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3
votes
1answer
104 views

minimizing $\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$

Suppose there are $n$ points $(x_i, y_i)$ for $i = 1,\ldots,n$. Please find another point $(x, y)$ to minimize function: $$\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$$
3
votes
1answer
327 views

Solving a system of non-linear (trig) equations:

I am having trouble trying to solve the following equations: $\sin(\alpha)+\sin(\beta)=\dfrac {1000} A$ $\sin(\alpha)+\sin(\gamma)=\dfrac {800} A$ $\dfrac {20(1+\cos(\alpha-\beta))} {\cos(\beta)} ...
1
vote
0answers
135 views

Optimization: Minimizing Quadratic Vector Valued Functions

I have a vector valued function $F: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m}$, which consists of quadratic taylor approximations. So one could say that function F consists of stacked approximations ...
0
votes
0answers
92 views

How to optimize a function with several variables

I need to develop code to optimize a set or variables based on the following conditions. I don't have the source of function. The function gets a point (x,y) and generate a mapped point (x',y') ...
1
vote
0answers
45 views

Formulating square packing as a form of optimization

I was looking at square packing problem which is defined as: Given a number N... Find the smallest square that can pack N unit squares Each square can be associated with a 3 dimensional point ...
1
vote
2answers
31 views

how do i find $\max\{x+z\}$ and $\max\{1+y^2\}$ where ?$x\ge0 $,$ y\ge0$,$ z\ge0$ and$xy+xz+yz=1 $

how compute $\max\{x+z\}$ and $\max\{1+y^2\}$? such that $x$,$y$,$z$ satisfied $$\begin{cases} xy+xz+yz=1 & \\ x\ge0 \\ y\ge0\\ z\ge0\\ \end{cases} $$ i face with this problem when i try ...
1
vote
1answer
113 views

Gradient descent/ nonlinear optimization intuition needed

all. I'm taking an introductory AI class, and we're using the gradient descent algorithm to find the optimized/ lowest cost of a set of thetas (variable coefficients) to best fit a regression line. In ...
3
votes
3answers
93 views

How do one solve a nonlinear combinatoric problem?

I am an undergraduate CS student and I am struggling with a problem. $Qx = b$ where $Q$ is a constant $m \times n$ matrix (with $m>n$), $x$ is a $n \times 1$ vector and $b$ is a $m\times 1$ ...
2
votes
2answers
424 views

How can I solve Lagrange multiplier equation with multi constraints?

This site is really awesome. :) I hope that we can share our ideas through this site! I have an equation as below, $$ min \ \ w^HRw \ \ subject \ \ to \ \ w^HR_aw=J_a, \ w^HR_bw=J_b$$ If there is ...
0
votes
2answers
118 views

Fastest Algorithm for NLP 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
495 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 ...
0
votes
0answers
61 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
votes
1answer
67 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 ...
6
votes
1answer
154 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 ...
3
votes
0answers
65 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 ...
0
votes
0answers
57 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
vote
2answers
173 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
255 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
275 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
votes
1answer
260 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 ...
2
votes
0answers
65 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 ...
0
votes
1answer
54 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
votes
0answers
64 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
vote
2answers
451 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 ...
0
votes
1answer
42 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 ...
1
vote
3answers
161 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
1answer
98 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 ...
3
votes
1answer
152 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 $x,y,z \geq 0$. ...
1
vote
0answers
81 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
106 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
265 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
vote
1answer
76 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
0answers
38 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) = ...
5
votes
0answers
194 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
votes
1answer
621 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 $B.C$. ...
0
votes
2answers
52 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 $G(x)$ ...
0
votes
2answers
77 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 ...
0
votes
1answer
572 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
votes
1answer
147 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
votes
0answers
60 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: ...
0
votes
1answer
54 views

Distributed Robust Optimization

Consider the following constrained optimization problem $\mathcal{P}$. $$ \min_{x \in X \subseteq \mathbb{R}^n} f(x) \ \text{sub. to: } g(x,y) \leq 0 \ \forall y \in Y \subseteq \mathbb{R}^m $$ ...
1
vote
1answer
68 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
votes
0answers
53 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 ...
0
votes
1answer
242 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 ...
3
votes
1answer
399 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
votes
1answer
32 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
132 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
votes
1answer
145 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 ...
2
votes
0answers
32 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 $c \mu - \log\det(B)$ ...
6
votes
3answers
478 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 ...