer
121 views
Fixed point stability of piecewise linear system
I have an autonomous system of nonlinear equations of the form:
$$Mx'' + C(\omega)x' + K(\omega)x + F_{nl}(x) = 0$$
where $M$ is the mass, $C$ the damping and $K$ the stiffness matrix. ...
1
vote
1answer
61 views
Least squares and (non-)linearity of parameters
I have a question about least squares and about what happens, if the function that we minimize, $E(P)$, is not linear in its parameters $P$.
Assume we want to minimize a function (the exact terms are ...
1
vote
1answer
54 views
Transform the sample to make it more similar to a given
$X=\{x_{i}\}$ and $Y=\{y_{i}\}$ are numeric samples: $y_i \ge 0, x_i \ge 0, i \in [0..N]$.
I need to find the mapping $F(X)=\{F(x_i)\}$ with fairly simple formula such that:
Euclidean distance ...
1
vote
1answer
362 views
Optimizing Nonlinear Constraint Equations with Discrete Variables and Multiple Objective Functions
I have the following constraint functions:
$$g_{i_{min}} \leq y_{i+1}-y_{i} \leq g_{i_{max}}$$
$$y_{i_{max}}-y_{i} \geq h_{i}$$
$$v_{i_{min}} \leq \Biggl[\frac{(y_{i+1}-y_{i})^{3} ...
1
vote
0answers
27 views
Is duality theory in optimization as useful as it seems?
I have been reading a lot about nonlinear optimization and duality, and it seems that duality theory is extremely useful. I feel that I am missing some of the negative aspects/difficulties associated ...
1
vote
0answers
19 views
A fundamental question for the linearity of saddle point condition
I have a simple question about the linearity of saddle point condition. We have $$f(h_0,h_1,h_2)=k_1f_0(h_0,h_1)+k_2f_1(h_0,h_2)$$
and we also know that
$$f(\hat{h}_0,h_1,h_2)\leq ...
1
vote
1answer
69 views
How to draw a fixed length curve?
Is it possible to draw a curve with some specified length between two points? I'm considering damped sines like WolframAlpha or Bezier curves.
1
vote
1answer
61 views
Convex Optimization of quadratic function with inequality constraints
How would I solve the following problem?
$$\min_{x\in\mathbb{R}^n} x^T A x$$ subject to the constraints $$x_i\geq 1,\,i=1,\dots,n,$$
where A is positive semidefinite and symmetric. Is it possible to ...
1
vote
0answers
41 views
Formulation of a problem as semidefinite programming
I would appreciate some help with this problem:
$R$ is a positive semidefinite matrix $\in{R}^{n\times n}$, $A \in{R}^{n\times m}$.
I need to formulate this optimization problem as semidefinite ...
1
vote
0answers
23 views
Boltz method for solving normal equations
Recently I came across an interesting comment in a geodetic paper which follows as:
"Initially, the
normal equations were solved using the Gaussian method
of successive elimination. This method, ...
1
vote
0answers
31 views
Sequential problem for n=1, non linear regression
I am trying to understand an example in my stats course notes, the example relates to calculating the best value for the next experiment.
The function of the line is very simple:
$$ln(Y_i) = ...
