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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How to find the minimal value of this function under such constraint

$f(x,y)=x-\sqrt{y-x^2}$ with $a<x<b$ and $x^2<y<c*x-d$. What I did is, first take partial derivative at $x$ and $y$ respectively, however, there is no critical point because fy is always ...
3
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2answers
151 views

Singular Values of Matrix as Optimization Problem

Assume that $A$ is a positive semidefinite symmetric matrix. It is known that $$\max_{||y||\leq1} \quad y^TAy$$ Has an analytical solution which is the maximum eigenvalue of $A$. This isn't hard ...
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42 views

Complex Non-liner First order ODE problem

Good day people I am modelling a "water bottle rocket" using basic Continuum Mechanics. I have found a equation describing the acceleration of the rocket. I need to integrate this function to find ...
2
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1answer
121 views

Extremum of functional of a complex function

consider functional $E$ defined by $$E[z]=\int F(x,z(x))dx$$ where $F$ is a complex-valued nonlinear function. How can we find the function $z(x)$ so that $$G=|E|^2=EE^*=\iint ...
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1answer
47 views

The most optimal way to solve this set of non-linear equations in high dimensions

So I have a series of non-linear equations which I wish to solve as fast as possible, to illustrate for the case of $n = 4$, I have the following equations: \begin{gather*} ...
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1answer
46 views

Is it possible to convert into linear programming problem

I have a problem of the form $$\sup_{x\in\Bbb{C}^n}\left\{\frac{\|Ax\|_\infty}{\|Bx\|_\infty}\right\}$$ where $A$, $B$ are matrices with different number of rows and $x$ is an $n$ dimensional vector. ...
2
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1answer
39 views

Conceptual Understanding of Non-Linear Optimization Problem

I'm in non-linear optimization, and I'm having trouble wrapping my head around what this problem is asking me for. If anyone could help with a conceptual explanation (not an answer!), it'd be greatly ...
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71 views

How to minimise an objective function which is not a direct function of the decision variable?

I have a problem with partitioning a water network by closing some pipes. I use some graph theory techniques to find some candidate pipes to close; but to select which pipes among them to close (my ...
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1answer
61 views

Prove that a multivariable function doesn't have global extremes

So my question is actually this. Say I have a function $F:\mathbb R^2\to\mathbb R$. If I find all the potential local extremes by finding the roots of the partial derivatives and I find that only one ...
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0answers
50 views

is there any infinity norm bound to simplify this

I have a problem of the form $$\sup_{x\in\Bbb{C}^n}\left\{\frac{\|Ax\|_\infty}{\|Bx\|_\infty}\right\}$$ where $A$, $B$ are matrices with different number of rows and $x$ is an $n$ dimensional vector. ...
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40 views

Constraints in optimization; redundant hardness?

This is not an accurate mathematical problem, and rather a philosophical and ambitious question. As far as I know, unconstrained problems are easier than constrained problems; right? This is mostly ...
3
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1answer
105 views

Does having a zero eigenvalue preclude a matrix from being indefinite?

If a $3\times3$ matrix has a positive eigenvalue, a negative eigenvalue, and a zero eigenvalue, is it then, by definition, indefinite? I think so, since the matrix has both a positive and a negative ...
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1answer
315 views

Converting Non-linear Programming Problem from Maximization to Minimization

I have a non-linear maximization problem and I want to convert it to be a minimization problem, can I do so by multiplying it by a negative sign, or is that wrong; and if that is wrong what should I ...
2
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2answers
273 views

What change of variables, if any, transforms this nonconvex problem into a convex one?

I'm looking for a convex reformulation, if any exists, of the following minimisation problem: Let $A$ be a symmetric, positive definite $n \times n$ matrix, and $b \in \mathbb{R}^n$. Minimise ...
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0answers
41 views

Effect of approximating a non-differentiable function on optimisation of minimisation

I am looking at a problem of constrained minimization, where the function to be minimized contains the Heaviside function, and as such is not twice continuously differentiable. My question is what ...
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0answers
36 views

Representing a 2D function as a sum of rectangles of arbitrary shape and orientation

Suppose I am given a non-negative function $f(x,y)$ defined for $x \in [0,1]$ and $y \in [0,1]$. I'd like to represent this function as a weighted sum $w_i$ of a small number of rectangular apertures. ...
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0answers
733 views

Significant improvement when I use lsqnonlin function with wrong sized X

I was trying to solve a nonlinear least-square optimization problem using matlab function lsqnonlin with default algorithm trust-region-reflective. Let the optimization problem be "minimize ...
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2answers
678 views

How to find roots of a non linear multivariable equation using numerical methods

I started a course in linear algebra and numerical methods but I couldn't understand how can we numerically find roots of a nonlinear multivariable equation. f: Rn -> R Find f(x)=0 where x is ...
2
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1answer
72 views

Initialization of Limited-memory BFGS (using libLBFGS)

I am using the package libLBFGS in order to minimize an objective function, for which the first derivative (with respect to the optimization variable) is known and computable. I use the default ...
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1answer
47 views

How to prove the demicountinuity of nonlinear operators?

Define a nonlinear operator $\mathbf{J}(\mathbf{x}):~\mathbb{R}^3 \rightarrow \mathbb{R}^3$ as $$ \mathbf{J}(\mathbf{x}):= |\mathbf{x}|^{-\alpha}\mathbf{x},~0<\alpha<1. $$ How to prove that ...
2
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2answers
274 views

Are solutions to optimization problems with smooth, continuous, and strictly concave objective functions and linear constraints always unique?

As an example, if I have a minimzation problem where my objective function is represented by a sphere in n dimensions (one dimension per decision variable), and all my constraints are linear, then ...
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2answers
66 views

$f(x_1,x_2)=\frac{x_1^2}{x_2}$ quasiconvex and/or quasiconcave or nothing on $\mathcal R\times \mathcal R$?

Related to the 3.16e question in Boyd's book. It asks what is $f$ in $\mathcal R\times R_{++}$. I am not interested in it but related thing when the domain is larger. So $f(x_1,x_2)=\frac{x_1^2}{x_2}$ ...
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1answer
89 views

First-order necessary condition for relative minimum point

I'm studying linear and nonlinear programming and I came across with the following proposition : given $\rm x\in\Omega$ we are motivated to say that a vector $\mathbf d$ is a feasible direction at ...
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2answers
24 views

Show that $x_1^2+x_2^2+(x_1-x_2)^3 \rightarrow \min$ has no solution

How can I show (preferably using the Bolzano-Weierstrass theorem), that $x_1^2+x_2^2+(x_1-x_2)^3 \rightarrow \min$ has no solution? I can see that it is true, but how can I show it?
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1answer
47 views

A non-linear optimization problem

I have the following optimization problem on the variables $a_1, ..., a_n$: $$ minimize \frac{\sum_{k=1}^{n}\max(k\cdot a_{k},1)}{\sum_{k=1}^{n}a_{k}} $$ $$ such\ that\ \ 0\leq a_k\leq 1\ \ \ (k=1, ...
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0answers
90 views

Levenberg-Marquardt, QR decomposition

Could anybody explain, how the Levenberg-Marquardt method may be solved using the QR decomposition? I know a current solution ...
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0answers
306 views

How to solve nonlinear constrained optimization in Matlab?

I have to solve a nonlinear constrained function in matlab, and I am not familiar with it's commands. the problem is: minimize $E(b,c)$ constraints: $k1< c\sqrt{b}< k2 ; c/6>k3$ Note: E(b,c) ...
2
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1answer
154 views

Prove or disprove the conjecture about the function below.

After thousands of numerical tests we stated the conjecture that their is exactly one local extremum of the function below. $$ {\rm f}\left(w\right) = {1 \over 2}\sum_{i = 1}^{n}\left({1 \over 1 + ...
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5answers
131 views

How to find the minimum of the function?

How to find the minimum of the following function $$ {\rm f}\left(w\right) = {1 \over 2}\sum_{i = 1}^{n}\left({1 \over 1 + {\rm e}^{-x_{i}\,w}} -y_{i}\right)^{2} $$ where $x_{i}, y_{i} \in \left(0, ...
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25 views

How does this polar function behave?

I came across this question in my textbook for Nonlinear Optimisation and I don't know what to do: Consider the function: $$ f(x_1,x_2)=(r-1)^2-\frac{1}{2}(r-1)^2\cos \left( \frac{1}{r-1}-\phi ...
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1answer
390 views

Definition of tangent cone in continuous optimization .

Looking at the definition of tangent cone in continuous optimization : If $M$ is a open subset of $\mathbb R^n$ $x \in M$, The tangent cone of $M$ at $x$ is defined by $$\mathbb T (M, x) = \big\{d ...
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0answers
76 views

Some problems with a proof of the Farkas Lemma

The following is a proof of the Farkas Lemma that is creating me quite some problems. [I presented the all proof simply to point out the notation used by the author.] My problem is with the last part ...
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1answer
155 views

Minimizing a linear function on a strictly convex set.

All the theorems that I know considering the uniqueness of a solution to a minimization/maximization problem requires the strict convexity/strict concavity of the objective function. But consider the ...
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2answers
38 views

Maximum of convex functions

how can i proof that: If $f_1, . . . , f_m$ are convex functions,than function $F(x) = \max(f_1(x), \dots , f_m(x))$ is convex? thanx for help.
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1answer
164 views

Third and higher order optimality conditions?

In the derivation of first and second order optimality criteria for a vector $X^*$ to be a local optimum to an unconstrained problem, we ignore the higher order terms of Taylor's expansion as we ...
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0answers
74 views

How do I find out if a critical point of a function is a maximum or a minimum?

If I've found the critical point of a function defined in some constraint (perhaps using Lagrange multipliers and the like); how do I find out if it's a relative/global maximum/minimum of a function ...
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0answers
25 views

How to check if steepest gradient method will converge?

So I have this function $ f(x,y) = x^4 - 2x^2 +x + 4y^2 $ and I want to know if the steepest gradient method will converge if I pick an arbitrary point and apply said method. My initial thought ...
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3answers
62 views

Minimization of $log_{a}(bc)+log_{b}(ac)+log_{c}(ab)$?

I am trying to find the minimal value of the expression: $log_{a}(bc)+log_{b}(ac)+log_{c}(ab)$ I think experience gives that the variables should be equal, if so then the minimal value is 6, but ...
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0answers
52 views

Using Newton's method to find an optimized matrix

I'm trying to apply Newton's method to find a local optimum of the matrix $\Sigma$ to minimize the objective function: $f(\Sigma) = -\sum_{n=1}^{N}\left(-\ln{2\pi} - ...
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1answer
28 views

Help in Linear summation of n(any given number)

The original formula is this. We're computing the complexity of an insertion sort. How did the first formula turned into the second formula?
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1answer
49 views

Uniqueness of Solution to non linear polynomial equations given by lagranges method

When considering Lagrange's method of multipliers for finding maximal solutions to a set of non-linear equations, I have reached a set of 4 equations in 4 real unknowns, $(a,b,c,\lambda)$: ...
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0answers
35 views

Decomposition of a symmetric semi-definite matrix into sums of sparse symmetric semi-definite matrix

I'll first provide the background. Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables $x^{(1)},\ldots,x^{(n)}$. We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...
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62 views

Fitting a sine using linear regression

If I have two functions $s_1 = A_1 \sin(\theta+\phi)$ and $s_2 = A_2 \cos(\theta+\phi)$ is it possible to fit a sine or a cosine using linear regression? I usually have much less that a period ...
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32 views

Nonlinear optimisation of Expectation

I am preparing for my exams and I can't get my head around the following question. I know there exists a general method for solving these problems but I don't know where to start. I would greatly ...
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2answers
63 views

$\sum\limits_{k=0}^{19} \sqrt{1+u_k^2} \rightarrow \min$

Solve $\sum\limits_{k=0}^{19} \sqrt{1+u_k^2} \rightarrow \min$, such that $x_0 = 0, x_{20} = 5$ and $x_{k+1} - x_k = u_k$. I think I know how to solve problems like these recursively, but I ...
4
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2answers
86 views

Maximize $\prod\limits_{i=1}^n m_i$

Someone visits a market where one hundred different types of fruit are sold. All types cost $1$ euro per pound. The utility the buyer receives from buying $m_1$ pounds of the first type of fruit he ...
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0answers
61 views

KKT minimization problem

Solve $x^2 - 2y \rightarrow \min$ subject to $\max\{3x^2, e^y + 2\} + \sqrt{x^2 + y^2 - 2x + 1} \leq 6x + \sqrt{5}$ and $ \sqrt{x^2 + y^2 - 4x - 4y +8} -2x+2y \leq 0$ I tried computing the ...
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1answer
38 views

What is inexact steepest descent method

Is there anybody knowing what is the inexact steepest descent method for solving non-linear optimization problems? Any reference or formal definition available online? I was asked by someone, but ...
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1answer
195 views

KKT maximization problem

$x^2y \rightarrow$ max, such that $x^2 + 4xy \leq 1, x \geq 0$ and $y \geq 0$. I think I need to use the KKT conditions here. I did however not yet succeed in solving it, so could someone ...
2
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1answer
821 views

Taylor's theorem for vector valued functions

I'm reading about linear and nonlinear programming and on one page I have the following statment (I have highlighted the areas where I have problems and drawn questions for them in the bottom of it): ...