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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How to maximise functions of this shape $y=2\cdot3^{-x}$

How can I find the maximum of $2\cdot 3^{-x}$? I know its close to $1$ because I have seen its graph, but when I differentiate the function and set it equal to zero (to get a maximum) I get $-2\cdot 3^{-x}=0$. What does that mean? How can I solve…
Sophie
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How to minimize such an energy?

For the Energy, $$ Q(u)=\int_I(1+|u'(x)|^2)^{1/4} dx $$ where $u(0)=0$ and $u(1)=1$, $u$ is $C^1$ in $I$ and continuous up to the boundary, $I=(0,1)$. How to show the infimum of $Q$ is $1$?
Yourent
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For each of the following functions, what do the first- and the second-order optimality conditions say about wether 0 is a minimum on $\mathbb{R}$.

For each of the following functions, what do the first- and the second-order optimality conditions say about wether 0 is a minimum on $\mathbb{R}$. $f_1(x)=x^2$ $f_2(x)=x^3$ $f_3(x)=x^4$ $f_4(x)=-x^4$ So the dazzling thing is that the second…
onimoni
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On a quadratic optimization

The problem is formulated as follows: Given $0\neq x \in \mathbb{R}^n$, and $k\leq n$, consider the following optimization problem $$\min_{\textrm{rank}(C)=k}x^t(I_n-C)^t(I_n-C)x$$ where $I_n$ be the unit matrix of $\mathbb{R}^{n\times n}$ and the…
Roy Han
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Optimizing duration of activities

I would like to understand optimization through a simple application, and then progressing towards understanding more general concepts. My inquiry starts with its application: Optimizing duration of activities in a day. The length of the day and…
tropsobor
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Optimization problem with many inequality constraints

I have a maximization problem with 4 variables and six inequality constraints. This problem is already solved and I have access to its solution. However, I would like to solve it analytically (not numerically) without looking at the solution. I am…
PaulS
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Maximizing $\frac{\sum w_i a_i}{\sqrt{\sum w_i^2 b_i}}$ with $\sum w_i = 1$

Intuitively, $\frac{\sum w_i a_i}{\sqrt{\sum w_i^2 b_i}}$ looks like a ratio between a $L_1$ norm and $L_2$ norm. To provide motivation, we consider i.i.d. $(x_i, y_i)$ and $\text{Var}[x_i] = \sigma_{i}^2, \text{Var}[x_i] = \tau_{i}^2,…
david
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Understanding Pareto Optimality for a multiple criteria optimization problem

In the book (Anders Klarbring and Peter W. Christensen, "An introduction to Structural Optimization"), the Pareto optimality has been briefly mentioned which is a method used in the case when we have an optimization problem that involves several…
user134613
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Starting and finishing on different points (minimum moves)

A road is constructed with n lines (n>2) sharing exactly one point in the center. There are n cars, each at a unique endpoint. Cars starts facing the center Cars can move forwards and backwards, each considered 1 move. Cars can't pass each other…
Prim3numbah
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Finding a matrix $M_{n\times m}$ that minimizes a sum conditional on $M1_m=c_m$ and $1_n^tM=c_n$

I am trying to implement a paper, and one of the problems that I need to solve is the following: The domain is $$\Pi(\pi_0, \pi_1)=\{w\in\mathcal M_{K_0,\,K_1}(\mathbb R^+) : w1_{K_1}=\pi_0 \text{ and }1_{K_0}^tw=\pi_1\}$$ and I want to find…
AnotherSherlock
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When optimizing a $C^2$ function with $L$-Lipschitz continuous gradients, show that $\alpha_k \ge \frac{2 \tau (1 - c_1)}{(1 + \theta) L}$

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a $C^2$ function with $L$-Lipschitz continuous gradients such that for some $L > 0$: \begin{gather*} {\lVert \nabla f(x) - \nabla f(y) \rVert} \le L {\lVert x - y \rVert}, \quad \forall x,y \in \mathbb{R}^n…
clay
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Fitness function example

I promised my son that I will understand myself and then explain him particle swarm optimization (PSO) with flock of birds or school of fish as an example. Every tutorial on the web starts with animals for the motivation but none uses a fitness…
Tims
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Are minimizing $f(x)$ and $k\cdot f(x)$ the same?

Suppose I want to minimize an objective function $f(x)$ with respect to some constraints $g(x) = b$. Is it true that minimizing $f(x)$ and $k\cdot f(x)$ is equivalent in the constrained optimization when $k$ is a positive constant? I feel it is true…
shani
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How to solve the following constrained optimization problem?

Consider the following optimization problem, where $t_i$ is known $$ \max_{0\leq a_1,\cdots,a_k\leq 1} \sum_{i=1}^k a_it_i $$ I want to ensure that the following inequality holds after optimization is completed $$ a_1\leq a_2\leq \cdots \leq…
korangar leo
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whether unbounded objective function can be studied as an optimization problem

An optimization problem typically deals with an objective function to be maximized or minimized that can very often be written in a quadratic form. If the objective function can be written as a positive definite quadratic function, it can be…
feynman
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