# 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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### The Jeep Problem with Equally Spaced Stations

Consider the following problem. A jeep can carry a maximum load of fuel of 1 gallon, and it travels $l$ miles with $l$ gallons of fuel. The jeep moves along a straight line, and is required to cross a ...
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### From constrained to unconstrained optimization

I have the following convex optimization problem: \label{prob} \begin{aligned} &\underset{{\bf W, \xi}}{\text{min}} & \frac{1}{2} ||{\bf W}||_2^2 + \sum_{i=1}^n C_{y_i}\max(0,...
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### How to minimize $(p_1^2 + (1-p_1)^2)^n$ where $p_1 = 1-(1-(k/n))^N$

Consider $S_{n,N,k} = (p_1^2 + (1-p_1)^2)^n$ where $p_1 = 1-(1-(k/n))^N$. If we fix $N$ and $n$, how do we find a $k$ which minimizes $S_{n,N,k}$? We assume that $1 \leq k < N$ if that makes a ...
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How do you compute the gradient of a function that involves a maximum? For example, I have the function: $$f(\vec{t}) = v(1-\exp(-\lambda\cdot \max(t_0,t_1)))$$ With $v$ and $\lambda$ constant, for ...
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### How to solve an inverse problem $d=Ax_1 + Ax_2$

In the optimization problems, there is an operator, $A$, which transforms the model, $x$, to the data domain, $d$. Generally, we don't know the model and we are trying to find it according to the ...
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### Approximation of non-differentiable optimization problems with max function

The book by D. Bertsekas "Constrained optimization and Lagrange multiplier methods", Ch. 5.1.3 describes at p. 312 a method that is used to solve non-differentiable optimization problems by ...
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### How can I solve an optimization problem $x^T A x$ with constraint $x^T x = 1$?

Let $A \in \mathbb{R}^{n \times n}$ be a positive definite matrix. \begin{align} &\operatorname*{minimize}_{x \in \mathbb{R}^n} & & x^T A x \\ &\text{subject to} ...
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### A trigonometric problem when calculating distance to the boundary of a convex hull

Suppose we have a sphere and a point outside of the sphere. We denote the point outside as $v$ and the origin of the sphere as $x$. The convex hull of the sphere and $v$ should be like an ice cream ...
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