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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Find all the points satisfying the Fritz John conditions

Consider the problem $$\min \>x^2+y^2 $$ $$s.t.\> x^2-(y-1)^3=0$$ Find all the points satisfying the Fritz John conditions Solution The FJ conditions are $$2x+\mu_1 2x=0$$ $$2y-\mu_1 ...
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34 views

A nonlinear optimization problem with difficult Kuhn-Tucker system of equations

I know about the sufficient optimality theorem Kuhn-Tucker, and this problem can use the Kuhn-Tucker theorem directly, but ridiculously, I got stuck on the system of equations to find one root for ...
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10 views

how does sequential axis search work?

I see that some algorithms that need to search for a global minimum in multiple dimension space, say find x and y to minimize f(x,y), instead of searching in x,y simultaniously, starting from initial ...
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23 views

What are the general algorithm and precise mathematical language that can optimise the nodes in a graph?

Recently I came across this via social media Out of curiosity (and because I am a visual learner) using the paragraph in the article, I end up drawing some kind of mixed graph, as shown I then ...
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1answer
22 views

Linear Matrix Inequality - “HOW TO”

I have a tough time understanding how to use Linear matrix Inequality to solve simple inequality problems. I would appreciate a simple "How to" on the following examples. $$ \bar PA + A \bar P - ...
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27 views

Non-binding constraints with positive shadow prices (matlab)

The output of fmincon indicates positive shadow price for linear constraints, although the corresponding constraints are not binding. What could be wrong mathematically? I've checked the code but ...
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1answer
40 views

Transportation problems

i'm a master student at the deparment of statistics. And i will prepare a presentation on transportation problems in the course of optimization (or linear programming / mathematical programming) I ...
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1answer
59 views

Weakly unimodal function using Golden Section Search

I was going through the Golden Section Search https://en.wikipedia.org/wiki/Golden_section_search and as I understand it should work for every unimodal function. Here, the definition of unimodal ...
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9 views

Nonlinear Programming

I have the following non-linear programmig problem that I have arrived at after various manipulations. I have to find the set of values for $x$ and $y$ that satisfy the following: $$ x^{n}+y^{m}=C $$ ...
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39 views

About a nonlinear problem using sufficient optimality Kuhn-Tucker theorem

I'm learning nonlinear programming and got stuck on this problem. Here is the problem: Find the minimum of $\theta(x,y) = x^2 + y^2 - 2xy + 3y + 5$, with $x, y \in R$ which satislies $g(x,y) = x^2 ...
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34 views

dual feasibility of Kuhn-Tucker condition?

minimize $f(x)$ subject to \begin{align} f_i(x) & \le 0, \quad i \in \left\{ 1,\ldots,m \right\} \\ h_i(x) & = 0, \quad i \in \left\{ 1,\ldots,p \right\} \end{align} Then the Lagrange ...
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1answer
43 views

Is mathematical programming an analytical or numerical technique?

Is linear programming, mixed-integer programming, integer programming, nonlinear programming, etc. numerical or analytical techniques? I always thought they were numerical methods because you can't ...
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17 views

Optimization of Inputs to Monte Carlo Simulation Based on Outputs

I have an optimization process that seems to work, but I want to better understand why it works and whether there's a better way to do what I'm trying to achieve. Basically I am optimizing two (or ...
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38 views

min-max optimization problem

how do you solve the following optimization problem to find the global solution? $~~~~~\underset{y}{min} ~ \underset{x}{max} f(x,y)$ subject to $~~~~~g(x)<0$ with knowing that both g(x) and ...
2
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1answer
33 views

Some True or False questions on Nonlinear Optimisation (Exam Preparation)

I am currently preparing for a Nonlinear Optimisation exam and am working through some old question papers and came across these True or False questions: When minimizing a convex function over ...
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1answer
21 views

Mistake in my NLP using Lagrange Multipliers?

I have the following NLP \begin{matrix}\text{Minimize:} &x^2+y^2+z^2 \\ \text{Subject To:} & x+2y+z-1=0 \\ &2x -y -3z-4=0\end{matrix} I need to solve this using the Lagrange ...
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42 views

Constrained LQR with a fixed terminal state. Can MPC be applied to this problem?

I am interested in solving the constrained LQR problem with discrete finite time when the target $x$ value is given, but the final $u$ could be anything s.t. constraints. $$\text{minimize }J = ...
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2answers
15 views

Linking between Square Matrix and Positive Definite Matrix?

I'm not mathematically trained. A module I'm taking this semester needs me to: Show that a square matrix with only diagonal values that are all positive is a positive definite matrix. What is the ...
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11 views

Find $\tau$ to minimize $\sum_{i=1}^{T/\tau} max \{ m_1, …, m_i \} $, such that $m_i = \{ d_{i\tau+1}, …, d_{(i+1)\tau} \} $

Suppose $T$ numbers $\{ d_1, ..., d_T \}$ are chosen by an adversay. We have divided the numbers into blocks of size $\tau$, i.e.$\{ d_1, ..., d_\tau \}$, plus $\{ d_{\tau+1}, ..., d_{2\tau} \}$, ...
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17 views

Singular value decomposition:Canonical Correlation, Reformulation of the objective function

The CCA method aims to find two loading vectors or projections $\alpha$, $\beta$, the linear combinations of variables in $X$ and $Y$, to maximize the correlation between $\alpha ^t X$ and $\beta^t ...
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33 views

Prove that a function is convex

In order to show that if the epigraph is convex then the function $f(x)$ is convex I did like this. Let $x_1,x_2 \in C$, C is a convex set. Then the points $(x_1,f(x_1))$ and $(x_2,f(x_2))$ are in ...
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1answer
99 views

Proving Convergence of a Derivative Free Method

Hopefully this question is on topic for the mathematics community (rather than the statistics) since the optimization method I am using relies on a statistical model. Well I am building an algorithm ...
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1answer
25 views

Nonlinear optimization related to symmetric functions

Suppose $f(x,y)\geq 0$ is integrable and symmetric in $x$ and $y$, i.e.$f(x,y)=f(y,x)$. Consider the following nonlinear optimization problem $$\max F(a,b)=\int_0^a\int_0^bf(x,y)dxdy,$$ Subject to ...
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25 views

Optimization involving integrals with varying limits

What are the common methods and tools to tackle optimization problemsinvolving integrals. To be precise lets consider the following optimization problem that I came across with: ...
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1answer
24 views

Positive definite and semi definite in non linear programming [duplicate]

How can I prove the following. Suppose that A is a square matrix and suppose that there is another matrix B such that $A=B^TB$. a)Show that A is positive semi definite b)Show that if B has ...
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1answer
26 views

Karush-Kuhn-Tucker NLP

Consider the nonlinear program Minimize: \begin{align}f(x,y) = \frac{1}{2}x^2 - 10xy + 10y^2\end{align} Subject to: \begin{align}2x +y^2 &\le 5 \implies g_1(x,y)=2x + y^2 -5 \le0 \\ ...
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120 views

How is the Lagrangian related to the perturbation function?

Given a convex programming problem $$\begin{align*} \text{minimize} &\quad f(x) &\\ \text{such that} &\quad g_i(x) \leq 0 & i=1\dots k\\ & \quad h_j(x) = 0 & j= k+1\dots n ...
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31 views

How to make a non-linear problem linear?

I have the following constraint which is the product of multiple binary variables: $1- \prod_i^n (1-(c_i x_i)) >= T$ where $x_i$ is a binary variable, $c_i$ is a constant and $T$ is a constant ...
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34 views

solving a collaborative filtering problem

I was reading this paper Bell RM, Koren Y. Scalable collaborative filtering with jointly derived neighborhood interpolation weights. Proc - IEEE Int Conf Data Mining, ICDM. 2007:43-52. ...
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91 views

Nonlinear LS regression

• Problem formulation I have to fit the following nonlinear model to a dataset: $$f(x)=\frac{C_1 \cdot a}{a^2 + C_2 \cdot x^2}$$ $a$: fitting parameter $C_1, C_2$: Given constants I can't apply ...
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38 views

references: L-BFGS rate of convergence

I was trying to find results about the rate of convergence for the L-BFGS algorithm (in the nonlinear case). What I end up with so far is that the BFGS-Algorithm converges Q-superlinearly this 50 ...
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19 views

Linear space transform transformation based on covariance?

I have a linear space of n dimensions with non-overlapping groups characterized by different variation (different covariance matrices). Is there a way to deform non-linearly the space according to an ...
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20 views

quasi-newton method converges in at most n+1 iterations

Given $B_{k+1}$ be obtained from $B_k$ using the symmetric rank-one update formula. Assume that the associated quasi-Newton method is applied to an n-dimensional, strictly convex, quadratic function, ...
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22 views

Nonlinear Least Squares vs. EKF

What is the relationship between nonlinear least squares and the Extended Kalman Filter (EKF)? I've learned both topics separately and thought I understood them, but am now in a class where the EKF ...
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24 views

Is it true that there exist exactly ${k\choose n}$ bases that lead to this basic feasible solution?

Let a matrix $A=\left(A_{ij} \right)_{k\times n}$, with $A_{ij}\in\mathscr{R}$, and $$\mathrm{P}=\{ \mathtt{X}\in \mathscr{R}^n \,|\, A\mathtt{X}\ge b\}. $$ Suppose that at a particular basic ...
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1answer
51 views

KKT condition - minimization problem

$y^2-8 \ln(x+4)\rightarrow$ min, such that $-x^2 -y^2+9 \geq 0, y \geq 0$ *I have to find all possible optimal points.* Lagragian function is: $L(x,y,γ_1,γ_2) = y^2 - ...
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1answer
18 views

Linearizing a constraint for ILP

I have binary variables $x_{ij}$. One of my constraint is $$\sum\limits_{i}\sum\limits_{j} x_{ij}*f_i(\sum\limits_{j}x_{ij})\leq B \ $$ where my $f_i()$ is implemented as a table. Will it be ...
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12 views

Problem with nonlinear equation system

I need to calculate two coefficients: k and n. This involves solving two equations for k and n: $$\frac{(8 \pi d)}{\lambda }{kn}=\frac{\text{$\triangle $I}}{I}$$ $$\text{$\triangle ...
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19 views

How to find Common (invariant) Subspace between more than two Hankel Matrices?

Note: I am not a mathematician but a control engineer. A general nonlinear $n_{a}^{th}$ order discrete-time state-space model is described by the following equations: \begin{align} ...
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25 views

Choosing a non-convex global optimization algorithm based on the number of permitted steps

Can anyone comment on the most suitable approach for the following optimization problem: We are given finite bounds for a set of $n$ real-valued parameters of an unknown deterministic function. The ...
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28 views

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
39 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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56 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*} ...
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1answer
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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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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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26 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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32 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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16 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
40 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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31 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 ...