# Tagged Questions

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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### A question about a proof in nonlinear programming book

I have a question about the proof of Proposition 1.2.1 (Stationarity of limit points for gradient methods) in the nonlinear programming book (2nd edition) by Bertsekas. At the beginning of the proof ...
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### Distance between a plane and set of points

Suppose $m$ data points belonging to a class represented by matrix $A$. Therefore, the size of matrix $A$ is $m\times n$. In addition, suppose $w\cdot x + b=0$ be equation of a plane in ...
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### If this problem is not unbounded, what's wrong in this dual derivation?

In a paper with 100 citation, Robust Support Vector Machine Training via Convex outlier Ablation, a convex relaxation is used. In this paper, a form of robust svm proposed: \begin{align} \min_{0\leq ...
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### Function Optimization

Assume any recursive function like: (just for example, my rekursive function is just too big to write) $x_{n+1}=\frac{(x_{n}-3)^{5}x_{n}^{2}}{a\sqrt{x_{n}}}$ (or any other non-linear function) Is ...
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### finite differences nonlinear least squares

I am facing to following nonlinear least-squares problem: $$\min_{u,\gamma} \frac{1}{1000} \int_{ \gamma(x,y)^2} + \int_{ [u(x,y)−u (x,y)]^2} + \int_{ [∆u(x,y)−\gamma(x,y)u(x,y)]^2}$$ ￼where the ...
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### How to find a pareto optimal solution in a smart way (3 variables)

$\max\left( { 3x }_{ 1 }+4{ x }_{ 2 }+2{ x }_{ 3 } \right)$ ${ x }^2_{ 1 }+{ x }^2_{ 2 }+{ x }^2_{ 3 }\le 1$ ${ x }_{ i }\ge 0$ I have to find a Pareto Optimal solution, but I can't solve this ...
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### Closest line to point after non-linear map

I have a map on a vector space $M(\vec{r})$, defined as below. All components (vectors, matrices, everything) are reals in the unit range $[0,1]$. The map $M(\vec r)$ is defined as the sum of an ...
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### Non-linear least squares and Bundle Adjustment

In METHODS FOR NON-LINEAR LEAST SQUARES PROBLEMS, 2nd Edition, April 2004 by K. Madsen, H.B. Nielsen, O. Tingleff on page 17 it states: Given a $f: R^n \mapsto R^m$ with $m \geq n$ We want ...
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### If $H$ is positive definite and $s^Ty>0$, then $s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1$

Let $H\in\mathbb{R}^{n\times n}$ be symmetric and positive definite $s,y\in\mathbb{R}^n$ with $s^Ty>0$ How can we show, that $$s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1\;?\tag{1}$$ ...
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### How to obtain the optimal lagrange multiplier vectors if the globally optimal solution for a nonconvex QCQP is found.?

I am using a blackbox solver to solve the following nonconvex QCQP to global optimality. $$\min_x x^TQ_0x + c^T x \\ s.t. \quad x^TQ_1x+c_1^Tx=b_1 \\ Ax=b \\ l\leq x\leq u$$ where $Q_0$ is ...
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### Linearisation of nonlinear system

I'm asked to write the linearisation of this nonlinear system around the equilibria: $$\begin{cases} x_{t+1}=-x_t+2x^2_t \\ y_{t+1} =-2x_t^2-y_t\end{cases}$$ The two equilibria are therefore: ...
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### Minimize $w=9y_1+4y_2$ subject to linear inequalities

Minimize $w=9y_1+4y_2$ subject to : $4y_1+9y_2\geq 360$ $y_1+4y_2\geq 40$ $y_1\geq 0,~y_2\geq 0$