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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8 views

Hessian matrix of the mahalanobis distance wrt the Cholesky decomposition of a covariance matrix

I'm stuck with the following problem: I have to compute the second derivative (hessian matrix) of the mahalanobis distance $$ [x-\mu]^{T} \Sigma^{-1} [x-\mu] $$ wrt to the Cholesky decomposition of ...
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9 views

Chemical Reaction Networks: Applications

I have recently read a bit on chemical reaction network theory. I was wondering whether the mathematical concepts have cross field applications like neural networks. For example, can I apply chemical ...
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0answers
9 views

How to linearize the multiplication of optimization variables?

I have to linearize the objective function of my optimization problem which is in the form: $$\text{Maximize }\prod_{n=1}^{N}x_n $$ where, $x_n\ge 0$ and $N>2$.
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25 views

Sum/product of two functions of two variables are to be minimized

I have two functions $f(x,y)$ and $g(x,y)$ whose sum/product (whichever is possible) is to be minimized. The values of $x,y$ can vary in the interval $0<x,y<1$ (hence none of them can have a ...
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1answer
21 views

Least Squares Sensitivity to data

Let ($x_1$,$y_1$),...,($x_n$,$y_n$) be my data set. I have a function $f(x,{\bf c})$ where ${\bf c}=(c_1,...,c_m)$ is a vector of $m$ parameters. I want to fit to the data using non-linear least ...
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3answers
73 views

A convex optimisation problem involving the Euclidean norm

Any ideas on how to approach the following optimisation problem? $$\begin{array}{ll} \text{maximize} & \|Ax\|_2^2+\|Bx\|_2^2+\|Cx\|_2^2 \\ \text{subject to} & \|x\|_2 = 1\end{array}$$
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17 views

Quasiconvex objective function with nonlinear constraint

I have the following optimization problem: $$\min \frac{F}{A}+c_p D K \sqrt{A}+c_g \Theta^2+c_e a D + t e (1-\Theta) D+ \frac{t a}{A}$$ s.t. $a \leq [e (1-\Theta) D A] p_a$ $$\Theta \leq p_\Theta$$ ...
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28 views

Lasso with non-linear objective

I have a non-linear objective function that I want to minimize considering some constraints in order to obtain a sparse solution (lasso type). min f($\theta$) s.t. $\sum_i|\theta_i|\leq t$ $\theta_i ...
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1answer
18 views

floor/ceiling/round functions in the constraints of an optimization?

I have a constrained optimization problem in which I have to impose a "floor" or "ceiling" constraint to the solution. In fact I decided to use these nonlinear rounding functions because I needed to ...
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2answers
33 views

Change of variables in minimization

I have the following non linear programming to solve: $$\left\{\begin{matrix} \min & (x-y)^2 +e^z+e^{-z} \\ \text{s.t.} & xz=0 \\ & yz=0 \end{matrix}\right.$$ The book suggests to ...
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1answer
47 views

How can I experiment with Lagrange multiplier in QCQP?

Suppose we want to solve following optimization problem (it is a PCA problem in this post) $$ \underset{\mathbf w}{\text{maximize}}~~ \mathbf w^\top \mathbf{Cw} \\ \text{s.t.}~~~~~~ \mathbf w^\top \...
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39 views

Why the original MINLP and Linearized MILP are giving mismatched results?

I have an MINLP and its linearized formulation problem given below where the objective (nonconvex) and constraint C4 are nonlinear. We linearized them by applying some known techniques. However, when ...
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19 views

Book Recommendation for Infinite Dimensional Stochastic Optimization Problem in Discrete Time

Let $X(k)$ be i.i.d. discrete random variables and for all $k=0,1,2,...,N-1$, let $X:=(X(0),X(1)...,X(k))$ and $f := (f(0), f(1),...,f(k))$ with $f(k)$ be the decision function at time $k$, I want to ...
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1answer
45 views

Minimizing a strictly convex function with inequality constraint

So we've been learning about the Kuhn Tucker conditions in my non-linear optimization course and I've been having trouble with this problem: QUestion: description here Question: a strictly convex ...
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147 views

Optimal decomposition of discrete function into sum of factorised terms

I am trying to solve the following optimisation problem. Let $x_i \in \{1, \ldots, N_i\}$ be discrete variables, and $f(x_1, \ldots , x_n)$ any real-valued function. I want to decompose $f$ into a ...
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0answers
22 views

What does it mean if the KKT conditions do not result in a real solution for a convex problem?

Given a convex optimization problem with equality and inequality constraints, the KKT conditions are sufficient and necessary conditions for optimality. What does it mean if the KKT conditions do not ...
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19 views

modelling a composite objective function (max + argmax) as an (integer) linear program

Suppose $\mathbf{x} = [x_1, x_2, \ldots, x_n]$, where $x_i \in \{0, 1\}$ are binary variables. We know for a fixed $\mathbf{w}$ the following problem is an Integer Linear Program: $$ \arg\max_{\...
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20 views

Smallest enclosing cylinder

I have a set of 3D points that approximately lie on a cylinder. This cylinder is straight and can be oriented in any direction. I would like to compute the minimal enclosing cylinder for the set; taht ...
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0answers
20 views

linearize average success probability constraint

In my optimization problem, I've a constraint to calculate the average success probability of a path. $x_{i,j}$ is binary variable defined as: $$ \begin{align} \label{eq3:1} x_{i,j} = \begin{cases} ...
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1answer
170 views

Optimize nonlinear model over time period (w/ Kuhn-Tucker Conditions)

I'm working on a nonlinear model that includes a time interval of 12 months. The goal is to maximize the total net benefit (NB) over the entire time period given the constraints listed below. I've ...
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34 views

Maximum of Contour Line

Consider the potential $U(x,y)=ay^{2}+b(e^{x-y}-1)^{2}+c(e^{x+y}-1)^{2}$ where $a$, $b$ and $c$ are known constants. I want to move through a contour line of this potential $U(x,y)=k$, say $y=g(x)$. ...
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13 views

reference request quadratic optimization problem

I have this problem and it seems similar to something people must have studied in quadratic optimization/non-convex optimization. $\min_{a,b \in [0,1]^n} a^TM b\\ \text{subject to. } a^TQb\geq \alpha$...
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19 views

Non linear functional optimization under constraints

For some given positive functions $l(t)>0$ and $h(t)>0$, such that $h(t)>l(t)$, I want to solve this functional optimization problem on $a(t)$: $\min_a\int_0^T[l(t)\cdot\min(a(t),0) + h(t)\...
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33 views

Does $\min\{x_1x_3+x_2^2 | x_1^2+x_2^2+x_3^4 = 4\}$ has an optimal solution?

Does $\min\{x_1x_3+x_2^2 | x_1^2+x_2^2+x_3^4 = 4\}$ has an optimal solution? I think continuous function over closed and bounded domain has an optimal solution but I am not sure. Can anyone give me ...
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1answer
24 views

Solving an optimization problem with a linear objective and quadratic constraint

The title is general, but what I am specifically interested in, is how to solve the following problem: $$\text{Maximize } c $$ $$\text{Subject to:}$$ $$a+b+c<0$$ $$b^2-4ac<0$$ $$a,b \in \...
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50 views

What are some applications of vertex separators?

What are applications of finding a vertex separator that minimizes a cut in a graph. To clarify the problem I am talking about is is given a graph of n vertices and a partition $m_1,m_2,..,m_k $of ...
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35 views

Constrained Non-Linear Least Squares Solver

I need C# code for solving constrainted non-linear least squares problems. I'm prepared to write the code myself, but I need to understand the algorithm first. Can anyone describe a constrained non-...
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1answer
35 views

Intersection of a helicoid and a line

I have a Helicoid described by the following parametric equations: $$x = u\cos(v)$$ $$y = cv$$ $$z = u\sin(v)$$ The helicoid revolves around the y-axis: Eliminating $u$ and $v$, we obtain the ...
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21 views

How to programatically solve the optimal control problem?

I have to programatically (write a program) find a control function $u(\cdot)$ to minimize the following functional: $$ J(u,x) = \int_0^T { f_0(x(t), u(t), t)}dt + \Phi(x(0)) \rightarrow \min$$ ...
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1answer
36 views

Local minimums and maximums of function of three variables

I got such function: $f(x_1, x_2, x_3) = x_1 x_2 x_3(4-x_1-x_2-x_3)$ I need to find all local minimums and maximums of this function. I calculated partial derivatives and I got that the only points ...
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38 views

Quadratic problem with two vectors linked by one quadratic constraint

I would like to find $$\min_{w,b} w_iA_{ij}w_{j} + b_iB_{ij}b_j + 2\alpha_iw_i + 2\beta_ib_i $$ Constrained to: $$ w_i > 0 $$ $$ \sum w_i = 1 $$ $$ b_i=w_i^2 $$ Where $A$ and $B$ are positive ...
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2answers
43 views

how to solve this non linear ode

$$ y y'(x) +y(x)^2(\sqrt {x^3}+{7\over4}\sqrt {x^5}+{1\over2}\sqrt {x^7})-{1\over2x}=0 $$ How to solve this equation?? I searched text book , and I only found bessel, legandre. But they are not same ...
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2answers
36 views

Solving Optimization Problem (Orthogonal Projection) Using Projected Sub Gradient / Dual Projected Subgradient

Given the following optimization problem (Orthogonal Projection): $$ {\mathcal{P}}_{\mathcal{T}} \left( x \right) = \arg \min _{y \in \mathcal{T} } \left\{ \frac{1}{2} {\left\| x - y \right\|}^{2} \...
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1answer
42 views

Is Lagrangian Multiplier Equivalent to Brute Force for binary decision variables

I have a set of variables $x_{i} \in \{1,k\} $ in a non linear optimization problem. As this variable has only two possibilities I have encoded this into a constraint. I assumed having equality ...
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44 views

How can I solve this optimization problem?

How do I solve this optimisation problem? $$W = \left(\frac{n(X-Y-Z)p}{Zq}\right)^{1/a},\, a>0$$ $\operatorname{Max}\{ W\}$, subject to $0\leq n \leq 1$, $0\leq Y \leq X$ and $Z \leq Z_{max}$ ...
2
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1answer
37 views

Falsi regula using maple

Hi I'm using falsi regula algorithm in maple. For first function it worked fine : restart; epsilon := 1e-3: f := x->x^3+x^2-3: a:= 1: b:=2: step:=infinity: while abs(step) >= ...
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11 views

Backpropagation in a Convolutional Layer

Is my assumption correct that in backpropagation we use cross correlation for both: gradient for weight update and gradient of the error signal? So in Matlab I may use these equations?: calculate ...
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1answer
89 views

Constrained optimization problem of 4 variables!

I am stuck with this problem. I thought of trying to first solve the problem with weak inequalities for all the constraints using Kuhn Tucker conditions, and checking for solutions at which the ...
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1answer
24 views

How to solve this problem through bisection search or any other method?

I have an optimization problem in the form $$\text{Minimize}\hspace{1mm}D$$ $$\text{subject to}$$ $$\sigma_1+\sigma_2=\sigma$$ $$\rho_1+\rho_2=\rho$$ $$\epsilon\le\rho_i\le c_i\hspace{1mm},i=1,2$$...
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2answers
67 views

How do I solve the following equality-constrained quadratic program?

I am trying to minimize: $$(x_1-k_1)^2 + (x_2-k_2)^2 + (x_3-k_3)^2 +\ldots+ (x_n-k_n)^2$$ subject to following equality: $$B = 1 + x_1 + x_2 + x_3 + x_4+\ldots+x_n.$$ Is there a closed form ...
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33 views

Index of a stationary point of constrained optimization

For an unconstrained optimization problem with objective function $F(x,y,z)$ the index of a stationary point is well-defined: If $(x^*, y^*, z^*)$ is a point where the gradient of $F(x,y,z)$ vanishes, ...
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32 views

How to derivative the function which have constraint like $x^2+y^2+z^2 = 1$

For example , we have function like $f(x,y,z) = 1 - 2\times x^2 + y + z$ with constraint $x^2+y^2+z^2 = 1$ when I need to compute the derivative of $\frac{∂f}{∂x}$ , should it be $\frac{∂f}{∂...
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23 views

Nonlinear optimization with constraints; is changing variables an reliable approach?

I have a optimization problem as follows, $$ \begin{array}{cll} [\hat{x_1},\hat{x_2},\hat{x_3}] = & \text{argmin}_{x_1,x_2,x_3} \sum_{i = 1}^N \sum_{t = 1}^T \left[ \ln(f_{i,t}(x_1,x_2,x_3)) + \...
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1answer
68 views

Weight Modification for Computationally-Efficient Nonlinear Least Squares Optimization

There was a time where I could figure this out for myself, but my math skills are rustier than I thought, so I have to humbly beg for help. Thank you in advance. I am solving a weighted nonlinear ...
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2answers
80 views

Problem with finding Karush-Kuhn-Tucker points and checking for global or local minima.

I need to solve the following optimization problem $$\begin{align*} & \mathrm{Min}:\quad f(x_1,x_2)=x_1-10x_2\\ & \mathrm{subject \ to}: \quad x_1^2 -x_2 \geq 0\\ & \qquad \qquad \...
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1answer
17 views

Which optimization method when Hessian is singular?

I am trying to optimize a non linear function of four variables for which the Hessian matrix is always singular (pairs of columns / lines are colinear). I wanted to use a Newton method until I checked ...
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3answers
87 views

Hottest and coldest points on a heated circular plate (use Lagrange multipliers)

A circular plate given by the relationship $x^2 + y^2 \leq 1$ is heated according to the spatial temperature function $T(x,y) = 2x^2 + y^2-y$. Find the hottest and coldest point on the plate using ...
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1answer
34 views

How to convert a non-linear constraint to a linear constraint for integer programming?

I have non-linear scheduling model and I want to convert it to a linear model. But I have no idea about how can I do it. The nonlinear constraint is: For each $i, i'\in I$ and $j, j' \in J$ and $q, ...
1
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1answer
55 views

Optimization problem: $\min \limits_{\mathbf{q}} \sum_{n=1}^N q_n$, s.t. $\frac{c_{nn} q_n }{\sum_{m \ne n} c_{nm} q_m } \ge a$

\begin{array}{rl} \min \limits_{\mathbf{q}} & \sum_{n=1}^N q_n \\ \mbox{s.t.} & \frac{c_{nn} q_n }{\sum_{m \ne n} c_{nm} q_m } \ge a, \forall n \in \{1,\ldots,N\} \end{array} For this ...
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0answers
13 views

How to solve an optimization problem with objective function as a time average expectation

I have an optimization problem with the objective as $$ \overline{h(x)}=\lim_{T \to \infty} \sum_{t=0}^{T-1} E[f(x)] $$ where $$ h(x)=\frac{f(x)}{g(x)} $$ and $f$ is convex and $g$ is linear. I ...