find an optimal value for t(time) on interval [t_min, t_max]

I want to find minimum time (t) between the interval [t_min, t_max]. I have found different algorithms, like steepest descent method, simplex method. And I am not sure how to approach this methods, say steepest descent algorthim, and how to select the tolerance value. so any help would be appreciated if I get how to implement steepest descent method to find optimal t on the interval [t_min,t_max].

Thanks in advance

Hashir

[Sam Chak]

Advised to provide a toy problem if it is inconvenient to show the original mathematical problem.

[Torsten]

@Hashir Roshin Valiya Parambil

The minimum time in [tmin,tmax] is tmin.

If you mean something different, you will have to describe your problem more clearly.

[VAHA]

@Torsten, I'm sorry, that was my mistake. I need to find an optimal time between tmin and tmax. Optimal time should not be either tmin nor tmax, but between those intervals.

[VAHA]

@sam chak, Suppose, an object needs to start from one point and reach at the target, with minimal energy consumption. But tmin is lower limit for which the travel is not feasible but tmax is more expensive. I need between them.

Answer 1

[Sam Chak]

@Hashir Roshin Valiya Parambil

I think I get what you mean. However, without understanding the nature of your problem, it is difficult to give a particular suggestion. Try reading about Brachistochrone curve and the Calculus of variations.

[VAHA]

@sam chak, I know the Brachistochrone problem. Here my problem, is to find an optimal time to steer my vehicle from an initial position to target location with minimum fuel consumption. So my actual problem is to find the minimum fuel consumption, that is constrained optimization. But before that, I need to find optimal final time , and that time must be between tmin and tmax. Tmin is the time where no fuel is used , and tmax is the time where complete fuel is used. Tmin and tmax is a fixed quantity based on the boundary conditions. And my optimal time, t should be tmin<=t<=tmax. I hope this explanation will help you understand my problem.

[Sam Chak]

Okay @Hashir Roshin Valiya Parambil. I understand your problem.

Since you understand the mathematics of Calculus of Variations and the Brachistochrone problem, I presume that you are really good at math but that doesn't solve your problem.

If this is 2D flat surface problem, I guess you need to find the shortest path to the target location, and naturally, that will be the minimum fuel consumption required. Read about the following:

1. Travelling salesman problem
2. Dijkstra's algorithm
3. A* search algorithm