A non-linear optimization problem includes an objective function (to be minimized or maximized) and some number of equality and/or inequality constraints where the objective or some of the constraints are non-linear. Use this tag for questions related to the theory of solving such problems or for ...

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how to minimize this convex function?

$x_i$ and $y_j$ are variables. I intend to minimize this function and obtain the optimal value of $x$ and $y$: $\begin{align} ...
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240 views

maximizing a function of a positive semi-definite matrix with bounded trace

I need to maximize a function $f(A)$ where $A$. With the constraints that $A$ is positive definite and has a trace $tr(A) \leq K$. $tr(A)=K$ will work for my problem too. I can differentiate towards ...
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38 views

On solving non-linear programming problem and the relevant software

I have a non-linear programming problem, in which all the inequality is linear and only the optimization goal is in a non-linear form. The problem is as following. $x_j$ is the variables and $a_{k,j}$ ...
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146 views

Distinction between linear and nonlinear model

[I have already asked this question on CrossValidated but until now received no answer] I have read some explanations about the properties of linear vs nonlinear models, but still I am sometimes not ...
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67 views

Approximating the optimal value of a function involving a Gaussian integral

Consider the following function $$ f(\lambda) = \alpha (1+\lambda^2) + (1-\alpha)2\int_\lambda^\infty (x-\lambda)^2 \phi(x) dx $$ where $\alpha \in (0,1)$ and $\phi$ is the standard normal probability ...
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88 views

Methods to minimise multilinear functions with trilinear, quad-linear and higher-linear terms?

My goal is to minimize functions such as $$f_1(\mathbf{p})=p_1p_3p_7+p_1p_4p_7+p_2p_3p_7+p_2p_4p_7-p_1p_3p_5p_6-p_1p_4p_5p_6-p_2p_3p_5p_6-p_2p_4p_5p_6$$ and ...
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51 views

how to solve this optimization problem with functions?

Suppose $\theta\in[0,1]$ and $s(\cdot)$ is strictly increasing in $\theta$ with $0\leq(0)<s(1)\leq1$ and $s(0)+s(1)>1$ How could I solve the program, $\max_{s(\cdot)\in ...
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64 views

Linear Complementarity Problem - multiple solutions, which one will it find?

If I have a inequality constrained system: w = Mz + q <= 0, z<=0, z^T w = 0 that for some given properties M and ...
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146 views

Dimension analysis and explaining the $\varepsilon$

Reference of this post (page no 6 from equation 39) The time averaged total energy, $\bar E$, has the following $\varepsilon$ expansion in $D$ dimension: \begin{equation} ...
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1answer
297 views

optimization of a non-differentiable, component-wise step function

I would like to estimate the (local) minimum of a function $c:R^N \mapsto R^+$ where: $c$ is only differentiable almost everywhere, there exists a component $j$, such that $\frac{\partial ...
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1answer
491 views

Optimizing trigonometric and nonlinear functions

First, Please, keep in mind that I'm a programmer not mathematician, and I have a fair mathematical background. I used optimization in Java to fit some observations to a trigonometric function, I ...
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163 views

Optimization for constrained problem

I'm reading about Lagrange multipliers from a Pattern recognition book appendix and on one point the following is stated: $\begin{align} ...
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284 views

iterative method to solve nonlinear equations

I'd like to know whether there are any methods like the Gauss-Seidel method to solve nonlinear equations. For example, I'd like to solve $f(\textbf x) = 0$, where $f(\textbf x)$ is a nonlinear ...
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1answer
56 views

Approaches to fitting noisy oscillatory data?

I have observations $\hat{f}$ from data at points $\mathbf{x}=\{x_1,\ldots,x_N\}$, that is modeled as a known oscillatory form $f(k\ x)$ (for example, the sinc function), where $k$ controls the ...
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2answers
2k views

Which optimization algorithm converges faster?

everyone. I'm having a large scale unconstrained optimization problem. If I treat the unconstrained problem as a constrained problem with infinity constraints, I should be able to use both the ...
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1answer
46 views

Directive on Dimensionality Reduction

I have a data set (24 data records) which is in $\mathbb{R}^{13}$ and I need to project it to a lower dimension (at least to $\mathbb{R}^{3}$). My objective of the dimensionality reduction is to ...
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1answer
67 views

A minimization problem

Define $$L(w,u)=\frac{1}{2}\|w-u\|^2+\beta \left\|\frac{w}{x}\right\|,~w,u\in \Bbb{R}^n$$ where $$\frac{w}{x}=\left(\frac{w_1}{x_1},\ldots, \frac{w_n}{x_n}\right)$$ $$\|x\|=\sqrt{x_1^2+\cdots+x_n^2}$$ ...
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1answer
48 views

A minimization problem [duplicate]

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 $\beta$, ...
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1answer
1k views

Optimization of a function of two variables.

Suppose we have to minimize a function $f(\mathbf x,\mathbf y)$ where $\mathbf x$, $\mathbf y$ are vectors in Euclidean space. The function is convex in $\mathbf y$ when $\mathbf x$ is kept constant ...
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86 views

Steepest Descent/Newton

Suppose these over-determined system of equations: $$ |\mathbf{x}^T\mathbf{v_n}| = A, \qquad n = 1,2,\cdots,N-1 $$ $$ \mathbf{v_n}= [1 \quad w^n \quad w^{2n} \quad \cdots \quad w^{(N-1)n}]^T , ...
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784 views

An example of the Sequential Quadratic Programming (SQP)

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 ...
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1answer
130 views

Minimize a nonlinear sum subject to a quadratic constraint

Currently I am solving an optimization problem that could be written as follows: $$\min J= \sum_{i=1}^N {(q_i^H\Lambda q_i)}^{\frac{1}{3}} $$ subject to $\{q_i\}_{i\in [1..N]}$ forming an ...
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122 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: $$\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$$
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592 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)} ...
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207 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 ...
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100 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') ...
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72 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 ...
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2answers
33 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 ...
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168 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 ...
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102 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 $m\times 1$ ...
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1k 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 ...
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159 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) = ...
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942 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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81 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 ...
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84 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 ...
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1answer
217 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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90 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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103 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 ...
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216 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.
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1answer
389 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 < ...
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2answers
440 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 ...
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373 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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83 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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76 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 + ...
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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 ...
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2answers
704 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
44 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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246 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 ...
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1answer
132 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 ...
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209 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 $x,y,z \geq 0$. ...