Mathematics Stack Exchange
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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.

In mathematics, computer science, economics, or management science, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

An optimization problem can be represented in the following way: given a function $f:A\to\mathbb{R}$ from some set $A$ to the real numbers, we want to find an element $x_0\in A$ such that $f(x_0)\le f(x)$ for all $x \in A$ ("minimization") or such that $f(x_0)\ge f(x)$ for all $x \in A$ ("maximization").

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Scheduling grid optimization

I am trying to optimize the programming of multiple TV channels for a given week. For each show (a day, a time and a TV show) it is possible to forecast in advance the number of people that will watch the show. I would like to use these forecasts to…
Arnaud
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First-order condition for one kind of optimization

I encounter an optimization problem like this: $$\min_{w(x)}{\int {w(x)f(x|e)dx}}$$ subject to $$\int {v(w(x))f(x|e)dx} - g(e)=u$$ $w(x)$ is a function and suppose it has desirable differentiability. $f(x|e)$ is the density function of x conditional…
Zariski
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Optimization: constructive solution

Consider the following program \begin{align} \max_{x,y \geq 0} f(x,y)\tag{1} \end{align} I wanna construct a solution with $y^* = 0$ and $x^* > 0$. Suppose FOCs satisfy \begin{align} f_x(x^*,y^*) = 0\\ f_y(x^*,y^*) < 0 \end{align} and SOCs also…
clueless
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optimization equivalence

Given the functions $f_1(r,x)$ and $f_2(r,y)$: $[0,1]\times \Bbb R \to \Bbb R ^+$, solve the following problem $$\underset{r,x,y}{\text{argmin}}\; f_1(r,x)+f_2(r,y) \\ \text{subject to}\; x^2+y^2=1$$ When is this equivalent to…
teucer
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Minimization of a piecewise affine function of $2$ variables

How does one minimize the following function? $$f(x,y) = |kx + ly + c|$$ where $x,y \in \mathbb N$.
Priyansh Goel
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Evaluate $f(x_0)+f(y_0)$

Let $$f(x)=3(x-2)^{\frac{2}{3}}-(x-2),~0\leq x\leq 20$$ Let $x_0$ and $y_0$ be the points of the global minima and maxima, respectively, of $f(.)$ in the interval $[0,20]$. Evaluate $f(x_0)+f(y_0)$ Note that $$f'(x)=2(x-2)^{-\frac{1}{3}}-1=0$$…
Argha
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Minimum of $\sum \left|7x-1\right|$

How can I evaluate the minimum of $$ \left|7x-1\right|+\left|7y-5\right|+\left|7z-1\right| $$ if $x,y,z$ are non negative reals such that $ x+y+z=1$ and $y^2 \le 3xz$? Without softwares help..
user207096
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Solution Technique to Optimize Sets of Constraint Functions with Objective Function that is Heaviside Step Function

I have the following constraint inequalities and equalities: $$Ax \leq b$$ $$A_{eq}x = b_{eq}$$ The problem is that the objective function, which I am asked to minimized, is defined…
Graviton
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Am I allowed to do this chain-rule type thing when optimizing wrt to a functional?

So we have a function $f(x) = g(x)h(y(x))$ that is convex in $y$ which we want to optimize by choosing the appropriate $y(x)$. I have seen the following done in engineering books, but it just looks so cheesy and pitfall-y. $\frac{\partial…
Benjamin Lindqvist
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Confusion regarding lagrange multiplier

I have certain confusion regarding lagrange multiplier for inequality constraints. I was going through this book by Bishop (Pattern Recognition and Machine Learning) and it says that if I have a function f to be maximized and constraint $g \leq c$ .…
user31820
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Confusion regarding Lagrange multipliers

I was studying Lagrange multipliers. However, I have some confusion. Let's say I have a function $f(x,y)$ to be minimized and I have some constraints $g(x,y) = 0$. If I minimize the function $$ L(x,y,\lambda) = f(x,y) + \lambda g(x,y) \>, $$ then…
user31820
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How one can find equations for eleven squares problem?

In the book Which way did the bicycle go was given that one can put eleven squares $S_1,\ldots S_{11}$ of side length 1 inside a square of side-length $3.877083$ if any pair of $S_i,S_j$ has no common interior points. It was said that this can be…
student
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Linear assignment problem with maximizing the product

I am trying to solve a variation of linear assignment problem: The problem has an equal number of agents and a number of tasks. Any agent can be assigned to perform any task, and for each agent we know the probability that he will succeed…
Vladimir Fomenko
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Is there a published numerically stable version of the Recursive Least-Squares algorithm?

I have implemented in MATLAB the recursive least-squares algorithm given in, for instance, Hayes's excellent Statistical Digital Signal Processing and Modeling (p. 541ff). However, when I run the algorithm on real data with a forgetting factor…
lindelof
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equality constraints in robust optimization

Is it possible to write the robust form of equality constraints in linear programming models? Or it is possible only for inequality constraints?
user223276
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