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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To solve a non-linear equation system with huge amout of variables analytically or numerically

I have such a nonlinear equation system $x_i=\frac{\sum_{j\neq i}a_{ij}\times\sum_{k\neq i}x_k}{\sum_{k\neq i}x_k-\sum_{j\neq i}a_{ij}}$ where $a_{ij}$s are known coefficients in $[0,1]$. And $x_i$s ...
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2answers
43 views

Why is an affine set convex?

I wanted to know why do we say that an affine set is convex? From what I understood, if we take two points $x_1$ and $x_2$ $\in \mathbb{R}$, then, the affine set $A$ defined by these two points will ...
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2answers
88 views

Lagrange's method question

Find the extreme values of the function $f(x,y,z)=x^2+y^2+z^2$ subject to the condition $xy+yz+zx=3a^2$. I tried to solved it by Lagrange method and got $3$ equations. \begin{align*} &2x+k(y+z)...
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1answer
24 views

Estimating errors from optimization? (Genetic algorithm or otherwise)

I have a vector of observations $\vec x_{\text{obs}}$ that have been measured with known uncertainties $\vec \sigma_{x}$. I have a model $f$ that takes parameters $\vec \theta$ and produces values $...
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35 views

nonlinear Lasso with constraints

Just encountered a very strange problem in my research. It's a nonlinear lasso with constraints. The optimization is $\min $ $\sum_{t=1}^{T}f\left( y_{t},x_{t};\beta \right) +\lambda \left\Vert \...
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1answer
30 views

Can an unfeasible solution be optimal in an LPP

In a linear programming problem, Is it possible to have an unfeasible solution that is optimal?
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37 views

Semidefinite programming formulation for a simple minimization

I am trying to formulate following problem (with some constraint) as a semidefinite programming problem (SDP), \begin{equation} \text{minimize } ~~ -a^T B^{-1} a \end{equation} where $B$ is a ...
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18 views

Theory of Adaptive Interaction

Does someone has knowledge on the theory of adaptive interaction? I have read that is a simple and effective way to perform gradient descent in the parameter space. I need to implement a adaptive PID ...
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1answer
71 views

KKT condition just binding and inactive

I have read the textbook saying that if both KKT and Lagrangian multiplier $\lambda$ are $0$, then the constraint is just binding, whereas if KKT multiplier is equal to 0, and Lagrangian multiplier is ...
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34 views

Constrained optimization of $||S - ABA^T||$

Given asymmetric matrix $S_{n\times n}$ we want to decompose it into $A_{n\times k}$ and $B_{k\times k}$ such that $S\approx ABA^T$. (Constraints: columns of B sum up to one while all elements are non-...
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69 views

Finding the value of coefficients of a equation using non-linear least square method.

I have the following data: x: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 y: 22 36 42 51 57 64 68 71 75 79 85 87 88 91 94 97 99 99 103 104 105 107 108 109 ...
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73 views

KKT conditions for a maximization problem

I have an optimization problem \begin{equation} \mathbf{w}^*= \text{argmax} \sum_{d=1}^{D}\log\left(\frac{|\mathbf{\hat{f}}_{d}^{H}\mathbf{w}|^{2}+A_d}{|\mathbf{\hat{f}}_{d}^{H}\mathbf{w}|^{2}+B_d}\...
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50 views

Division of linear functions in convex polytope

I am a computer scientist, and find myself needing the following lemma: If f(x)=(g(x))/(h(x)), where g and h are linear and positive with domain the convex polytope d, then extrema of f occur at ...
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44 views

Sufficiency of KKT problem

I have a continuous, twice differentiable function $f:\mathbb{R}^2\rightarrow \mathbb{R}$. My objective is to maximise the function $f$ subject to quasiconcave function $g_i(x,y)\le 0,\, i\in\{1,2\}\, ...
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31 views

Optimization when the function is not known, how generally it is performed?

I've a question on optimization (it is a general question). I have a problem $P$ where basically i have the analytic expression of the objective function, which is a multivariate polynomial (positive),...
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2answers
71 views

$g$ is coercive for $g(x)=x^TAx+b^Tx+c$

Suppose $A$ is a symmetric positive definite matrix $A\in \Bbb{R}^{n\times n}$, $b \in \Bbb{R}^n$, and c is a real number. Let $$g(x)=x^TAx + b^Tx + c$$ Show that $g$ is coercive. Because $A$ is ...
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70 views

Choose of the parameters related to the Wolfe-Powell conditions

Let $f\in C^1(\mathbb R^n)$ $x\in\mathbb R^n$ and $d\in\mathbb R^n$ with $$\langle\nabla f(x),d\rangle<0\tag{1}$$ Then, $t>0$ is said to satisfy the Wolfe-Powell conditions with parameters $\...
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34 views

mathematical optimization via lagrange multipliers (or other method)

Let \begin{eqnarray} g_k(x_1,\ldots, x_K) = x_k - \log \sum_{j=1}^K \exp x_j. \end{eqnarray} Notice that, for $i\neq k$ \begin{eqnarray} \frac{\partial g_k}{\partial x_i} = -\frac{\exp x_i}{\sum_{j=...
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18 views

Levenberg-Marquardt fitting with multiple $y$ for each $x$?

I am trying to implement LM for my research project. The functions is in the form of $f(x,params) = y$. The issue is that for each value of $x$, there are five values of $y$, even for same $params$. ...
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38 views

Computational methods to minimizing the norm of a matrix monomial.

Linear optimization solves the problem $$\min_{\bf x}\{\|{\bf Ax - b}\|_2^2\}$$ Edit: Some clarification Doing the derivation of the optimum, first expand the norm: $$\|{\bf Ax - b}\|_2^2 = ({\bf ...
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2answers
35 views

$f (x_1,x_2)=x_1^2+x_2^3 $ Finding if a local minimum exist

Why is $(0,0)$ not a local minimizer for $f (x_1,x_2)=x_1^2+x_2^3 $? Because $(0,0)$ is a critical point and Hessian matrix at $(0,0)$ is positive semi definite. Therefore isn't it a local minimum ...
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117 views

Finding global maximizers and minimizers

I want to find if global maximum or minimum exists in $ f(x,y)=e-^{(x^2+y^2)}$ I found that (0,0) is the only critical point. In the Hessian matrix $H_{(f)}(0,0)$ was negative definite and so (...
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169 views

How to determine Coercive functions

A continuous function $f(x)$ that is defined on $R^n$ is called coercive if $\lim\limits_{\Vert x \Vert \rightarrow \infty} f(x)=+ \infty$. I am finding it difficult to understand how the norm of ...
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Is s-energy, specifically logarithmic energy, based on function -log(d) or log(d) with distance d?

There was an earlier question about s-energy; I think I got the point. However, in various problem settings I see one is interested maximising pairwise distances between points and, there they say, ...
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23 views

Are the number of Employed bees and onlooker bees always the same in Artificial Bee Colony optimization?

I have been recently introduced to this Artificial Bee Colony algorithm. Lets consider the initial guess and assume that half of the bees are employed and the rest half are onlookers. Once the ...
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57 views

An optimization problem involving a probability density function

I have three time-series $\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}$. I would like to find a linear combination of the time series, that is, some scalars $a_{1},a_{2},a_{3}$ such that the sum $$\...
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1answer
140 views

How can linearize the product of decision variables in ILP?

Here, we have something like this: R + (1-R)T + (1-R)(1-T)S + (1-R)(1-T)(1-S)Q = 1 where R, T, S, Q are binary decision variable How can I convert this ...
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60 views

Stuck on step in Lagrangian Problem

For $w,E$ column vectors, $i$ the vector of ones, and $\Sigma$ - an $n\times n$ positive definite symmetric matrix, I am trying to solve the following maximization problem: $$ \max_{\{ w\}} \left\{ \...
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20 views

Non-linear programming problem from a trapping region in a nonlinear ODE system?

I have the next function... wich describes a trapping region from a glycolysys model of Sel'kov... I need to minimize $z=h(u)=u^3(\frac{1}{A}-\frac{B}{A^2})+u^2(\frac{\epsilon B}{A^2}-1-\frac{\...
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1answer
24 views

Computationally inexpensive method to find a rotation which minimizes the norm of two tensor difference.

So I have two matrices ${\bf T}_1$ and ${\bf T}_2$, they are tensors in the sense that they can be built as $${\bf T} = \sum_{\forall i} a_i({\bf v_i}{\bf v_i}^T)$$ with positive real weights $a_i$ ...
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37 views

Global optimization methods where constraints are lipschitz functions

Is there any global optimization methods where objective function is nonlinear (not lipschitz) but constraints are lipschitz functions?
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27 views

About finding the common diagonalizing similarity transformation.

Say I have $2k$ matrices $M_{a_1b_1}$, $M_{a_2b_2}$,..,$M_{a_kb_k}$ and their negatives. Here $M_{a_ib_i}$ is such that it has $0$ everywhere except that it has $1$ at $(a_i,b_i)$ and $(b_i,a_i)$ ...
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53 views

Find the critical point and show it is not a global minimizer (using Hessian)

Consider the function $f(x,y) = x^3 + e^{3y}-3xe^y$. Show that $f$ has exactly one critical point and that this point is a local minimizer, but not a global minimizer. I have attempted this, but it ...
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1answer
26 views

Is it possible to solve this 2D geolocation problem?

I have $N$ equations: $$R_i=\alpha_i\frac{T_i}{(x-x_i)^2+(y-y_i)^2}, i=1..N$$ $T_i>0$, $0 < \alpha_i < 1$ and so $R_i>0$. $R_i$, $x_i$ and $y_i$ are known quantities; $x$, $y$, $T_i$ ...
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67 views

Local global minimizers and maximizers

I want to find the local and global minimizers and maximizers of the following two functions. 1) $f(x)=x^2e^{-x^2}$ 2) $f(x)=x+ \sin x $ These are my answers. 1) $f(x)=x^2e^{-x^2}$ $f'(x)=...
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3answers
43 views

Global Optimization of a well-defined function with gradient information

I try to minimize the function $$ f(x_1, …x_n)=\sum\limits_{i}^n-a_icos(4(x_i-b_i)) +\sum\limits_{ij}^{edge}- cos(4(x_i-x_j)) $$ $$x_i,b_i\in (-\pi, \pi)$$ where $\sum\limits_{ij}^{edge}$ only sums ...
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162 views

Minimum of the sum of two functions

I want to show that trying to find the minimum of the sum of two or more functions of two different groups is a not convex problem. For example: $ \min\limits_{Y,Z} f(X,Y,Z)=...$. Moreover the values $...
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1answer
48 views

Is it linear or nonlinear, time-invariant or time-varying?

The equation of motion can be expressed as $M(t)\ddot{q}(t) + D(t)\dot{q}(t) + K(t)q(t) = f(t)$ where $q(t)$ is the defection, $M(t)$, $D(t)$, and $K(t)$ are the mass, damping, and stiffness ...
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52 views

Maximizing a convex function under constraints

Consider the following non-convex problem: \begin{equation*} \begin{aligned} & \text{maximize} & & f(X) \\ & \text{subject to} & & f(X)\le b\\ &&& A_kX = c_k, \ k=...
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Distributing resource based on Efficiency

I am trying to form an optimization problem where I have $k$ nodes who transmits packets with rate $x_k$. The objective is to maximize the rate. $\hspace{28mm} \text{ Maximize } \sum_k \log x_k$ ...
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Sufficient conditions for an optimization problem

Given an optimization problem \begin{equation} \max{F(x)} \text{ subjected to }T(x)=u \end{equation} Where $F:\mathbb{R}^n \rightarrow \mathbb{R}$ is concave function and $T:\mathbb{R}^n \...
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variable transformation in optimization

I have an optimization problem with two sets of parameters, $x_i \in [0,1]$ and $y_k \in [-\frac{\pi}{2},\frac{\pi}{2}]$ where $i,k \in \{1...n\}$ are indices. One way to solve this problem is using ...
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41 views

Solving Nonlinear System for two variables

I have an optimization like below: $\text{ minimize } \sum_k - w_k\log_2 x_k $ $\text{subject to: } x_k \leq q , k =1,2, \cdots, N .$ $\hspace{20mm} 0 \leq w_k \leq 1$ I can form the Lagrange ...
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When is the Lagrangian dual function smooth?

Consider a nonlinear optimization problem of the form \begin{align} \min_{x}&\quad f(x)\\ \nonumber \text{subject to } \quad&h_i(x) = 0,\,i=1,\ldots,I\\ \nonumber \quad&g_j(x) \le 0,\,...
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1answer
64 views

Finding KKT conditions for nonlinear optimization problem.

I have an optimization like below: $\text{ minimize } \sum_k - \log_2 x_k $ $\text{subject to: } x_k \leq q , k =1,2, \cdots, N .$ I can form the Lagrange of the problem as below: $L(x, \...
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35 views

Solving Nonlinear system with logarithmic objective function

I have my objective function as : $\hspace{25mm} \text{Minimize} \sum_k- \alpha_k \log_2 W_k$ $\hspace{25mm} \text{subject to}: 0\leq W_k \leq q', 0 \leq \alpha_k \leq 1 $ $\hspace{25mm} k =0,...
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1answer
167 views

How can I solve a nonlinear optimization problem where constraint contains exponential term?

I have the optimization problem as below: $\hspace{10mm} \text{Maximize} \sum_{k} \alpha_k {R}_k $ $ \hspace{10mm}\text{subjcet to:} $ $ \hspace{10mm}\exp \left[ - (2^{{R}_k } -1) \left( \frac{\...
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29 views

Optimization problem: solving one implies solving the reverse?

I am looking to solve an optimazation problem $Maximize_{x} [A(x)]$ s.t. $B(x)\geq B_0$, where $B_0$ is a constant. If I solve this problem (i.e, finding the optimal $x^*$ that optmize while ...
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118 views

Who knows Krotov's Method in Optimal Control Theory

I'm finishing my PhD thesis about applications of optimal control theory in the field of energy harvesting. In the course of my PhD I dealt with different ways to compute optimal controls, and I found ...
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27 views

How to Find the Maximum of a Function Represented by a Back-Propagation Neural Network?

First, I train a standard feed-forward neural network over a training set of data points. I get an approximate function, say $F(x)$, represented implicitly by that neural network. 1. How do I ...