Are you sure that is what the problem says? R*P is a number while C is a vector. The sum C+ R*P is not defined.

Or do you mean I= (C+ R)*P?

This product is NOT "dot product". You appear to be using [A, B]*[X, Y]= [AX, BY], a "coordinate wise" product.Here is an example with n = 2 (I am writing these vectors transposed so I can type it easily). C is given as [100 120]. P is given as [50 40]

[I1 I2] = [100 120] + [R1 R2] * [50 40]

With n = 2, this breaks down to a system of 2 equations that I was able to solve:

I1 = 100 + (1 - R2) (50)

I2 = 120 + R2 (40)

I set I1 = I2 since this would maximize the minimum value between I1 and I2 (however even this approach would break down under certain conditions).

100 + 50 - 50R2 = 120 + 40R2

150 - 50R2 = 120 + 40R2

30 = 90R2

R2 = 1/3, R1 = 2/3

Is there any way to generally solve this problem? Algorithmic solutions would be helpful as well.