MPEC: Mathematical Programs with Equilibrium Constraints
Mathematical Programs with Equilibrium Constraints (MPECs) are formulations that can be used to model certain classes of discrete events. MPECs can be more efficient than solving mixed integer formulations of the optimization problems because it avoids the combinatorial difficulties of searching for optimal discrete variables.
SIGN Operator
! MPEC formulation for SIGN function
! y = SIGN(x) returns a value y, where:
! 1 if the corresponding element of X is greater than zero
! -1 if the corresponding element of X is less than zero
Model sign
Parameters
x = -2
End Parameters
Variables
y >= -1, <= 1
s_a >= 0
s_b >= 0
End Variables
Equations
! test sign operator, y = sign(x)
x = s_b - s_a
minimize s_a*(1+y) + s_b*(1-y)
End Equations
End Model
Absolute Value (ABS) Operator
! MPEC formulation for ABS function
! y = ABS(x) returns a value y, where:
! y = x if the corresponding element of X is greater than zero
! y = -x if the corresponding element of X is less than zero
Model abs
Parameters
x = -2
End Parameters
Variables
y
s_a >= 0
s_b >= 0
End Variables
Equations
! test abs operator, y = abs(x)
x = s_b - s_a
y = s_a + s_b
minimize s_a*s_b
End Equations
End Model
Minimum Selector (MIN) Operator
! MPEC formulation for MIN function
! y = MIN(x1,x2) returns a value y, where:
! y = x1 if x1 < x2
! y = x2 if x2 < x1
Model signum
Parameters
x1 = -2
x2 = -1
End Parameters
Variables
y
! slack variables
s_a >= 0
s_b >= 0
End Variables
Equations
! test min operator, y = min(x1,x2)
x2 - x1 = s_b - s_a
y = x1 - s_a
minimize s_a*s_b
End Equations
End Model
Maximum Selector (MAX) Operator
! MPEC formulation for MAX function
! y = MAX(x1,x2) returns a value y, where:
! y = x1 if x1 > x2
! y = x2 if x2 > x1
Model signum
Parameters
x1 = -2
x2 = 4
End Parameters
Variables
y
! slack variables
s_a >= 0
s_b >= 0
End Variables
Equations
! test max operator, y = max(x1,x2)
x2 - x1 = s_a - s_b
y = x1 + s_a
minimize s_a*s_b
End Equations
End Model
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