Appendix B : Separable Programming Technique for Nonseparable Function.
We will consider only two dimensional function but the methodology can be easily
extended to multi-dimensional function. Consider a nonseparable function z  F ( x, y )  0
defined on X max  x  X min  0 and Y max  y  Y min  0 .
The basic idea of separable
programming is that F ( x, y ) can be approximated by sufficient number of pieces of linear
functions. 22
We define a grid of values of ( x, y ) (not necessarily equidistant).
values of ( x, y ) at the grid points are denoted by ( X p , Y q ) .
The
For a equidistant grid,
( X p , Y q ) is defined as
X p  X min 
Y q  Y min 
p
( X max  X min ) , ( p  0, 1, 2, ..., P )
P
q max
(Y  Y min ) , ( q  0, 1, 2, ..., Q )
Q
(B1)
(B2)
Each grid point is associated with a nonnegative weighting variable 0   pq  1 where
P
Q
 
pq
 1 . The function value z  F ( x, y ) can be approximated by linear combination
p 0 q 0
of the function values of the nearest 4 grid points surrounding ( x, y ) .
P
Q
z   F ( X p , Y q ) pq
(B3)
p 0 q 0
The values of  pq are designed so that only 4 of them associated with the nearest 4 grid
points surrounding ( x, y ) are nonnegative and the others are zero.
76
In order to make such
array of  pq values, we introduce two Specially Ordered Sets Type 2 (SOS2) variables  p
and  q with respect to ( x, y ) .
P

p
1
(B4)
q
1
(B5)
p 0
Q

q 0
P
X
p
p x
(B6)
p 0
Q
Y 
q
q
y
(B7)
q0
Note that  p and  q have only two consecutive nonnegative values and all the others are
zero. Then,  p and  q are related with  pq as follows:
Q
 p    pq
(B8)
q 0
P
 q    pq
(B9)
p 0
GAMS/CPLEX, GAMS/GUROBI and GAMS/XPRESS have the capability of SOS Type 2
formulation. When
F ( x, y ) is not strictly positive, a positive constant should be added
and subtracted from Eq. B3. One two dimensional term F ( x, y ) produces PQ continuous
variables, P+Q SOS2 variables and P+Q+4 constraint equations.
77