The nonlinear-optimization tag has no wiki summary.
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17 views
Fastest Algorithm for QP with linear constraints
I have an minimization problem of the following form:
(Im not a mathematician, i come from the programming side, so excuse me if i have not the perfect standard of writing the formulas)
$Z(x) = ...
1
vote
2answers
37 views
Solve the Lagrangian dual problem
Consider the (non-linear) optimization problem ($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 ...
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0answers
25 views
Optimization problem - maximizing number of satisfied linear inequalities subject to quadratic constraint
I am wondering if anything 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
$$A z ...
2
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1answer
36 views
Formulate the Langrangian function of a non-linear optimization problem and solve it for $y\geq0$
Consider the (non-linear) optimization problem ($P$)
$$max \quad3x_1 + 4x_2$$
$$s.t. \quad x_1^2 + x_2^2 \leq 25$$
$$ \quad x_1,x_2 \geq 0$$
Formulate the Lagrangian function ...
1
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0answers
27 views
Is duality theory in optimization as useful as it seems?
I have been reading a lot about nonlinear optimization and duality, and it seems that duality theory is extremely useful. I feel that I am missing some of the negative aspects/difficulties associated ...
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0answers
23 views
maxima of the sum of unimodal functions .
I have a set of unimodal functions. Each function has real roots. All roots of each function lie outside a certain limit points. These limit points are the same for each function. Each function is in ...
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0answers
19 views
A fundamental question for the linearity of saddle point condition
I have a simple question about the linearity of saddle point condition. We have $$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 ...
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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 ...
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0answers
17 views
an upper semi-continuous semi-strictly quasi-concave function
May you help me?
I was confused that why "an upper semi-continuous semi-strictly quasi-concave function is quasi-concave".
If there is a picture,it may be better.
I really need your help ～
Thank ...
1
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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
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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
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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 ...
0
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1answer
25 views
Positive semidefinite Matrix examples query
This might be really dumb question but I've just started dealing with such matrices. I would like to know why $$\begin{bmatrix} 0 & 1 \\ 1 & x \end{bmatrix}$$ cannot be a positive semidefinite ...
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0answers
34 views
Minimization/maximization of Sum-Product cost functions
I am a little rusty in optimization. Typically in digital detection/decoding one encounters a sum-product cost function. Mathematically, this may represented by
\begin{align}
J(X)=\sum_i\prod_j ...
2
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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 ...
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0answers
14 views
Hessian approximation in BFGS optimization technique
BFGS (Broyden–Fletcher–Goldfarb–Shanno) optimization method is used where Hessian of object function is hard (or impossible) to calculate. Here consequtive sequence of Hessian approximation are built ...
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1answer
24 views
KKT formulation
How to reformulate the following problem
$$min \frac{1}{2} (x_1-a_1)^2+ \frac{1}{2} (x_2-a_2)^2$$
$$s.t. \mathbf{1}^Tx=1$$
$$ ||x||_2\leq2$$
as the following system of KKT conditions:
$$(1 + ...
2
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0answers
44 views
Nonlinear optimization of constraint parameter - subdifferential?
Disclaimer: I discovered that the FAQ suggests to post research-level to mathoverflow instead of math.stackexchange. I "moved" the question accordingly, cp. post at mathoverflow. Sorry for the ...
1
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2answers
87 views
Classifying local behavior of fixed points using eigenvalues from linear stability analysis of 3D system
I've learned about classification of fixed points of 2D systems using linear stability analysis and I am wondering how if at all I can apply the same process to analyzing local behavior of fixed ...
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1answer
12 views
Non-Linear Programming: Max. $(a-\alpha)Q -(b-\beta)Q^2$ subject to $Q\geq 0$
I am required to work on the following optimization problem:
Max. $(a-\alpha)Q -(b-\beta)Q^2$ subject to $Q\geq 0$
How do I do so?
My FOC is $(a-\alpha) -2(b-\beta)Q-\lambda(-Q)=0$ and
...
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3answers
67 views
What is the dual of this optimization problem?
Consider the points $x_1, \ldots, x_N \in \mathbb{R}^n$, and a (locally bounded, convex) function $f: \mathbb{R}^n \rightarrow \mathbb{R}$.
I am looking for the dual of the following optimization ...
0
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1answer
57 views
Gateaux derivative
I have the following definition of Gateaux differentiability
$f$ is Gateaux differentiable at $x_0$ if there is a continuous and linear operator $T$ so that
$$ \lim_{t \rightarrow ...
2
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1answer
42 views
Linear stability analysis on a constrained three-dimensional system of ODE
Let $\begin{cases} \dot x = f({\bf u}) \\ \dot y = g({\bf u}) \\ \dot z = h({\bf u})\end{cases}$ be a well-defined nonlinear system with ${\bf u} = (x,y,z)$ and restricted to domain $x,y,z \geq 0$. ...
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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 ...
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
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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' ...
1
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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 ...
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0answers
27 views
Geometric Interpretation of Kuhn-Tucker
Use the Kuhn–Tucker Theorem to decide which of the following equations/inequalities hold:
(a) $g(x_0) < 0$.
(b) $g(x_0) = 0$.
(c) $\nabla g(x_0) · (x_0 − x^∗) < 0.$
(d) $\nabla g(x_0) · (x_0 ...
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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, ...
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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:
$$ln(Y_i) = ...
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0answers
43 views
Stochastic gradient descent for nonconvex functions
I am trying to optimize a nonconvex function of the form
$$f(x) = \sum_i g_i(x) - h_i(x)$$
where x is a vector of variables, and $g_i$ and $h_i$ are both convex. While I am aware that such a ...
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0answers
158 views
I need help about some compactness arguments
I need help to find some compact sets for my engineering problem. Through this page I learned quite much about it however since I have neither read a book yet nor have an experience I am not able to ...
2
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1answer
142 views
Python numerical solution for a nonlinear second order ODE with two boundary conditions
I want to solve numerical the next equation, in Python
$$u''(x) = \left( a - \Big(b\big(u(x)^{2}\big)\Big) \right) \big(u'(x)\big)^{3}$$
it is a nonlinear second order $ODE$ with two $B.C$. ...
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2answers
41 views
How do I derivate a function in the Norm?
I have a funtion
$$f(x) = \frac12\|G(x)\|_2^2$$ where $G(x): \mathbb R^n \rightarrow \mathbb R^n$ is a twice continuous differentiable funtion. I want to determine $\nabla f(x)$ in terms of $G(x)$ ...
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2answers
62 views
Anyone saw this interesting function before?
Say $\theta\in\Re^n$ and $\theta_i\in(0,1)$ for all $i$. Define
$$
f(\theta) = \frac{1}{n}\sum_i^n\{(1-\theta_i)\log(1-\theta_i)+\theta_i\log\theta_i\}
$$
It is easy to see the minimizer of ...
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0answers
27 views
Non linear optimization Gradient method
Let $f(x)= x^2 -\frac{x^3}{3}$. Ok so i found the local min is at 0 and i was given $x_0=1,\alpha = \frac{1}{2}$, I dont understand how i am suppoused to find $x_k$ such that $x_{k+1}=x_k-\alpha ...
0
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1answer
57 views
Rate of Convergence for Gradient Descent (Example)
I am trying to determine the rate of convergence for $f(x,y) = 5x^2 + 5y^2 − xy − 11x + 11y$. Would anyone be able to provide guidance as to how I might go about doing this? Should I select my own ...
0
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1answer
28 views
Nonlinear method to solve an equation with the error function in it
My question is to find a method to solve the following non-linear equation. I know it should be an iterative method, but I don't know what would be the best method to use. Any help is highly ...
0
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0answers
56 views
Minimum of some functions
Denote $U=\{(x_1,x_2,...,x_n):0<x_j<1 (1\leq j\leq n),\sum_{j=1}^nx_j=1\}$.
Let $f_i=f_i(x_1,x_2,...,x_n)$ ($1\leq i\leq n-1$) be $n-1$ real functions which satisfy:
...
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1answer
37 views
Distributed Robust Optimization
Consider the following constrained optimization problem $\mathcal{P}$.
$$ \min_{x \in X \subseteq \mathbb{R}^n} f(x) \ \text{sub. to: } g(x,y) \leq 0 \ \forall y \in Y \subseteq \mathbb{R}^m $$
...
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1answer
39 views
Distance between a point to a $2d$ ellipse in $3d$ ambient space
Suppose we are working in the 3D Euclidean space. We are given an arbitrary point $p$ and a 2d ellipse:
$$E=\{x:x^TQx\leq1,x^Tq=0\},$$
where $Q$ is a positive definite matrix and $q$ is an ...
2
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0answers
49 views
Linearization of a dynamic system, $\small 10 \frac{d^2 y}{dt^2} + \frac{d y}{dt} = u^2(t) $
I have a dynamic system, with in signal u and out signal y. The system is described with the following differential equation
$$
10 \frac{d^2 y}{dt^2} + \frac{d y}{dt} = u^2(t)
$$
The task is to ...
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1answer
55 views
Lipschitz constant for optimization of multivariate function
I intend to implement an optimization algorithm which requires the computation of the Lipschitz constant. My function is a multivariate function with more than 50 variables. I am wondering whether ...
2
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1answer
85 views
Lagrange Multipliers for Function Spaces
For some constant $A > 1$ I am trying to solve the constrained minimization problem
minimize $F(u)$ in $C$
subject to $H(u) = 0$.
Here $F(u) = \int -u dx$ and $H(u) = \int \sqrt{1 + (u')^2} dx - ...
3
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1answer
28 views
existence of solution of $Ax= \max(b-x,0) $
How do you prove the existence of a solution to the linear system:
\begin{equation}
Ax= \max(b-x,0)
\end{equation}
A is an $n\times n$ matrix and $b$ is a vector in $\mathbb{R}^n$. $x$ is the ...
2
votes
2answers
90 views
optimality of quadratic programming problems
Suppose we have a general quadratic programming problem:
\begin{align}
\min_{x}\,\,&c^Tx+\frac{1}{2}x^TQx,\\
\mbox{s.t.}\,\,& Ax=b,\\
&x\geq0,
\end{align}
where $Q$ is positive ...
0
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1answer
40 views
Sion's minimax theorem
Sion's minimax theorem is stated as:
Let $X$ be a compact convex subset of a linear topological space and $Y$ a convex
subset of a linear topological space. Let $f$ be a real-valued function on ...
1
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0answers
21 views
Convex formulation of a nearly convex optimization problem
The following problem has come up in my studies of logarithmic norms.
I wish to find $\mu \in \mathbb{R}$ and a positive semidefinite $B$ so as to minimize the convex function $c \mu - \log\det(B)$ ...
4
votes
2answers
128 views
Finding good approximation for $x^{1/2.4}$
I would like to a good (8 bits accuracy) approximation for $x^{1/2.4}$ in the range $[0, 1]$. This transform is used for converting linear intensities to SRGB compressed values, so it's important that ...
1
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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 ...
