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").

22512 questions
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Lagrange multipliers and steepest-descent direction

Let $h: R^n \to R$ be a continuously diferentiable function. I know how to prove that there exists a solution to the steepest-descent direction of h at a, and the solution is given by $$ \frac{Dh(a)}{|Dh(a)|}.$$ But I wonder if there is a way to…
Bayesian guy
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Optimal stacking with varying heights

Given a set of 24 boxes with varying heights. How can one optimally stack the boxes into 3 columns, with the least difference in column height. The width of all boxes is equal. The # of boxes in each column does not have to be equal. Best case…
wdm
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How would I maximize $Ae^{-(x-b)^2}+Be^{-(x-c)^2}$?

Someone me asked this and I was unable to answer. How can I maximize the function $f(x)=Ae^{-(x-b)^2}+Be^{-(x-c)^2}$? Progress: For $A=B$, this is the same as maximizing $-(x-b)^2(x-c)^2$, which is easy. If $A\neq B$, I'm not sure how to factor in…
add
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Separation problem for cutting plane algorithm

I have a question regarding the separation problem that comes up when using a cutting-plane method. The separation problem is solved with a greedy type algorithm. Let's say that the initial problem have one constraint and x is binary. (Knapsack…
Ulrich_Peters
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Solution by dynamic programming

I want to find the solution by dynamic programming of the following problem. Minimize $f(x_1,x_2)=e^{-x_1^a}+e^{-x_2^a}$ subject to $e^{-x_1}+e^{-x_2}=b, x_1\geq x_2>0, a,b \in (0,1]$ I started with the second stage min$e^{-x_2^a}$ subject to…
Quema
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Finding argmax subject to norm constraint via parametrization and Lagrange multipliers

got a two-part question here. Given (symmetric, positive definite) matrix $A$ = $$\begin{bmatrix} 2 & -1 \\ -1 & 2 \\ \end{bmatrix} $$ solve the problem: $arg max$ $g(x) = (Ax,x)$ subject to $\lVert x\rVert^2 = 1$. Use two methods: (1) Use…
BigT
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Principle "slice" of a function over hyperdimensional space

How can I get the principle 2d "slice" (of possibly limited size) of a function $R^N→R$ such that the variance of values inside is maximized? For example, when trying to optimize the matrix $W$ in order to minimize the…
ZisIsNotZis
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Optimize stocks for the investment

I don't know the problem name exactly, but it should be related to stock investment strategy. I have budget A = 30 Consider this list: myList = [[10,15], [11,13], [9,10]] for each element the first is expected return value, and second is the price.…
Aryan Firouzian
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Maximization of a weighted average

Suppose that $f$ is a weighted average of the functions $g_i$, $$f(x) = \sum_{i=1}^N \lambda_i \cdot g_i(x)$$ (where each $\lambda_i >0$). Under what conditions is the maximum value of $f$ a weighted average of the maximum values of the $g_i$? $$…
JDG22
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How to minimize $\frac{1}{y_1} + \frac{1}{y_2} + \cdots + \frac{1}{y_k}$ subject to $y_1+y_2+ \cdots + y_k = a$?

We seek how to minimize $$ \frac{1}{1-x_1} + \frac{1}{1-x_2} + \cdots + \frac{1}{1-x_k} $$ for $x_1,\ldots,x_k \in (0,1)$ subject to $$ x_1+x_2+ \cdots + x_k = c $$ for some constant $c \in (0,k)$. We expect this minimum occurs when…
Rebecca J. Stones
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Extremum Problem

How big is the circle? My first steps: $ x^2 + y^2 =r^2$ $ f(x)=y=e^{-x^2}$ Substitute $y^2$ in $x^2 + y^2 =r^2.$ So, $x^2 + e^{-2x^2} =r^2$ Is this way correct? Because after calculating the first derivative $2x -4xe^{-2x^2}$ and…
Lars Baumann
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Relaxation of a (possibly infeasible) QP that is always feasible while remaining a QP

I am currently working on a problem that requires a solution to a standard QP of the form $$ \min_{z \in \mathbb{R}^n} z^TQz + q^Tz \quad s.t. Az \leq b, \, Q \succ 0, $$ with $b \in \mathbb{R}^m$ that that could potentially be infeasible ($Az \leq…
Torsten Koller
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Calculating maximum amount of objects by weight in a container

first time posting, with a probably very simple question for you. This is something that I need for work: I have a container to fill with exactly 250 objects, and a maximum weight capacity of 2339g I have two kinds of objects to fill the container…
Deagol
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minimize $L(i, n)$ for a given $n \in \mathbb{N}$

Given some fixed $n \in \mathbb{N}$, I want to minimize the following function of $i$: $$L(i, n) = \Bigl\lceil \frac{n}{i} \Bigr\rceil + i$$ within limits $1 \leq i \leq \bigl\lceil \frac{n}{2} \bigr\rceil,\; i \in \mathbb{N}$ Can somebody help me…
Truth-seek
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Optimality criteria and gradients

I'm working on the following exercise: Let $F = \{x \in \mathbb{R}^n \mid Ax = b, x > 0\}$ where $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$. Further, let $f: F \to \mathbb{R}$ be a differentiable function. We assume that the following…
3nondatur
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