# Tagged Questions

Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

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### not really sure which method to use

let there be $A(2,4,6)$ , $B(6,2,2)$; On the $x$ axis, find a point $P$ such that the sum of it's distances from points $A$ and $B$ would be minimal. I'm not really sure which method I should use ...
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### converting from max to min in an optimization function?

I have the following maximization objective function related to a svm Then the author says that: is the same as minimizing: ||w||^2, why is this? and that our final optimization function is: ...
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### Condition for maximizer of convex combination to be expansion mapping

I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$ $$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\}$$ such ...
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### Replacing a max constraint in a binary program

For $x \in \{0,1\}$, I want to express $x = 1 \Leftrightarrow \exists k: y_k = 2$ where $y_k \in \{0,1,2\}$, i.e. $x \leq 0.5\max_k\{y_k\}$ using binary decision variables but I can't figure out how ...
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### Lagrange Multipliers, maximize $f=xy$ restricted to $g=x^2+y^2=r^2$

So I have to solve the system of equations $$\cases{\nabla f = \lambda \nabla g\\x^2+y^2 = r^2}.$$ Then $y=2\lambda x, x=2\lambda y$. Sorry if this is obvious, but how can I get $x$ and $y$ only as ...
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### Global Max and Mins

Given $f(x,y) = x^2+y^2-14x-20y$ and restrictions $x≥0,\; 0≤y≤42 \;\text{and}\; y≥x$, I need to find the max and min. By finding partial derivatives and setting them to $0$, I get the min to be -149....
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### Global max/min of surfaces

Given $f(x,y)=4x^3+4x^2y+3y^2$ and restrictions $x,y≥0$ and $x+y≤1$, I'm trying to find global max and mins. I found the partial derivative and found the critical point $(0,0)$ by setting those to $0$...
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### Basis pursuit denoising

The Lagrangian form of basis pursuit denoising $\min_{w} ||w||_{\ell_{1}} + \lambda ||Aw-x||_{\ell_{2}}^{2}$ can be solved using proximal gradient descent. Proximal methods also can be used to ...
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### How do I answer questions that ask to construct examples with some given properties?

Give examples of sets and functions with the following properties: an open set $D_1 \subseteq \mathbb{R}$ and a continous function $f:D_1\rightarrow \mathbb{R}$ such that $f$ has both a maximum and ...
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### find travel time given path and velocity field

As I was studying refraction, I began wondering what path would light take when entering a non-homogeneous transparent medium, i.e. a certain material in which the refraction index $n$ varies (...
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### Calculation of an expression ($\max_{U}\min_i \sum_j |U_{ij}|^2 |e_i^j|^2$)

There is an orthonormal basis $\{e_i\}(i=1,\ldots,n)$ in $\mathbb{C}^n$, each of them is represented in form of column vectors $$\begin{pmatrix} e_i^1\\ \vdots\\e_i^n\end{pmatrix}.$$ My purpose is to ...
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### Methods to translate global constraints to local constraints

Are there any general methods for (global) optimisation which can translate a global optimisation problem to a "local" one? Or in other words, translate global constraints to local constraints. To ...
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### How to argue that $f(x,y,z)=-x^2-xy-y^2+4yz-8z^2+2xz$ has a global maximum?

Let $f(x,y,z)=-x^2-xy-y^2+4yz-8z^2+2xz$. I know $f$ has a local maximum at $(0,0,0)$ but how do I argue that this is also the global maximum. The solution provided simply states it is a global ...
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