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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Binding inequality constraints in linear programming with quadratic constraints

I am trying to maximize the following objective function: $a_{1}b_{1}x_{1}+a_{2}b_{2}x_{2}+a_{3}b_{3}x_{3}+a_{4}b_{4}x_{4}$ The quadratic constraint is given by $b_{1}^2 x_{1}^2 + b_{2}^2 x_{2}^2 + ...
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71 views

Nature of the Hessian of the dual function?

I originally posted this over at MathOverflow but it did not receive much (...any) attention. I'm hoping someone can point me in the right direction over here. Consider a nonlinear optimization ...
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1answer
32 views

Nonlinear integer programming problem

I am trying to maximize the following function $$ f(m,n) = \frac{m \log 3 + n \log 2}{\sqrt{m^2+n^2}} $$ where $ n $ and $ m $ are integers, not both $ = 0 $, although one could be $ 0 $. This is ...
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53 views

trust region - choice of scaling matrix

According to many resources, TR algorithms often suffer from bad scaling. The simplest remedy is to use scaling matrix D in following way \begin{align} \min_d \ f + g'd + \frac{1}{2}*d'Bd \\ \...
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22 views

Can the low-rank approximation problem be formulated as the following convex model?

Given a three-order tensor $\mathcal{Y}$, our aim is to find a tensor $\mathcal{X}$ to approximate it and $\mathcal{X}$ should satisfy the following property: $\mathcal{X}$ can be well approximated ...
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39 views

Quasiconvexity of a function on the positive orthant using rays

It is seen in Boyd's book on Convex Optimization book that to show a function $f:\mathbf{R}^n\rightarrow \mathbf{R}$ is quasiconvex, it is enough to show that $f$'s restriction to a line is ...
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54 views

Proving equivalent optimization problems

Consider the problems $\min f(x) , x \in X$ and $\min g(x), x \in X$. two optimization problems are said to be equivalent if an optimal solution to one, is also optimal to another. I would like to ...
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28 views

Convexity of n $4u^4x^{2.5}y^{-5}$ over $x>0, y >0$ and $u>0$

I am trying to find if the function $4u^4x^{2.5}y^{-5}$ is convex over $u>0, y>0$ and $x>0$. The thing which comes to my mind immediately is to check the positive semi-definiteness of the ...
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School District Boundary Optimization--Interpreting the Objective Function

I’m looking for a little help on a new problem. I’m in a linear programming class and trying to work on a project exploring methods on nonlinear optimization and I came across the following question ...
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39 views

Convert non-linear into linear

What I tried: Let $$u_1 = x_1^3$$ $$u_2 = x_2 x_3$$ $$u_3 = x_3^3$$ Then we have $z = u_1 + u_2 + u _3$ s.t. $1 \le u_3 \le 343$ $u_3^{1/3}$ should be integer $u_1 \in \{0, 1\}$ $u_2 \in \{0, ...
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24 views

Convexity of the Pareto front: formal definition

Does anyone have a reference to a formal definition of what convexity of a Pareto front in multiobjective optimisation means? All literature I've seen uses the term without defining it.
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Convergence rate for such modified method of steepest descent

We consider only the quadratic case. $f(x,y)=\frac{1}{2}x^TQx. $ And suppose we can choose $x_0$ to make $g_0$ in the span of its eigenvectors $e_i$, where $g_k=Qx_k$, being the gradient in each ...
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25 views

Numerical Optimizer Matlab Calibration

I am trying to mimimize the following function in order to calibrate the Libor Market Model $$\sum_{i=1}^{n} \left(\sigma_i^{market}-\sigma_i^{Reb}\left(a,b,c,d,\beta\right)/\sqrt{T_i}\right)^2,$$ ...
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19 views

Minimum bounding rectangle is aligned with the convex hull

To start off, here's the problem I'm trying to solve: Suppose we have a finite collection of points in 2D. We would like to find the minimal bounding rectangle (MBR) for these points. By definition, ...
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1answer
38 views

Matrix optimization over a quadratic function

I want to find matrices $F$, $G$, and $H$ minimizing $\begin{bmatrix} x^T & y^T& z^T \end{bmatrix} \begin{bmatrix} I & 0& 0 \\ 0 & F &0 \\ 0 & G &H \end{bmatrix}^{...
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1answer
28 views

Lagrange multiplier - maximum not on tangent contour

I am trying to validate how Lagrange multipliers work. Looking to maximize $f(x,y)=1-x^2$ along curve $x^2 + y^2 = 1$, the solutions are $f(0, -1) = f(0,1)=1$. However, according to Lagrange ...
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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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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} \displaystyle\sum_{i=1}^n\ln\left(1+a_i\...
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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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31 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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46 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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136 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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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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26 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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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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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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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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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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143 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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34 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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50 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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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) ; $\frac{dx3}{dt}=...
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21 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 [0,...
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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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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
183 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 $x^\star_k\...
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54 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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33 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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73 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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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
147 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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38 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 ...