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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29 views

Lagrange multiplier method

Question 1: Could somebody please refer me to an introduction to Lagrange multipliers which is easy to read but still in full generality? Question 2: I am interested in particular in the following ...
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18 views

How can I compute fast the minimum of a linear plus Kulback-Leibler on the unit simplex?

Given $a, x^0 \in \mathbb{R}^n$ I wish to compute $$\min_{x \in \Delta_n} a^t x + \sum_{i=1}^n x_i\log(x_i/x^0_i) - x_i +x^0_i $$ where $\Delta_n$ is the unit simplex $\{x \in \mathbb{R}^n \mid ...
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25 views

Gauss-Newton Non-Linear Squares Optimisation

I doubt this is solvable at all, but I thought I will give a try. Essentially I am trying to extend Gauss-Newton algorithm to 2nd Taylor term. ...
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16 views

Book on duallity and sensitivity in nonlinear optimization

I am looking for a recommended book on duallity and sensitivity in nonlinear optimization, as duallity and sensitivity is a well studied topic in LP , I am struggeling to find books in this subject ...
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13 views

Maximize a concave function under nonconvex constraints

I have to maximize the rate, which is a concave function, under certain constraints, where one of them is not convex; My optimization problem is: $\max_{\mathbf{P}_{2,n}} \frac{B}{L} \sum_{k=1}^L ...
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15 views

How do I know that method of steepest descent works?

Here is the definition of the method of steepest descent given in the book "The mathematics of nonlinear programming" by Peressini. Suppose $f(x)$ is a function with continuous partial derivatives on ...
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23 views

Global stochastic maximization of a multi-parameter function

I have a function $F:\mathbb{R}^n\to[0,1]$ such that $$ F(\lambda) = \mathbb{E}_x[f(\lambda;x)] = \int f(\lambda;x)\mu(x)dx,$$ and I want to find $\tilde\lambda$ that maximizes F, i.e. ...
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gradient descent to solve binary non linear optimization problem [on hold]

I am trying to code a solution for an optimization problem that has binary matrix which has to be optimized,since the problem is not convex and has binary variables,i am finding it hard to solve ...
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16 views

Solving Nonconvex Programming using Genetic Algorithm

Consider such a programming problem, \begin{equation} min \| \Psi(\boldsymbol{x}) \|_p \quad s.t. \quad \boldsymbol{Ax} = \boldsymbol{b} \end{equation} Where $\boldsymbol{x} \in \mathcal{R}^{n}$, ...
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42 views

How to maximize $\sum\limits_{i=1}^n u_iln(x_i)$?

How to maximize this? $$ \sum\limits_{i=1}^n u_iln(x_i), $$ where $u_i,x_i$ are real numbers, $n$ is a positive integer, $0 \leq u_i \leq 1, 0 < x_i < 1, \sum\limits_{i=1}^n u_i = 1, ...
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75 views

No clear analytic method to prove unique maximum? ($2^{-x}+2^{-1/x}$)

Prove that $f(x) = 2^{-x}+2^{-1/x}$ has the unique local maximum $(1,1)$ for $x>0$. Do not use computer software. Proving that $(1,1)$ is a maximum is easy, but I'm having trouble with the ...
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27 views

How to prove a solution is indeed a constrained minimum?

I'm reading the following example on Heath's Scientific Computing (page 266, second edition if anyone has it). "Minimize $f(x_1,x_2)=2\pi x_1(x_1+x_2)$ subject to $g(x_1,x_2)=\pi x^2_1x_2-V$" ...
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14 views

Linear programming (or possibly nonlinear) formulation

The problem is like this; The construction company is considering erecting three office buildings. The time required to complete each of them and the number of workers required required to be on the ...
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20 views

Epsilon constraint method - Pareto optimal solution representation

There's a course that I do remotely and I have a homework question which I have no idea how to answer. I did look up a lot in google and did not find any good examples - only loads of information and ...
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1answer
30 views

Non-uniqueness of worst-case (max-min or min-max) optimization

I have a worst-case optimization problem, where i want to maximize the minimum from the uncertainty set (uncertainty is given as an ensemble of 100 realizations, so an ensemble based approach). It is ...
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9 views

Can this equation have an explicit solution?

Given $n > 0$, $0 \leq i \leq n$ is an integer, $D = diag(d_1, \dots, d_n)$ is positive definite, $e_i$ is the $i$th column of a $n \times n$ identity matrix, $u \in R^n$ such that $B = D + u * ...
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38 views

How do we plot nonlinear differential equations

If this is not nonlinear I apologize, I'm still learning differential equations. I am attempting to make a stream plot of a predator-prey model of eccentric closed curves by using the following ...
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51 views

good extensive library for covex optimization

I need a java library to solve a convex optimization problem, kindly give me few names that are extensively used. I have googled my query but since I am new and have no prior experience in solving ...
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34 views

Optimization on fixed sum

Consider this following scenario. Suppose I have $N$ cents, and I want to dispatch these money to $n$ people, each got $x_i$ cents. In order to simplify this problem, we assume the cents are ...
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14 views

Dual norm of quasi norms

The dual norm $\Omega^*$ of the norm $\Omega$ is defined for any vector $\mathbf{z} \in \mathrm{R}^N$ by \begin{equation} \Omega^*:= \underset{\mathbf{x} \in \mathrm{R}^N}{max } \quad \mathbf{z}^{T} ...
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19 views

How to optimize these parameter

How to optimize the following respect to lambda1 and lambda2: $\sum_{i} f(i) * log(\lambda_1 g(i) + \lambda_2 h(i))$ f(i), g(i), h(i) are known funtions Find lambda1 and lambda2 that satisfy ...
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19 views

Necessary Conditions for Saddle Value point

This questions is from the Kuhn-Tucker paper "Nonlinear Programming" in Section 2 Lemma 1. I don't understand how those conditions are necessary for a saddle point. I always thought that a saddle ...
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39 views

How to convex problem with multiple constraints? $\underset{\mathbf{A}}{\text{minimize }} \|\mathbf{A}\|_F^2$

IF $\mathbf{X=AS}$ where $\mathbf{X} \in R_+^{n \times m}$, $\mathbf{S} \in R_+^{r \times m}$ are known variable and $\mathbf{A} \in R_+^{n \times r}$ is unknown variable. here $R_+$ denotes the set ...
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39 views

Non-convex function with global minimum [duplicate]

I am working on a complicated objective function which I suppose is not convex. But when I use a global optimization tool that can find all its local minimums, it will always converge to the same ...
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1answer
40 views

Maximum likeliood estimation of variances of transformed variables

I use MATLAB's fminunc function in order to find the minimum of a negative log-likelihood function $f(\overrightarrow{\theta})$, parametrized by 3 parameters lets say ...
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28 views

How can I minimize a quadratic on the unit simplex?

How can I compute $$ \min_{x \in \Delta_n} \frac{1}{2}\lVert Bx\rVert^2 + x^tAy$$ with $x \in \mathbb{R}^n, y \in \mathbb{R}^m, A_{m \times n}$, $B_{n \times n}$ where $\Delta_n$ is the unit simplex ...
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18 views

Maximizing the frobenius norm subject to constraints $\underset{\mathbf{S}}{\text{maximize }} \|\mathbf{S}\|_F^2$

IF $\mathbf{X=AS}$ where $\mathbf{X} \in R_+^{n \times m}$, $\mathbf{A} \in R_+^{n \times r}$ are known variable and $\mathbf{S} \in R_+^{r \times m}$ is unknown variable, How to solve the below ...
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18 views

KKT Sufficient condition when optimal solution is intuitively at the boundary

My optimization problem is: $\operatorname{arg\,max}_P \sqrt P$ subject to $P \le \upsilon_\tau$ where $P \in \mathbb{R}^+$ and $\upsilon_\tau \in \mathbb{R}^+$ Intuitively, because $\sqrt P$ is ...
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8 views

optimization of formulas involving binomial coefficients

I encountered such a problem. We need to find the min value and max value of $f(x,y)$. $x$ and $y$ are integers $\in[0,n]\times[0,n]$ and $(x,y)\neq (0,0)$ or $(n,n)$. $$ ...
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How can I solve $\min \{ \langle A(x),y\rangle + f(y) \text{ s.t. } y \in S^n, \operatorname{tr}(y) =1, y \geq 0\}$?

I'm trying to solve the problem $$\min \{ \langle A(x),y\rangle + f(y) \mid y \in S^m, \operatorname{tr}(y) =1, y \geq 0\}$$ where $x \in \mathbb{R}^n$, $y \in S^M$, that is, it's a symmetric $m$ by ...
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171 views

Global and local maxima in a weighted sum of logarithms of linear functionals?

Is is possible to describe, and locate efficiently, the maxima of the function below in the parameters $\mathbf{x}$ $$\sum_{i} p_i \log( N + \sum_j x_j[B_j +(A_j-B_j)\delta_{ij} + min(A_j,B_j) ]) ...
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43 views

Minimize a non-convex function subject to linear dynamics constraint

I want to solve the following problem: $$\min\limits_{\bf u} \frac{\bf c^T {\bf x} (T_f)}{\| \bf c\|\|{\bf x} (T_f)\|}$$ subject to $$\dot{\bf x} (t) = A {\bf x}(t) + B {\bf u}(t)$$ $$x(0) = x_0$$ ...
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33 views

Find a line such that sum of perpendicular distances of points to the line is minimized

Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...
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46 views

Is the optimum of this problem unique?

I have a convex optimisation problem: $$\min_{x_i} \sum_{i=1}^N a_i \exp(-x_i/b_i)\quad\text{ s.t. }\sum_{i=1}^N x_i=x\text{ and } x_i\ge 0$$ The KKT conditions are: $$\lambda=\begin{cases} ...
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61 views

Solving non-linear optimization using generalized reduced gradient (GRG) method

Consider the following elementary maximization problem: \begin{align} f{=}\mathrm{argmax}_{y_{l,c}, p_{l,c}}~\sum_{l=1}^{L}\sum_{c=1}^{C} y_{l,c}\text{log}_2\left(1+\frac{p_{l,c}}{I_{l,c}}\right) ...
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34 views

Question about duality in nonlinear optimization

Let $f(x)$ and $h(x)$ be functions from $\mathbb{R}^n$ to $\mathbb{R}$ and consider the minimization problem $$ {\rm minimize} ~~~ f(x)$$ $$~~~~~~~~~{\rm subject ~to}~~h(x)=0.$$ Suppose the minimum is ...
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59 views

Minimize Product of Sums of Squared Distances

The Question Given two sets of vectors $S_1$ and $S_2$,we want to find a unit vector $s$ such that $$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...
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22 views

Solving three-variable nonlinear equation systems

A physical problem which I've been studying leads to the following nonlinear equation system to be solved: $$\alpha\cdot79\cdot A_1 +(1-\alpha)\cdot 1025 \cdot B_1 = C_{11}$$ $$\alpha\cdot145\cdot A_1 ...
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How to find more than two coefficient for single variable nonlinear equation?

I don't have good knowledge on mathematics, but now I faced one problem with maths. That is, I have a data set which contains only one independent and one dependent variable. Now I have a equation ...
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how to write largest circle inscribed inside a triangle as an optimization problem?

can someone show me how to write this problem as a convex optimization problem.Find the largest disk that can be bounded by $X \geq 0$ , $Y \geq0$ and $X+2Y\leq1$. My institution is to cast to ...
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45 views

Minimize the squared dot product of two specific vectors

Do you think there exists a efficient algorithm(non brute-force) for the following problem. I search the optimal solution for the following problem: Given a vector $u=(u_1, u_2,..., u_k)^T$ with ...
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52 views

Is standard eigenvalue optimization problem convex

For any arbitrary symmetric matrix A , is the standard eigenvalue problem convex $ \lambda_{max}(A)= \max_{\|x\| \leq1} x^{T}Ax$
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33 views

Tucker's theorem from Farkas lemma

I am trying to understand the proof of Tucker's theorem using Farkas lemma but there are some points that are not clear to me. The proof I am following is in this paper at page 16. What I do not ...
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1answer
46 views

Non-linear system vs minimisation problem

If you have a non-linear system of equations which can be formally written as : \begin{equation} \begin{cases} F_1(\mathbf{x})=0\\ F_2(\mathbf{x})=0\\ \ \ \ \ \vdots\\ F_n(\mathbf{x})=0\\ \end{cases} ...
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12 views

Broyden's Method mismatched dimensions

Compute the first two iterates $\mathbf{x}^{(1)}$, $\mathbf{x}^{(2)}$ using Broyden's Method for the initial point $\mathbf{x}^{(0)}=(1,4)$ and the function $f(x_{1},x_{2})=3x_{1}^2+x_{2}^2-x_1x_2$ ...
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49 views

Performing Second Derivative test on multivariate function

I have two functions $f=xy^2$ and $g=x^2+y^2$. When optimizing $xy^2$ on the circle $x^2+y^2=1$ I get 6 critical points but when I try to perform the second derivative test, it equals 0, meaning that ...
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29 views

what is the good book to learn nonlinear programming

I would like to learn nonlinear programing. what is the best book to do so and I prefer if the solution manual of the book is available. thanks
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102 views

Optimization over vector spaces. Generalized KKT.

I am looking for the extension of the theorem I found in the book by Luenberger called "Optimization by vector space methods." Here is the statement of that theorem from Luenberger: Generalized ...
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24 views

Calculating the Hessian for nonlinear objective functions

In least squares minimization, we minimize the sum of squares of the residuals: $f = ||b-Ax||_2 ^2 $ The Hessian of this matrix is given by $2A^TA$, and from the Hessian we can do the second partial ...
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34 views

How to solve coupled nonlinear ODEs with a algebraic constraints?

Here is the problem I'm currently facing right now, I have set of ODEs which I can solve numerically given the initial condition. But I'm not sure how to go about giving algebraic solution with ...