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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Maximizing $y^H ( I - X pinv(X) ) y $ with respect to matrix $X$. How hard can it get?

Assume $X$ to be a tall block-diagonal matrix where each block is a collumn vector. Assuming $X^+ = (X^H X)^{-1}X^H $ to be the pseudoinverse of the matrix $X$, find $X$ which maximizes $$y^H ( X X^+ ...
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16 views

NLP, Recatungular or Polar? Sine or No Sine?

Let's say we have the following complex number, $V$: $V=V_i+jV_j=V_m\angle{V_a}$ Which type of representation is better in an NLP (polar or rectangular)? The polar form leads to one nasty ...
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29 views

Can we solve this system of inequalities analytically?

Let $A$ be positive real number and $k$ a positive integer. How to find the analytical solution of this system? Find the $a_i$ \begin{align} \begin{cases} ...
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1answer
62 views

Why does $f(x)=\frac{x^T Ax}{x^T x}$ always have a minimum value?

$f$ is defined for all $x\in\mathbb{R^n}-\{0\}$ nd $A$ is a symmetric matrix $n \times n$. I have to proof that $f$ has a minimum $f(x^*)$ and write a formula for $x^*$ using the spectral ...
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30 views

A supremum problem

Let $a=\underset{\|u\|_c\leq 1, \|v\|_r\leq 1}{\sup}v^TY^Tu$. If $\lambda<a$, $\underset{u_m, v_m}{\sup}v_m^TY^Tu_m-\lambda\|u_m\|_c\|v_m\|_r=+\infty$. While if $\lambda > a$, then ...
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2answers
44 views

Intersection of a power function with a line: how to compute?

How to compute $x$ from $$q x^p = 1 - x$$ where $x$ and $q$ are positive, while $p$ is a real number? When $p > 0$: it's two monotonic functions, one increasing and one decreasing, and having ...
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5answers
134 views

Nonlinear optimization: Optimizing a matrix to make its square is close to a given matrix.

I'm trying to solve a minimization problem whose purpose is to optimize a matrix whose square is close to another given matrix. But I can't find an effective tool to solve it. Here is my problem: ...
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18 views

How to regress certain non-linear data

How can I perform a regression onto data of that follows this shape: \begin{equation} U(x):=\sum_{i=1}^N\, a_ix^ie^{-b_ix} \end{equation} where the $a_i\in \mathbb{R}$ and the $b_i \in (0,\infty)$ ...
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1answer
18 views

Proving inequality from convexity of function

I am having trouble proving the following inequality for all $x,y>0$ from "The Mathematics of Nonlinear Programming" by Pressini, Uhl. The book states that it follows from the convexity of an ...
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13 views

Condition of two related matrices

I have a data matrix $\text{X} \in R^{n \times m}$ (n - number of variables; m - number of experiements) and two parameter vectors $\beta_{p} \in R^{p \times 1}$ and $\beta_{l} \in R^{l \times 1}$. ...
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32 views

What algorithms are applicable to solve a inequality constraint Quadratic Optimization?

Suppose that we have a quadratic optimization problem $$(QP) \qquad \min \lbrace\frac{1}{2}x^TQX+ q^TX\rbrace $$ s.t. $$AX=a;$$ $$BX\le b;$$ $$X \ge 0;$$ where $Q \in \mathbb{R}^{n \times n}$ ...
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7 views

Mapping from variable space to ccriterion space in Multiobjective Linear Fractional Programming

I would like to ask about the properties of the criterion-objective space of a Multiobjective Linear Fractional Program with two linear fractional objectives for maximization and linear constraints. I ...
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1answer
24 views

Global minimum of a parameteric function

Let $q:[1, + \infty) \subset \mathbb{R} \longrightarrow \mathbb{R}$ be a function defined as $ \qquad \qquad \qquad \qquad \qquad \quad q(x) = \left \{ \begin{array}{lcl} \delta_{1} & \text{ if } ...
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1answer
29 views

mutivariable unconstrained optimization using gradient search procedure [closed]

Multi-variable unconstrained optimization problem: Maximize the function, $$f(x)=2xy+2y-x^2-2y^2$$ using the gradient search procedure.
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20 views

What is the order of convergence of a vector?

I have a vector of sequences say $(1/k, (1/k)^k)$. I know that each elements of the vector converge to 0 but the way they converge is different. the First element converges sub linearly and the second ...
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12 views

Dual of the following non-linear program

I am new to optimization and understanding some concept. I understood how duality work and tried applying it some linear programs. I followed the same for non-linear programs but I end up wth a ...
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15 views

Which one is better to minimize SSE or MSE in ADMM?

I am minimizing the following ERM objective function. \begin{equation} \sum_i^m \ell(w;x_i,y_i) + r(w) \end{equation} within ADMM framework. ADMM convergence takes a long time (primal and dual ...
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2answers
134 views

Turn off the ovens! An optimization problem

The problem is more abstract, but can be illustrated nicely using ovens. A oven can produce any heat, but is most efficient when it produces $c$ heat. The inefficency increases quadratically as one ...
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1answer
32 views

Modeling integers in NLP

I was wondering why it is not OK to model binary (integer) variables of an optimization problem, in the following form x(x-1) = 0 What are the consequences for ...
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3answers
49 views

Convexity of a non linear optimization problem

I have a non linear optimization problem, namely: $$\min {\sqrt{(x-u)^2 + (y-v)^2 + (z-w)^2)}}$$ How can i show that the above function is convex. Doing via Hessian is a difficult task.
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23 views

Nonlinear Optimization

I have a nonlinear optimization problem, but constraints are ODE. Cost function is $J= x1+x1*x2+x1^2$ while constraints are, $\underline{x_i} < x < \bar{x_i}$ (for i=1,2,3) ; ...
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20 views

Maximization over minimum function

I want to solve the following optimization problem. Suppose we are given $p_r^i \in [0,1]$ for $r={1,2,...,N}$ and $i={1,2}$ such that $\sum_{r=1}^N p_r^i =1$ for i={1,2}. We want to find $x_r \in ...
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10 views

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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29 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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3answers
155 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
44 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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1answer
32 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
72 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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21 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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37 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
146 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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41 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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36 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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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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2answers
44 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
39 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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26 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
79 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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15 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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26 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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29 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
13 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
35 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
27 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
27 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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41 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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25 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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32 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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22 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 ...