Optimization plays a vital role in everyday life, operating behind the scenes in numerous familiar scenarios. It determines the quickest routes in navigation apps like Google Maps, compares prices in online shopping, and improves energy efficiency through smart thermostats. Airlines optimize flight and crew schedules, while fitness trackers customize workout and diet plans. Streaming services like Netflix adjust video quality to internet speed, social media platforms curate personalized feeds, and manufacturers align production and inventory with demand.
Unconstrained optimization refers to the process of finding the minimum or maximum values of a function without any restrictions or limits on the variables involved. This type of optimization seeks the optimal points in the function's domain where it achieves its extreme values freely.
Objective: To find the minimum or maximum of a function $y = f(x_1, x_2)$ without any restrictions on the variables $x_1$ and $x_2$. For example, to find the minimum of $y = x_1^2 + (x_2 + 3)^2 + 4$.
o
Visual Approach:
By plotting the function on a contour plot, we can visually inspect the levels of y and identify the low points in the function.
yy
A contour plot with discrete y values can help us derive corresponding x1 and x2 values by observing where the contours are tightest, indicating the function’s minimum.
yy
x1x_1
x2x_2