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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Why does Frobenius norm make BFGS scale-invariant?

On slide 11 here it is claimed that the weighted Frobenius norm leads to a scale-invariant optimization method. Similar claims about this norm can be found throughout the literature see 1,2,3. In ...
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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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9 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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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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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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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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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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26 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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28 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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84 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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179 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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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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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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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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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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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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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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41 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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51 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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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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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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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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35 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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66 views

How to obtain the optimal Lagrange multiplier vectors if the globally optimal solution for a nonconvex QCQP is found?

I am using a black-box solver to solve the following non-convex QCQP to global optimality. $$ \min_x x^TQ_0x + c^T x \\ s.t. \quad x^TQ_1x+c_1^Tx=b_1 \\ Ax=b \\ l\leq x\leq u $$ where $Q_0$ is ...
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550 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' ...
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1answer
41 views

Non-convex QCQP

Consider the following optimization problem: $$\begin{array}{ll} \text{minimize} & \mathbf{x}^{T} \mathbf{A} \mathbf{x}\\ \text{subject to } & \mathbf{x}^{T} \mathbf{P}_i \mathbf{x} > 0, \...
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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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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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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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Relaxation of non-convex QCQP with one quadratic and one linear constraint

According to Boyd we know that a non-convex QCQP problem with one quadratic constraint has strong duality with the relaxed SDP or Lagrange counterpart. (check "Convex Optimization" by Boyd, Appendix B)...
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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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40 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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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
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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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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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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How do I construct the Jacobian for use in a Levenberg-Marquardt algorithm.

I am working on a 3D reconstruction system and I am looking to use a Levenberg-marquardt algorithm to do bundle adjustment. I am not too sure about how LM works and what it requires. The model I am ...
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45 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 $Z \leq Z_{max}$ ...
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38 views

How to solve the following optimization problem with projection?

How to solve the following optimization problem with projection? \begin{alignat}{1} &\min_{u_+,u_-,s,l\geq 0} \frac{1}{\lambda} \langle A ,(a +u_+-u_-)(a +u_+-u_-)^\mathsf{T} \rangle+\mathbf{1}^\...
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2answers
48 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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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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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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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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588 views

Scaling factor and weights in Unscented Transform (UKF)

I'm trying to implement the UKF for parameter estimation as described by Eric A. Wan and Rudolph van der Merwe in Chapter 7 of the Kalman Filtering and Neural Networks book: Free PDF I am confused by ...
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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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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
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 \...