Tldr: Is it possible to somehow correct a coordinate system by the maximization function when using the simplex algorithm so it would be possible to just always follow the steepest slope from point to point instead of plugging in the slope into the maximization function?
Sidenote as to how I understand the simplex algorithm: When using the simplex algorithm to find the point on a body at which some function is maximized, we start at a random point on the body and calculate the slope to each adjacent point. We plug this slope into the maximization function. This results in a number that tells us how much we get into the direction of the optimal point on the body. When always following the biggest number, whe should eventually reach a point from where all adjacent points result in a negative number, i.e. a maximum.
Practical example: I tried to solve a production planning problem with three different products. The problem included 8 equations which served as boundaries and a function that should be maximized. (I left all these equations out of the question as I am not sure they provide more insight instead of just cluttering it.)
I visualized the 8 boundary equations as an 8 sided body in a three dimensional coordinate system (as each boundary equation results in a plane) in which the three coordinates represent the number of produced products respectively. The maximization function can be visualized as a plane in this coordinate system as well. My first instinct was to somehow "turn" the 8-sided body with the maximization function so that the three points at which the maximization function meets the zero points are at an equal distance (i.e. turn the plane and the body so that the plane meets the x, y and z axis where they are equal). This would basically eliminate the maximization function and we could just look for the point on the body which has the highest cumulative coordinates (I think).
So instead of calculating the slope to each adjacent point and plugging it into the maximization function to find the next point, we could just calculate the slope and always follow the steepest slope.
Is this possible? If so, how could it be achieved?
I am sorry if I used confusing terms as I learned math in another language and had to translate them into english to the best of my knowledge.