# 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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### The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is ...
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### minimizing $\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$

Suppose there are $n$ points $(x_i, y_i)$ for $i = 1,\ldots,n$. Please find another point $(x, y)$ to minimize function: $$\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$$
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### Optimal path around an invisible wall [duplicate]

The Problem On an infinite plane there are two points, $A$ and $B$, a unit distance apart. There is a $50\%$ probability that there is an invisible wall somewhere between the two points. The ...
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### Control on Conformal map

Let $\Omega$ be smooth simply connected open set of $\mathbb{R}^2$ such that $\overline{\Omega}$ is compact. We know that there exists a conformal diffeomorphism $\psi$ from $\mathbb{D}$ to $\Omega$. ...
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### Prove the supremum of the set of affine functions is convex

Let $\langle f_i \rangle _{i \in I}$ be a family of affine functions on a convex and compact set $\Omega \subset \mathbb{R^d}$ such that $f_i = a_i.x +b_i$ for $x \in \Omega$. Prove that f, defined by ...
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### Min problem by using Lagrange method

$$\min x^2+y^2$$ $$\text{s.t.}\ \ (x-2)^2+(y-3)^2\le 4 \ \ \ \text{and} \ \ \ x^2=4y$$ Please explicitly solve this question by using Lagrange multiplier method. I accept $(x-2)^2+(y-3)^2=4$ ...
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### Minimization of $\sum \frac{1}{n_k}\ln n_k >1$ subject to $\sum \frac{1}{n_k}\simeq 1$

Looking at an algorithm for minimizing $\sum_{k=1}^{m} \frac{1}{n_k}\ln n_k > 1$ subject to $\sum_{k=1}^{m}\frac{1}{n_k} = 1$ in which $n_k$ are positive and in general non-sequential integers, I ...
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### Difference: Newton's method, Newton-Rhapson method, Gauss Newton-method.

I would appreciate some clarification w.r.t. algorithms for solving nonlinear systems of equations. 1 - I don't understand the difference between Newton's method and Newton-Rhapson method. In [1], ...
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### What is a convex optimisation problem? Objective function convex, domain convex or codomain convex?

My teacher in the course Mat-2.3139 did not want to answer this question because it would take too much time. So what does a convex optimisation problem actually mean? Convex objective function? ...
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### Why nonlinear programming problem (NLO) called “nonlinear”? What does “nonlinearity” actually mean? Is it “not linear” or something different?

My teacher in the course Mat-2.3139 presented the same definition as in Wikipedia for the nonlinear programming problem here but he did not specify what the nonlinearity actually means or what it ...
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### A minimization problem

Define $$L(w,u)=\frac{1}{2}\|w-u\|^2+\beta \left\|\frac{w}{x}\right\|,~w,u\in \Bbb{R}^n$$ where $$\frac{w}{x}=\left(\frac{w_1}{x_1},\ldots, \frac{w_n}{x_n}\right)$$ $$\|x\|=\sqrt{x_1^2+\cdots+x_n^2}$$ ...
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### Find $\min x^TAy+b^Tx+c^Ty$ subject to $1^Tx=1^Ty=1,x\ge 0,y\ge 0$

The problem seems to be easy but I can't find a solution :( Problem: Given $A\in\mathbb{R}^{m\times n}, A\ge 0, b\in\mathbb{R}^{m}, c\in\mathbb{R}^{n}$. Minimize $f(x,y) = x^TAy+b^Tx+c^Ty$ subject to ...
Consider the compact sets $X \in \mathbb{R}^n$, $Y \in \mathbb{R}^m$, $A \in \mathbb{R}^n$, $M \in \mathbb{R}^{n \times m}$. For fixed $(\bar{a},\bar{B}) \in A \times M$, by the MiniMax Theorem we ...