Find the minimum sum of a series of equations

[alexleong]

I’m dealing with a series of equations to find out the values of x1 and x2 so that the sum of S0+S1+...+Sn will have the minimum value.
The x1 and x2 values are limited to –1<x1<1 and –1<x2<1.

S0 = 0
S1 = a1 – [B(1 – x1) + a0* x1 – S0*x2]
S2 = a2 – [B(1 – x1) + a1* x1 – S1*x2]
S3 = a3 – [B(1 – x1) + a2* x1 – S2*x2]
S4 = a4 – [B(1 – x1) + a3* x1 – S3*x2]
S5 = a5 – [B(1 – x1) + a4* x1 – S4*x2]
...
Sn = an – [B(1 – x1) + an* x1 – Sn-1*x2]
n
T = $$\Sigma$$Sn
n=1

Where
B is a constant.
T is the minimum sum of the equations.
Sn is the result of each equation.
a0, a1, a2...,an are the coefficients of the equation.

Hope you understand my question and thanks a lot.

 

[TomyDuby]

I see. Let me see:

S0 = 0
S1 = a1 – [B(1 – x1) + a0* x1 – S0*x2] = a1 - [B(1 – x1) + a0* x1]=
= (a1 - B) + (B - a0) * x1

Assume n = 1.

T = S1 = ((a1 - B) + (B - a0) * x1

This is a polynomial of first degree in x1. It has a minimum at x1 = + or -1 depending on the sign of B - a0.

Now calculate S2:

S2 = a2 – [B(1 – x1) + a1* x1 – S1*x2] =
= a2 – [B(1 – x1) + a1* x1 – ((a1 - B) + (B - a0) * x1)*x2] =
= a2 - B + (B+a1)*x1 + (a1 - b) * x2 + (B - a0) * x1 * x2

and for n = 2 T will be a polynom of second degree in x1 and x2. (the second degree comes from the product x1 * x2)

Similarly for higher n. The result, T, is a polynom in x1 and x2. It is not that difficult to find a minimum of such a polynom.

Anything else?

 

[alexleong]

hi Tomy,
First of all, thanks for your prompt reply. I still not quite get what you meant, how do I know the values of x1 and x2 where (S0 + S1 + S2) is the minimum? Please show me a simple example, let n = 2
S0 = 0
S1 = (a1 - B) + (B - a0)*x1
S2 = a2 - B + (B - a1)*x1 + (a1 - B)*x2 + (B - a0)*x1*x2

S0+S1+S2 = (a1 - B) + (B - a0)*x1 + a2 - B + (B - a1)*x1 + (a1 - B)*x2 + (B - a0)*x1*x2

Thanks

 

[TomyDuby]

Ok, so let me look at the case of n = 2. Consider function T of variables x1 and x2. From analysis of function of several variables we know that this function attains an extremum (that is either a minimum or maximum) only if the partial derivatives of T with respect of x1 and x2 vanish:

dT/dx1 = 0
dT/dx2 = 0

In your case T is a polynom linear in x1 and x2 but it has a term that is x1*x2.

To decide whether this extremum is a minimum or maximum we have to look at the second derivatives. Again from analysis of functions of several variables we know that function T has a minimum if see "[URL
http://en.wikipedia.org/wiki/Second_partial_derivative_test.[/URL]

Can you continue from here?