Minimize |(2 x-y, x+y)^T ^2-6 x over all x, y, where · denotes the Euclidean norm on ℝ^2.

Name: Problem 26, Chapter 7: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2020-10-27T18:41:50-08:00
Duration: 3 min 8 s
Channel: Noah Musser
Description: Minimize |(2 x-y, x+y)^T ^2-6 x over all x, y, where · denotes the Euclidean norm on ℝ^2.

step 1 Right, this is problem number 10. In 10, we're looking for the minimum. These two equations are

Minimize |(2 x-y, x+y)^T ^2-6 x over all x, y, where ·...

Key Concepts

Euclidean Norm

The Euclidean norm is a measure of the length or magnitude of a vector in Euclidean space. It is commonly defined as the square root of the sum of the squares of the vector's components, and it is fundamental in expressing distances and magnitudes in geometric and analytic contexts.

Quadratic Forms

Quadratic forms are expressions involving squared variables possibly combined with cross-product terms, which together form homogeneous polynomials of degree two. They are central to many optimization problems because their structure often implies convexity, making it possible to find global minima or maxima analytically.

Unconstrained Optimization

Unconstrained optimization refers to finding the extremum (minimum or maximum) of a function without any restrictions or constraints on the input variables. This approach is frequently applied in problems where the solution must be determined solely by the behavior of the function itself, often leveraging calculus-based methods.

Gradient and Critical Points

The gradient of a function, comprising its partial derivatives, plays a key role in optimization by indicating the direction of steepest ascent. Setting the gradient to zero yields critical points, which are potential candidates for minima, maxima, or saddle points in the function's domain.

Completing the Square

Completing the square is a technique used to transform quadratic expressions into a form that clearly exhibits their minimum or maximum values. This reorganization allows for a straightforward identification of the vertex of the quadratic function, thereby facilitating the optimization process.

Transcript

00:01 Right, this is problem number 10.

00:03 In 10, we're looking for the minimum.

00:06 These two equations are both true, so we know we're looking to combine them to make a quadratic of some kind so that we can find a minimum amount.

00:14 That minimum is going to be this place where the derivative of the q is equal to zero, right? but we can put this on the terms of one variable.

00:25 We can't do it with two with the tools we have right now, so we're going to go ahead and we're going to make some substitutions.

00:31 Let's go ahead and say this is the same as y equals 3 minus x.

00:36 And that'll allow us to substitute this entire piece right here in 4 .y.

00:42 So let's rewrite that.

00:43 Q equals x squared plus 2 times this expression squared.

00:53 So let's go ahead and do a little bit of simplification to get through it.

00:57 We're going to have, i'll do it in steps, i guess.

01:00 So i have nine insides plus the outsides are going to be minus 6x plus x squared.

01:10 And we got q here again equals.

01:14 We're going to do this in a whole other line.

01:15 Make sure we don't lose anything.