Nonlinear programming problem, mathematical programming, Summation Function.

[malamenm]

I was wondering if you might have some insight into a problem, where we consider an optimization problem:

max ∑ from j=1 to n of f_j(x_j) such that ∑ to n of xj <=B
x_j>=0, integers

where B is a positive integer and f_j is real to real

I am trying to formulate a solution using dynamic programming and to figure out the time complexity of this method.

Im a bit confused about the dynamic programming approach, how would you implement it for a function such as f₁(x)=sqrt(x) if n=5 and B=10kind regards

 

[Aaron Bex]

Dynamic programming is a powerful technique that can be used to solve optimization problems such as the one you have presented. In this particular case, the algorithm would need to find the optimal value of x1, x2, x3, x4, and x5 such that ∑ from j=1 to n of fj(xj) is maximized, while also ensuring that ∑ to n of xj <=B.

One approach to solving this problem with dynamic programming is to use a two-dimensional array where each entry contains the final value of the sum when xj is equal to some value from 1 to B. We then fill the array from left to right and top to bottom, starting with the first row. At each entry in the array, we calculate the maximum value of the sum by considering the two possible cases - either including the current xj or excluding it. The time complexity of this approach would be O(B^2 * n).

For example, if n=5 and B=10, and f1(x)=sqrt(x), then the process of filling the array would involve calculating the following values:

f1(1) + max(f2(1), f2(2)) + max(f3(1), f3(2)) + max(f4(1), f4(2)) + max(f5(1), f5(2))