# Questions tagged [optimization]

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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### Interpretation of costates in Pontryagin's maximum principle in optimal control

In my optimal control class, my professor proposed the following numerical solution approach to solving the necessary conditions for Pontryagin's principle: He says that, after computing $\lambda(t)$ ...
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### How many steps are needed to display at least 100 digits on a 4-functions calculator?

Inspired by this great question and its answers we had the privilege to contemplate on Mathematics.SE Suppose that at the beginning there is a blank calculator and you choose to write a digit, let us ...
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### Application of Fenchel Duality on a linearly constrained convex set in Hilbert Space [closed]

I am self-studying Luenberger(1969) Optimization by Vector Space Methods, and have a very hard time following the steps in Example 3 of Section 7.12, which is an application of the Fenchel Duality ...
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### Minimizing average distance to closest station inside a square

I have the following problem: I have a square of area M. In the square, I can place P stations however I like. After that, I link every point in the square (and on its edge) to the closest station. My ...
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### Maximizing $\sum_x p(x) (f(x) - g(x) \log p(x))$ over discrete probability distribution $p(x)$

I want to solve this maximization problem over discrete probability distributions $p(x)$, where $f(x), g(x)$ are functions of $x$ and we have $g(x) > 0$. $$\sum_x p(x) (f(x) - g(x) \log p(x))$$ I ...
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### Maximum likelihood and non linear optimization

I have to give a talk about maximum likelihood (statistics) i wanna demonstrate to the class the nonlinear optimization approach for it .Most of the distributions have closed formulas for the ...
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### How to solve an optimization problem with a sum constraint?

I am currently working on an optimization problem that can be mathematically represented as: \begin{aligned} \min_{\{f_{k}\}} & \sum_{k=1}^{K} \frac{q_{k}}{f_{k}} \\ \text{s.t.} & \sum_{k=1}^{...
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### Largest inscribed Circle to solve Bellman's lost-in-a-forest problem

I came across Bellman's lost-in-a-forest problem , which is an unsolved minimization problem in geometry. According to Wiki , there is a need of a general solution in the form of a geometric algorithm ...
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### Open Ended Problem: Estimating Productivity in a Factory

This is a statistical problem I am working on and I would like to request some guidance from the respected community. Suppose there is a factory that receives food orders. The food sits in a ...
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### Function being L-smooth does not imply Lipschitz continuity of gradient

it is a standard argument that if a function is L-Smooth (i.e. locally bounded by a quadratic function: $f(y) \leq f(x) + <\nabla f(x),y-x> + \frac{L}{2}||y-x||^2$), differentiable and convex, ...
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### Optimizing the placement of circles in a partition for maximal radius [closed]

While trying to design an automated loom, which requires me to design circular actuators for each individual thread, I ended up needing to find the ideal arrangement of these actuators on a 5in x 3in ...
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### Setting coefficients to zero in optimization problem

My textbook (optimal control by Kirk) states that, when a function $f(y_1,y_2)$ is subject to a constraint, you cannot simply set $\partial{f}/\partial{y_1}$ and $\partial{f}/\partial{y_2}$ equal to ...
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### Dual value function approach in stochastic optimal control: recover original value function?

I take as my reference Optimal Investment by Rogers (2013). Denote the agent's value function, which is convex, by $V(w)$, where $w$ is the state variable. Define the dual variable $z = V_w$, where ...
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### Fair Cost Distribution for Group Travel

Optimization Problem: Fair Cost Distribution for Group Travel Problem Statement I have an interesting optimization problem related to fair cost distribution for group travel. Here are the details: We ...
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### Non-convex programming

I want to solve a non-convex optimization problem of the form : \begin{array}{cl} \displaystyle \min_{x} & f(x)\\ \textrm{s.t.} & c(x) = 0,\\ \end{array} where $f$ is a concave smooth function ...
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### Is the polytope of the dual of a bounded feasible linear program the dual of the polytope of that linear program?

If you have a bounded feasible linear program. Subject to $Ax \geq B$ and $x \geq 0$. The feasible region is a polyhedron. If you you have a set in $R^n$, you can apply a duality transform, sending a ...
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### Calculate the normal cone to $X=\{f: [0,1] \rightarrow [0,1]\mid f \text{ increasing}\}$

Let $X:=\{f: [0,1] \rightarrow [0,1]\mid f \text{ increasing}\}$. We endow it with $L_2$ norm, and thus $X \subset L_2([0,1])$. We can show that the set $X$ is closed and thus compact, and also ...
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