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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Solving a bi-level formulation with no closed form for its second level

I was wondering if I can receive your suggestions regarding how to solve a bi-level formulation with no closed form for its second level. I can solve the second level, separately, using a heuristic projection algorithm. I studied some metaheuristics…
Mehrzad
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Why does the result of the Lagrangian depend on the formulation of the constraint?

Consider the following maximization problem: $$ \max f(x) = 3 x^3 - 3 x^2, s.t. g(x) = (3-x)^3 \ge 0 $$ Now it's obvious that the maximum is obtained at $ x =3 $. In this point, however, the constraint qualification $$ Dg(x) = -3 (3-x)^2 =…
bonifaz
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Could someone give a simple non-convex function with 3d plot similar to the one shown below?

I'm trying to make a 3d plot that is non-convex. I googled a little bit but didn't find a non-convex function that is as easy to understand as the mean squared error. I know the mean squared error is defined as ${\displaystyle \operatorname {MSE}…
JakeMZ
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functional minimization problem

Let $f:[0,1]\to [0,1]$. Define $\|f\|_1=\int_0^1f(x)dx$ and $\|f\|_2=\sqrt{\int_0^1(f(x))^2dx}$. How to find a $f$ such that $\frac{1-a\|f\|_1+\frac{(b\|f\|_2)^2}{2}}{b\|f\|_2}$ is minimized? ($a\in \mathbb{R}, b\geq0$)
Mango
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How to express this minimalization/optimalization problem (with example and pseudocode) mathematically

I'm stuck expressing the following scenario in mathematical terms. I have the code written in Python, an example can found at the bottom of this post. Scenario: I have a set of multiple data sets $D$. Each data set $d \in D$ has a day associated…
sandertjuh
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why are constrained problems hard to solve?

I was reading about solving optimization problems using meta heuristic techniques. In most of the cases, constrained related problems are converted to unconstrained using penalty functions. I was trying to understand as why constrained problems are…
gari
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how to get the values of integers A,B and C that give the closest value to N?

I have this formula : $N = 3 \cdot A \cdot B \cdot C + 4 \cdot A \cdot B + 4 \cdot A + 1$, where A B and C are integers that belong to the range $[0, 255]$ how can I choose the values of A,B,C in order to get a value of N that is close to a given…
Kamel Fakih
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maximum area of soap film bound by wire of unit length

A soap film has zero mean curvature at any point, and the area of any soap film bordered by wire is the surface of least area that spans the wire. What is the maximum total surface area of soap film that can be bordered by a connected frame formed…
Angela Pretorius
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What would be the gradient function

Suppose we have the function $f(x) = \frac{1}{2} x^T Qx - x^Tb.$ What would be $\nabla f(x)$? I think I am just getting confused about having both $x^T$ and $x.$ My intuitive answer would be: $\nabla f(x) = \frac{1}{2} Qx - b$ but I'm not sure if…
juliana paternoster
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Argmin: Understanding a Simple Calculation

I would have a short question: Let $A\in\mathbb R^{N\times N}$ be a symmetric and positive definite matrix, let $y\in \mathbb R^{N}$ and consider $$\alpha^{*} := \text{argmin}_{\alpha\in\mathbb R^{N}} \left\{\alpha^T \left( A^T A + \lambda…
Hermi
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Maximize area of a triangle with fixed perimeter

If perimeter of a triangle is $2d$, what is the length of sides so the triangle has maximal area? I found some solution using circle and angles, but I think I have to use derivatives. I need help.
user23709
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How close can get $x-y\sqrt{2}$ get to $1/3$.

Let $(x,y) \in \{1,\dots, 100\}^2$. Find the ordered pair such that $x-y\sqrt{2}$ is close to $1/3$ as possible. First, I noted that this is minimizing $|x-y\sqrt{2}-1/3|$, but I can't use Arithmetic-Geometric mean because $-1/3, -y\sqrt{2}$ are…
user797346
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Optimization with 'sort', or bilevel optimization with permutation matrix

Original problem: $$ \begin{aligned} \mathbf{w}^*&= \underset{\mathbf{w}}{\mathrm{Argmax}} ~ \text{Sort} \left[ R_1\mathbf{w} \right]^T \mathbf{p} - c||\mathbf{w}-\mathbf{w}_0||_1 \\s.t.~\\ ||\mathbf{w}||_\infty&\leq 1 \end{aligned} $$ Second…
user29988
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Trying to classify or find solution to a nonlinear optimization problem

What is the classification of this optimization problem: Find $X$ that maximizes: $$ \text{Find X that maximizes:} \,\,\, \sum_{i = 1}^n y_i^TXX^Ty_i \\ S.T. \,\,\,X^TX = I $$ I read that the optimal solution is related to the eigenvectors of the…
Frank
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Grid computing: how to redistribute calculation to optimise response time

I have set up an infrastructure for grid computation. That is, an operation of N iterations is distributed and independently calculated on different servers, and the final result is the aggregation of the individual results. Each server has a…
Luca Fagioli
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