Reduced Gradient Descent
Calculating the maximum semi-ellipsoid volume using Reduced Gradient Descent method with total surface area given as a constraint.
Reduced Gradient Descent and Newton methods are implemented from scratch. The method is designed to work with 3 variables and one constraint but can easily be modified to support more.
Algorithm
Reduced gradient is calculated as follows, with B - basic variable vector, NB - non-basic variable vector.
The step for basic variables is calculated according to the negative reduced gradient.
By solving the system of constraint equations we get non basic variables.
Examples
The following examples use the ellipsoid volume formula divided in half
and the total ellipsoid surface area formula divided in half as a constraint.
With surface area equal to 100
| Start X | Start volume | Time (s) | Iterations | Result X | Result |
|---|---|---|---|---|---|
| (2 3 4) | 50.24 | 0.333 | 79 | (3.9898 3.9898 3.9913) | 132.999 |
| (1 2 8) | 33.49 | 0.336 | 81 | (3.9898 3.9898 3.9913) | 132.999 |
| (0.8 0.8 0.9) | 1.21 | 0.350 | 80 | (3.9898 3.9898 3.9913) | 132.999 |
| (0.5 0.6 7) | 4.39 | 0.346 | 80 | (3.9898 3.9898 3.9913) | 132.999 |
Plot with starting point (2 3 4)
With surface area equal to 24.5
| Start X | Start volume | Time (s) | Iterations | Result X | Result |
|---|---|---|---|---|---|
| (1 2 3) | 12.56 | 0.368 | 12 | (1.9727 1.9777 1.9751) | 16.1304 |
| (1 2 5) | 10.47 | 0.375 | 5 | (1.9761 1.9761 1.9767) | 16.1585 |
| (0.5 0.5 10) | 5.23 | 0.364 | 7 | (1.9776 1.9776 1.9746) | 16.1649 |
| (0.7 0.2 0.8) | 0.23 | 0.334 | 10 | (1.9764 1.9737 1.9749) | 16.1263 |
Plot with starting point (0.7 0.2 0.8)
