Dynamic Programming: Solving Optimization Problems with Infinite Horizon

[yamdizzle]

How do we solve optimization problems with infinite horizon. I tried to look online for some guidance but nothing but just problems and no solution methods. For example how can I solve:

maximize a_t $\in$[0,1]
$\sum\frac{-2a_t}{3}$+log(S_T)
where sum goes from 0 to T-1
subject to: s_t+1 = s_t *(1+a_t)

Some sources say we can use backwards induction but doesn't really tell me how I can do so.
Can someone explain the methodology or direct me to somewhere that explains it.

Thanks

 

[gtoguy97]

!The method you are referring to is called dynamic programming, and it is a powerful tool for solving optimization problems with infinite horizons. Essentially, the idea is to break down the problem into smaller subproblems, which can then be solved independently. This is done by starting at the end of the horizon (T) and working backwards until you reach the initial state (t=0).At each step, you need to calculate the maximum value of the objective function, given the current state and the decisions available. This is done by considering all possible decisions, and choosing the one that yields the highest value. Once you have computed the optimal decisions for each subproblem, you can then combine them to obtain the optimal solution for the entire problem.For your example, you could start by calculating the optimal decision for time T-1, given the objective function and the constraint (S_T+1 = S_T*(1+a_t)). Then, you would work backwards, calculating the optimal decision for each time t from T-1 to 0.I hope this helps!