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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Matching of points in two discrete linear sequences with potentially missing points

This is a question that I've been thinking about in my research lately. I've gone down the route of a few linear-optimization techniques, but nothing particularly spectacular has come up. Anyway, ...
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2answers
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16 views

Convex optimization of a fractional objective function involving matrix determinants

I am interested in convex representation of the following fractional optimization problem. I have also described my approach in the following. However, as I am new to convex optimization, I am not ...
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11 views

Removing variables from convex linear program

I am solving linear program (possibly non-convex). Then we know that dual is always convex. Then I noticed that depending on objective functional I can sometimes remove particular variables from this ...
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3answers
105 views

How to find the root of a polynomial function closest to the initial guess?

I need some easy to implement and fast numerical method that finds the root of a nonlinear function (a polynomial in my case) closest to my initial guess. If I know that there is one root ...
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1answer
22 views

Failed attempts at fitting nonlinear Hill function (biochemistry) to data

I am trying to fit some data in Matlab to a Hill function of the form $y = \dfrac{1}{1+(K/r)^n}.$ I have data for $r,y$ and I need to find $K,n$. I have tried following the approach shown here in ...
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29 views

Solving a system of coupled nonlinear equations analytically rather than numerically

Let $x,y$ be two variables. Consider the following system: $$ \begin{cases} x=a_0+b (1+y^2/x^2)^{-1/2} \\ y=a_1+b(1+x^2/y^2)^{-1/2}, \end{cases}$$ where $a_0,a_1,b$ are parameters. I can solve this ...
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1answer
58 views

Using Calculus To Solve Optimisation Problems

I have a question regarding using calculus to solve an optimisation problem which is quite wordy. It is as follows: A researcher has funds to buy enough computing power for 7 years. Computing power ...
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20 views

Creating a configuration of points where each point is away from all other points by a pre-defined distance

Let's assume that the points $\in \mathbb{R}^2$ and there are only C=5 points (in practice, I may have $\mathbb{R}^{800}$ and 1000 points). The first out of the five points is fixed. We also have been ...
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31 views

Maximization and Minimization of $f(x,y)$

Find the extreme values of the function: $z=f(x,y)=x^2+(y-18)^2+90$ subjected to following constraint $x^2+y^2\leq196$ How to solve this? I used Lagrangian function but how to set up constraint ...
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1answer
106 views

Gradient of the dual function for a nonlinear program

I'm attempting to find a proof for a property from Floudas' Nonlinear and Mixed-Integer Optimization book. Consider a nonlinear optimization problem of the form \begin{align} \min_{{\bf x}}&\quad ...
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39 views

How can the lagrange multipliers in a simple constrained cost minimization problem be calculated? (for binding constraints)

Is there a simple algebric way to calculate the shadow prices (lambda) of the binding constraints given below? This is a cost minimization problem dependent on the generation output. The cost of ...
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31 views

Non-linear optimization programming

How many methods do we have for non-linear optimization problems, which the target function is linear but constrains are polynomial shape? Are there methods which can solve most of them? Or what ...
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10 views

Are there any optimization strategy suiting this framework?

For optimization problem: $min \quad f(x_1, x_2)$ Are there some strategies that are doing this sequentially, i.e. first solve $min_{x_1} \quad f(x_1, x_2^{k})$ to get $x_1^{k+1}$ then solve ...
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18 views

Convex problem with linear constraints

I wish to solve the following nonlinear program: $$\min_{\substack{x_i\ge 0\\x_1\le x_1+x_3\le x_2}}h_1 x_1+h_2x_2+h_3 x_3+k_1(\tau-x_1)^++k_2(\tau-x_2)^++k_3(x_2-x_1-x_3)^+$$ I have the KKT ...
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2answers
32 views

Optimization over min function

We want to maximize $\sum_{i=1}^N \sum_{j=1}^N \min(a_i,b_j)$ such that $\sum_{i=1}^N a_i =1$ and $\sum_{j=1}^N b_j =1$. I think the optimal solution is $a_i = 1/N$ and $b_j = 1/N$ for $i,j = {1,2, ...
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1answer
28 views

Identify if optimization problem is convex or non-convex?

I have formulated optimization problem for building, where cost concerns with energy consumption and constraints are related to hardware limits and model of building. To solve this formulation, I need ...
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24 views

what should i replace the product of nonlinear variables with when linearizing

I have a product of two continuous variables in a constraint of an optimization problem. I want to linearize the product and use $$x_1 \cdot x_2 = y_1^2 - y_2^2$$ I followed the steps mentioned in ...
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1answer
75 views

Is this illustration of Gauss Newton wrong?

In this illustration the value of each iteration is the minimum of the 2nd derivative. But the Wikipedia page says: the advantage [of the Gauss–Newton algorithm] is second derivatives, which ...
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13 views

Preconditioning vs Re-orthogonalise for Non-Linear Conjugate Gradient Method

I am using CG to solve an optimisation problem. Since my cost function is ill-conditioned, I am looking at improving the performance either by using Preconditioning or Re-orthogonalisation, especially ...
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24 views

Using least squares regression to apply nonlinear function to time series data

If you have a nonlinear function (see example), can you use a least squares regression approach to fit it to time series data ? Is this approach also valid for n variables? How many time points are ...
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28 views

Proving the global minimum of a continuous piecewise-convex function on a closed interval $[a,b]$

How can prove this clear fact that the global minimum of a continuous piecewise-convex function on a closed interval $[a,b]$, always happens at either extreme points ($a$ or $b$) or the minimum of ...
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19 views

KKT conditions for nonlinear problem

I need to state the KKT conditions for the following problem: Minimise $x_1^2 + 2x_2^2$ subject to $(x_1-1)^2 + x_2^2 \le 1$ and $x_2 = 1$. I have that these conditions are: $f(x^*) \le 0$ ...
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1answer
11 views

Feasible set and level sets

Consider the following problem: Minimise $x_1^2 + 2x_2^2$ subject to $(x_1)^2 + x_2^2 \le 1$ and $x_2 = 1$. Sketch the feasible set and the level sets of the objective function, and determine an ...
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2answers
25 views

How to introduce Levenberg-Marquardt?

How would you introduce the Levenberg-Marquardt algorithm: To someone who understand the concept of minimisation and derivative. By using intuition instead of equation if possible. For instance a ...
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1answer
22 views

Does the non-expansion property of the projection operator hold for all definitions of norm?

For convex problem, of course. I vaguely remember this holds for weighted norm also. But I am curious if there are some general conclusions about what kinds of norm will fit in this framework?
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1answer
26 views

Minimizing a quadratic function of 2 variables in quadratic region

Let $f$ be a real valued quadratic function of 2 real variables: $$f(x,y) = ax^2 + by^2 + cxy + dx + ey + f$$ How to minimize it? Subject to constraints: $$ 0\leq x \leq 1, \quad 0\leq y \leq 1 $$ ...
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34 views

Scaled proximal operator for proximal Newton method

The scaled proximal operator was introduced as an extension of the (regular) proximal operator: $prox^H_h(x) = \arg\min_y h(y) + \frac{1}{2}\|y-x\|^2_H$. (See ...
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22 views

How to solve the following convex constrained optimization problem?

\begin{equation}\label{constrained optimization} \begin{aligned} \min\limits_{\mathbf{X}}&\|X_{(1)}\|_{*}+\|X_{(2)}\|_{*}+\|X_{(3)}\|_{*}+\lambda\|Ax-b\|_2^2 &\ \ s.t. X_{ijk}=M_{ijk}\ \ ...
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28 views

Minimization/maximization of system of nonlinear equations

Consider a system of nonlinear equations of the following form: $$F_1(x_2, x_3, x_4...x_n)$$ $$F_2(x_1, x_3, x_4...x_n)$$ $$...$$ $$F_n(x_1, x_2, x_3...x_{n-1})$$ And we wish to simultaneously ...
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19 views

nonlinear optimization : restricting search space to “limited” preselected value

My function is nonlinear with respect to a scalar \alpha . However, the calculation of objective function is very time consuming, making optimization also very time consuming. Also, I have to do it ...
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1answer
48 views

Dynamic programming at a non linear programming problem

Could you explain to me how we can use dynamic programming in order to solve a non linear programming problem? What do we do for example if we are given the following problem? $$\max (y_1^3-11 ...
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25 views

minmizing frobenius norm($\min_{K \in \mathbb C^{n \times m}} \|KQK^*\|_F$) subject to an equality

In which known approach or algorithm a fat matrix $K\in \mathbb C^{n \times m} ,m>n,$ can be found: $\min_{K \in \mathbb C^{n \times m}} \|KQK^*\|_F$ subject to: $KK^*=I$ $Q\in \mathbb C^{m ...
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26 views

Minimizing summation using Karush–Kuhn–Tucker

Let $a_j, c_j , j=1,...n$ and b be positive constants. $$-$$ Minimize : $\sum_{j=1}^n \frac{c_j}{x_j}$ $$-$$ subject to:$\sum_{j=1}^n a_j x_j =b$ $$-$$ Write down the kkt conditions and solve the ...
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21 views

Optimization by KKT-method

I need to solve the following problem by KKT method. $$ \text{min} \ \ 2xy + 2yz + 2zx \\ \text{subject to} \ x^2 + y^2 ≤ 2, \ 2x + 2y + z = 0 $$ I have gotten as far as setting up the system of ...
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13 views

Using sequence of least squares solutions to solve non-linear problems?

I do know about the iteratively reweighted least-squares and have played around with it to some success finding non-linear solutions (like minimizing non-2-norms to achieve solutions which seem to be ...
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What is the moment matrix in converting polynomial optimization problem to a quadratic optimization problem

Happy new year, I have a function of the form below \begin{align} f(x,y,z)=\sum_i x_i y_i z_i + g(x)+h(z)\cr x,y,z \in R^n \end{align} where $h,g$ are quadratic functions. My difficulty lies in the ...
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32 views

Linear Optimization Problem with exponential variable

Hey Folks I've encountered an optimization problem which has a linear programming structure but it's coefficients are nonlinear function of another variable. here is the problem: $$\max ...
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2answers
61 views

How to efficently solve a convex optimization problem with positive semi-definite Hessian matrix?

Consider the following optimization: $$ f(x)= \min \sum_{i=1}^n \left(x_i-\sum_{j=1}^n x_j\right)^2 $$ Let $g_i(x)=x_i-\sum_{j=1}^n x_j$ , then $$ f(x)= \min \sum_{i=1}^n g_i(x)^2 $$ The ...
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1answer
48 views

Positive-Semi-Definite form of Variance?

first thing: I'm an informatics student and know some algebra. However, this seems to be a bit over my head, so please be gentle with me. ;) I have multiple sets of real variables. Let these sets be ...
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1answer
72 views

Maximize $ 2^{(-x)} + 2^{(-y)}$ subjected to certain conditions

I am reading through convex optimization and I came across this following problem: \begin{align*} \max \text{ } & 2^{-x}+2^{-y}\\ \text{s.t. } & \frac{1}{1+x}+\frac{1}{1+y}\leq b\\ & ...
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178 views

Monotonic Function Optimization on Convex Constraint Region

So I have the following function, which I want to maximize: $$f(x_1,...,x_n) = \sum_{i=1}^n\alpha_i\sqrt{x_i}$$ (where all $\alpha_i$ are positive), subjected to the following equality and inequality ...
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12 views

finding the least absolute inner product solution for some give vectors

Are there any efficient algorithms (like for PCA) that can find the orthogonal vector of a given bunch of vectors? Mathematically, let $\mathbf{b}_1, ...
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26 views

Is there any approach towards finding all infinite solutions of a set of nonlinear equations when the number of unknowns is more than equations?

I have 3 nonlinear equations with 4 unknowns, with some bound constraints. How can I see if there is a solution to the problem? I wonder if there is a similar approach in nonlinear equations like SVD  ...
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36 views

Maximizing Frobenius norm

I was wondering if anybody has any suggestions on the following problem: Let S be an $n \times n$ real symmetric matrix and $W$ is a real matrix of size $n\times d$; $1\leq d <n$. $$ \text{Find ...
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How to use the Karush–Kuhn–Tucker conditions?

From what I read, the Karush-Kuhn-Tucker conditions are a generalization of the Lagrange Multiplier Method. For the Lagrange Multiplier Method I have been able to find a serie of steps I must do to ...
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1answer
26 views

proving that a global maximizer does not exist

Show that no matter what the value of $a$ is chosen, the function $f(x_1,x_2)=x_1^3-3ax_1x_2+x_2^3$ has no global maximizers. Determine the nature of the critical points of this function for all ...
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Proving that a function is coercive

Let $f(x,y)=x^2-2xy+y^2$. I know this is not coercive as along the line $y=x$, when $||x|| ->\infty, f(x,x)=0$. But I don't understand what is wrong with the following way. ...
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34 views

Is $|x^3|$ convex?

Let $f(x)=|x^3|$ on I=$-\infty,+\infty$ Is this convex? How I did was f(x) = \begin{cases} x^3, & \text{if $x>=0$ } \\ -x^3, & \text{if $x<0$ } \end{cases} Then$ f '(x)$ = ...
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28 views

How to Prioritize Constraints of an optimization problem?

I have an optimal control problem of a vehicle the form: $$ \min_{u \in R} \ell (x(j),u(j)) = {\Vert x(j)- x_r(j)\Vert}_Q + {\Vert u(j)- u_r(j)\Vert}_R $$ subject to $$\dot{x}=f(x)$$ ...