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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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Suitable exact optimization method

I am looking for a suitable (exact) optimization method for the following problem (I have already solved the same problem using heuristic methods). For j number of binary variables (i.e.: $x_1,x_2,x_3,...x_j$), a variable A is defined as follows:…
Rat3dr
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What is the maximum vertical distance between the line $y = x + 42$ and the parabola $y = x^2$ for $-6 ≤ x ≤ 7$?

What is the maximum vertical distance between the line $y = x + 42$ and the parabola $y = x^2$ for $−6 \le x \le 7$? I have the answer at the back of my book but I am not sure how they got that. The book mentions somthing about derivatives.
Hidaw
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Why constraint optimization problem becomes infeasible when coefficient is zero

First of all thank you for your time and help. I am more on programming side and have average knowledge of calculus so less formal on mathematical notation. I have encountered a problem in non linear equality constraint optimization The objective…
user389718
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Convergence Newton's method

Hello and thanks for read. We have the problem $\min_{c(x)=0} f(x)$ with $c:\mathbb{R}^n\rightarrow \mathbb{R}^m$. If we use Newton's method for solve the problem we get a succession $(x_k,\lambda_k)_{k\in\mathbb{N}}$ such as for the last $k$ it…
Juan-Pablo Contreras
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Theorem about necessary conditions and sufficient conditions for optimality

I have problems understanding the following theorem. I don't understand whether (a) is neccesary condtion, sufficient condition or both; or (b) is neccesary condtion, sufficient condition or both. Hope someon can help
frankfranfrank
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generalising a maximisation problem with several constraints with corner solutions

Let $U(x,y)$ be a continuous, twice differentiable on its domain: $U_x >0; U_y \ge 0; U_{xx} >0; U_{yy} >0$ The problem is to find $x,y\ge 0$ to maximise $U(x,y)$ subject to $a = x + y + \tau I(y>0)$ where $a, \tau > 0$ are constants; $I(y>0) = 0$…
Khan
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$\max\limits_{x,y} ~~xy \exp(-Ax^2y),$ for $x > 0$ and $0 < y \leq 1$

Is there a closed form expressions for optimal $x$ and $y$ for $$z = \max\limits_{x,y} ~~xy \exp(-Ax^2y),$$ for $x > 0$ and $0 < y \leq 1$? If yes, how can we obtain it? $A$ is a positive constant. I can obtain optimal $x$ and $y$ numerically. I…
user168556
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Efficient techniques for multiobjective optimization for a beginner

I want to minimize a function $f(x)$ at multiple positions of $x$, e.g., from $x_1$ to $x_4$, $x_{17}$ to $x_{20}$, $x_{51}$ to $x_{54}$ etc. Each value of $x$ is an independent sinusoidal function. The value of $f(x)$ depends on amplitudes of $x_1,…
user372176
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Line fitting with respect to maximal $L^\infty$-distance to points in the plane

Given a set of points $v_1,\ldots,v_n\in\mathbb{R}^2$ I want to find a straight line $L\subset\mathbb{R}^2$ with minimal distance to the farthest point measured by the maximum norm, i.e. minimizing $$ \max_{i=1,\ldots,n}\mathrm{dist}_\infty(v_i,…
ken_why
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Maximin optimization

So I have the following optimization problem. $$\max_{x\in C} \min(f^{\top}x,g^{\top}x)$$ Here $C$ is some bounded constraint set defined by linear inequalities only. Suppose that $z^{*} = \max_{x\in C} \min(f^{\top}x,g^{\top}x)$. Let $x^{*}\in C$…
Calculon
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Absolute Extrema Multivariable Help Please

I been struggling with question for quite a while now and wanted some help. The question is to find the absolute max and min of the function $f(x,y) = x^3+y^3-3x-3y+2$, bounded by the region $x^2+y^2 \le 1$. I first found the partial derivatives…
Yonna
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Max profit question, regarding negative exponents with factors

$p= 10q^{-0,5}$ $C(q) = 5q$ In a monopoly the equation for maximum profits is: $p'(q)\cdot q + p(q) - C'(q) = 0$ First order condition: $-0,5 \cdot 10q^{-1,5} \cdot q + 10q^{-0,5} - q = 0$ can't figure out onwards from here. How do I do with these…
Hellin
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Solving one optimization problem with variable function parameters

Consider the following maximization problem with constraint: $$ \begin{cases} & \max f(x) + g(y)\\ & x f(x) + y g(y) = B \end{cases} $$ Functions $f(x)$, $g(y)$ and constant $B$ are defined. I'm interested in getting general solution for this…
Alex Velickiy
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Verifying an error estimate on a multidimensional function using its jacobian

This is taken from Nocedal & Wright's Numerical Optimization 2nd edition, pg 279 Theorem 11.3: Suppose $r:R^n\rightarrow R^n$ is continuously differentiable in a convex set $D\subset R^n$. Let $x^*\in D$ be a nondegenerate solution of the equation…
Paul
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Optimization problem involving step function

I've got to optimize the following function with respect to $\phi$: $q(\phi, x) = \frac{1}{n} \sum_{i=1}^{n}{H(y_i)}$ where $y_i = k - \phi l - x_i$ and $H(.)$ denotes the Heaviside function. $k$ and $l$ are constants, and $x$ follows either (1) a…
jush
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