Calibration with Positive
Mathematical Programming
(PMP)
Uwe A. Schneider
Strategy
• Modify the objective function such that the
first order conditions look exactly as the
first order conditions of the unmodified
problem with an additional constraint that
forces the solution to be equal to the
observations/expectations.
Original linear model
Max
Z  cj X j
j
s.t.
a
ij
X j  bi
j
Xj 0
First Order Conditions (FOC) of
original linear problem


L  X j , i    c j X j   i   aij X j  bi 
j
i
 j

1
L  X j , i 
1
x j
L  X j , i 
1
i
 c j   i aij  0
i
  aij X j  bi  0
j
Force solution to replicate
observed values
Max
Z  cj X j
j
s.t.
a
ij
X j  bi
j
Xj x
Xj 0
0
j
FOC when Solution is forced to
observed values


L  X j , i , j    c j X j   i   aij X j  bi    j  X j  x 0j 
j
i
 j
 j
2
L2  X j , i , j 
x j
L2  X j , i , j 
i
L2  X j , i , j 
 j
 c j   i aij   j  0
i
  aij X j  bi  0
j
 X j  x 0j  0
Objective function adjustment
Max
c X
j
j
 f X j
j
s.t.
a
ij
X j  bi
j
Xj 0
The following two conditions must hold:
f  X j 
x j
  j
Xj  x
0
j
Quadratic Term (often used in PMP)
f  X j   
jX
j
f  X j 
x j
2x
o
j
  j
X j  x0j
2
j
A more general term
f  X j    j
f  X j 
x j

Xj
  x
  j
X j  x0j
0
j

 1
Conclusions
• There are infinite alternatives to modify the
objective function, which all perfectly calibrate
the model
• Different alternatives do not affect the basic
solution but will impact scenario solutions
• If PMP is used, make sure you can justify the
value of the parameter alpha
• If alpha is not determined, try to use a relatively
small value which corresponds to a flat slope
• Try to avoid calibrating non-observed activities
because there is no justification for this