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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Maximizing the sum of rotated vectors

I came across an optimization problem as \begin{equation*} \underset{\{\theta_i\}_{i=1}^n}{\max}\quad\left\Vert\mathbf{y}+\sum_{i}^ne^{\jmath\theta_i}\mathbf{x}_i\right\Vert, \end{equation*} where $\mathbf{x}_i$ and $\mathbf{y}$ are $m$-dimensional…
Alex
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Why minimizing mean-square error maximizes Gaussian likelihood

Can someone help me to understand this slide? Minimizing mean-square error maximizes Gaussian likelihood $$f ( x | \theta ) = \frac { 1 } { \sqrt { 2 \pi \sigma ^ { 2 } } } e ^ { - \frac { ( \alpha - \mu ) ^ { 2 } } { 2 \sigma ^ { 2 } } }$$…
YohanRoth
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Multiple loan repayment => Optimization problem

What prompted this question was that I was reading advice on paying down multiple loans and it was suggested that the optimal strategy was to pay down the highest interest rate loan first, which does not seem entirely accurate. Instead it seems that…
user1543042
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How to prove a method converges quadratically and determine the rate constant

Apply Newton's method to $f(x)=(x-2)^4+(x-2)^5$ with initial guess $x_0=3$. We can observe that the sequence converges linearly with rate constant $3/4$. Now apply the iterative mathod $x_{k+1}=x_k-4f(x_k)/f'(x_k)$. This method should converge more…
i_a_n
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Optimization Problem: Shortest Path on River with Multiple Bends

I’m struggling to convert the following problem into an optimization problem. Consider a river. You’re traveling on that river. The river has multiple bends. Essentially, given a starting point on the river and an endpoint on the river, I need to…
user12888
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How to calculate the best points to buy and to sell?

There are pairs of buying and selling prices. For example: 101 99 100 98 102 100 105 102 101 99 ... I want to calculate maximum possible profit on historical data for the conditions: we buy and then we cannot buy until we sell it. Then we can…
user1034253
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optimizer of a supermodular function

Consider the problem $$\max_x \{xy+f(x)\},$$ where $x$ and $y$ are in $R^1$. Denote the optimal solution by $x^*(y)$. When multiple solutions exist, we take the smallest one. We are interested in the monotonicity of $x^*(y)$. Argument 1: since the…
Justin
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How to obtain a point that is nearest to other points?

Here is the question that I meet. $$\min_z{\sum_j{\max(0,\Vert z-f_j\Vert_2^2-m_j^2)}}$$ Intuitively speaking, I want to obtain a point $z$ that is nearest to all other points $f_j$, but, when the distance between $f_j$ and $z$ is less than $m_j$,…
Wei Zhu
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Optimisation on shape of a function

This is a basic level question as I have no prior experience with optimizing on shapes. I have a dataset and I am fitting a model to it. The equation of the model I want to fit is $$R(0,\theta)=\beta_0+\beta_1 \left[…
Dhruv Mahajan
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What mathematical operations are used to determine maximum resource utililzation?

I'm playing a game and would like to leverage mathematics to use optimal resource utilization. There are three troops I can build (ground, air, and horse). There are also three resources (wood, stone, and ore). The costs are troop wood stone …
corsiKa
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Different approaches of genetic algorithm

I wrote a code that implements a simple genetic algorithm to maximize: $$f(x) = 15x - x^2.$$ The function has its maximum at $x=7.5$, so the code output should be $7$ or $8$ since the population are integers. When I run the code ten times I get $7$…
Mohammad Nur
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Minimize sum of matrix columns based on selected set of rows

For an academic project, we are trying to solve the following problem. Given $k$ and a matrix $M$ with $r$ rows and $c$ columns, we want to find a set $K$ of $k$ rows such that $$\sum_{i=1}^c \min_{j \in K} M[j][i])$$ is minimal. It can be seen…
Jan N. van Rijn
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At each iteration of the column generation approach, must the value of the MP's objective function change?

I am implementing a column generation approach to my problem. The Master Problem (MP) is initialized with some initial columns. Then, the MP is solved and in sequence, the pricing subproblem is solved. Some columns with negative reduced costs are…
rbl
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Minimize L2 norm with binary weights

I have the following minimization problem: $\min_{b_i,\theta}\frac{1}{N}\sum_{i=1}^N\|y_i-f(x_i;\theta)b_i\|_2 $ where $b_i\in\{0,1\}$ I don't expect any solution to this but I would like to know how this is commonly called, i.e. under which terms I…
jmaxx
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Confusion related to a convex optimization problem

I was going through Stephen Boyd's lecture on convex optimization. However, I am a bit confused about a problem Given Minimize $f(x) = x_1^2+x_2^2$ subject to $f_1(x) = \frac{x_1}{1+x_2^2} \leq 0$ How come $f_1(x)$ is not convex? I was going…
user34790
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