Mathematics Stack Exchange
Mathematics Stack Exchange

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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Active restriction in optimization

Let $U$ be an open subset of $\mathbb{R}^n$ and $f:U\rightarrow \mathbb{R}$. Consider the problem constrained optimization problem of minimizing $f(x)$ subject to $x\in A$, where $$ A=\{ x\in U\ \mid g(x)\le 0,\quad h(x)=0 \} $$ and…
stefano
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Find (a, b, c) which minimizes $\max(a, b, c) - min(a, b, c)$ is infeasible?

Is the following objective infeasible ? $min: f(a, b, c) = \max(a, b, c) - \min(a, b, c)$ In case of fixing the RHS, function could be minimzed by maximizing the LHS. I have reformlated the problem, $max: f(a, b, c) = \min(a, b, c)$ $s.t \ \ \ g(a,…
user777758
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Show that $\min\{a_1,a_2,...,a_n\}$ is maximum when $a_1=a_2=...=a_n$.

Given $a_1,a_2,...,a_n\in\mathbb R$, and $a_1+a_2+...+a_n=A$. Show that $\min\{a_1,a_2,...,a_n\}$ is maximum when $a_1=a_2=...=a_n$. I feel this is quite a common sense but I don't know how to prove it. Thanks.
JSCB
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Optimal grass rolls placement on rectangular area?

My client sells rolls of synthetic grass to gardeners. Roll sizes are: width $2,3,4$ meters by $25$ meter length, each. Given any rectangular area, you may assume limit of $200$ meters, Provide an algorithm to find the rolls set that meets the…
Mulli
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Optimization of a function of 4 variables under constraint

I am currently trying to solve my own optimality problem. Let $f :(a,b,c,d) \in \mathbb{R^4} \mapsto 1 + d_1+d_2+d_3+d_4+d_5 \in \mathbb{R}_+$ where: $d_1 = \sqrt{a^2+b^2}$ $d_2=…
Axel
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classification of optimization problem

When I do some exercises of optimization, I always run into some defined problems like QP (quadratic programming), CCLP(chance- constrained linear programming) and so on. I want to know how to find the standard of classifying different optimization…
one user
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Error bound for regulary spaced nodes

In trying to find the error bound of polynomial interpolation for evenly spaced nodes I need to show $$|(x-x_1)(x-x_2)...(x-x_n)| \leq \frac{h^nn!}{4n}$$ I know that this maximum has to happen somewhere between the first two or last two nodes, but…
DWM024
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Perform NSGA 2 without variables

I have a data set with two columns. The variable names are cost 1 and cost 2. I want to minimize both cost 1 and cost 2 using the Pareto optimization method. So, while implementing NSGA II I have two objective functions i.e. cost 1 values and cost 2…
vp_050
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Relaxing an optimization problem

I have this optimization problem: $$\max_{a} \min_{\alpha} \ \sum_{i,j} w_{ij} (s_i - s_j)^2 I $$ $$I = 1 \text{ if } w_{ij} \ge \alpha \text{, and } (s_i - s_j)^2 \le a \ ; 0 \text{ otherwise} $$ Notes: $w_{ij} \text{ is normalized between…
rando
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How can you find the smallest radius for an arbitrary amount of circles at arbitrary points to cover all of an enclosed area.

I have an enclosed rectangle with dimensions l x w in the plane with N circles with centers (a,b) (different for each point). How could you calculate the smallest shared radius for the circles so all the points in the plane are within at least one…
William Freedman
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A proof problem in nonlinear optimization

To see the problem, the system has uploaded it on the top link. This is a problem in textbook Nonlinear Programming theory and algorithms, cuold anyone help me with the problem especially in the c. and d. part? Thank you in advance for your…
李明轩
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LICQ not satisfied for optimization problem?

what are the consequences when the linear independant constraint qualification (LICQ) does not hold? so when im looking for KKT points numerically, do i possibly get points that arent even optimums? or do i simply miss out on some optimums because…
user2276094
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Multiobjective optimization

I need some clarification on multi objective optimization. I would like to know if a problem has three objectives with completely different variables, should such a problem be solved as three independent single objective optimization problem or…
Honey Adams
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Finding the minimum possible value for a sum of an expression.

Determine the minimum possible value of the $\dfrac{a}{2b} + \dfrac{b}{4c} + \dfrac{c}{8a}$,where $a,b,$ and $c$ are positive real numbers. How should I go about writing equations to solve for values that can satisfy this expression? Is there…
Monica Sanders
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find the minimum

I want to find the min of this function $f(x,y)= x^2-x+y^2$ graphically under the constraint $x \geq 2$ and $x+y \geq 1$ i find the area of the constraints. I know how to find the solution analytically but i want to find it graphically. I notice…
Farouk Deutsch
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