## Mpec Examples

## Apps.MpecExamples History

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!!! Maximum Selector

* %list list-page%

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'''GEKKO MIN function (min2)'''

See [[https://gekko.readthedocs.io/en/latest/examples.html|GEKKO Documentation]] for additional examples.

See [[https://gekko.readthedocs.io/en/latest/examples.html|GEKKO Documentation]] for additional examples.

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from gekko import GEKKO

m = GEKKO()

x1 = m.Param(-2)

x2 = m.Param(-1)

# use min2

y = m.min2(x1,x2)

m.solve() # solve

print('x1: ' + str(x1.value))

print('x2: ' + str(x2.value))

print('y: ' + str(y.value))

(:sourceend:)

----

!!! Maximum Selector (MAX) Operator

* %list list-page% [[Attach:max.apm | MAX Operator Example]]

(:source lang=python:)

m = GEKKO()

x1 = m.Param(-2)

x2 = m.Param(-1)

# use min2

y = m.min2(x1,x2)

m.solve() # solve

print('x1: ' + str(x1.value))

print('x2: ' + str(x2.value))

print('y: ' + str(y.value))

(:sourceend:)

----

!!! Maximum Selector (MAX) Operator

* %list list-page% [[Attach:max.apm | MAX Operator Example]]

(:source lang=python:)

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(:sourceend:)

'''GEKKO MAX function (min2)'''

See [[https://gekko.readthedocs.io/en/latest/examples.html|GEKKO Documentation]] for additional examples.

(:source lang=python:)

from gekko import GEKKO

m = GEKKO()

x1 = m.Param(-2)

x2 = m.Param(-1)

# use max2

y = m.max2(x1,x2)

m.solve() # solve

print('x1: ' + str(x1.value))

print('x2: ' + str(x2.value))

print('y: ' + str(y.value))

'''GEKKO MAX function (min2)'''

See [[https://gekko.readthedocs.io/en/latest/examples.html|GEKKO Documentation]] for additional examples.

(:source lang=python:)

from gekko import GEKKO

m = GEKKO()

x1 = m.Param(-2)

x2 = m.Param(-1)

# use max2

y = m.max2(x1,x2)

m.solve() # solve

print('x1: ' + str(x1.value))

print('x2: ' + str(x2.value))

print('y: ' + str(y.value))

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!!! Absolute Value

* %list list-page%

to:

'''GEKKO SIGNUM function (sign2)'''

See [[https://gekko.readthedocs.io/en/latest/examples.html|GEKKO Documentation]] for additional examples.

See [[https://gekko.readthedocs.io/en/latest/examples.html|GEKKO Documentation]] for additional examples.

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from gekko import GEKKO

m = GEKKO()

x = m.Param(-2)

# use sign2 to define a new variable

y = m.sign2(x)

m.solve() # solve

print('x: ' + str(x.value))

print('y: ' + str(y.value))

(:sourceend:)

----

!!! Absolute Value (ABS) Operator

* %list list-page% [[Attach:abs.apm | ABS Operator Example]]

(:source lang=python:)

m = GEKKO()

x = m.Param(-2)

# use sign2 to define a new variable

y = m.sign2(x)

m.solve() # solve

print('x: ' + str(x.value))

print('y: ' + str(y.value))

(:sourceend:)

----

!!! Absolute Value (ABS) Operator

* %list list-page% [[Attach:abs.apm | ABS Operator Example]]

(:source lang=python:)

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# solve

m.solve()

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m.solve() # solve

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Model

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Model

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%width=50px%Attach:apm.png [[Main/Objects|APMonitor Objects]]

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'''GEKKO ABS function (abs2)'''

See [[https://gekko.readthedocs.io/en/latest/examples.html|GEKKO Documentation]] for additional examples.

(:source lang=python:)

from gekko import GEKKO

m = GEKKO()

x = m.Param(-2)

# use abs2 to define a new variable

y = m.abs2(x)

# use abs2 in an equation

z = m.Var()

m.Equation(z==m.abs2(x)+1)

# solve

m.solve()

print('x: ' + str(x.value))

print('y: ' + str(y.value))

print('z: ' + str(z.value))

(:sourceend:)

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!!!! Reference

Mojica, J.L., Petersen, D.J., Hansen, B., Powell, K.M., Hedengren, J.D., Optimal Combined Long-Term Facility Design and Short-Term Operational Strategy for CHP Capacity Investments, Energy, Vol 118, 1 January 2017, pp. 97–115. [[https://www.sciencedirect.com/science/article/pii/S036054421631814X|Article]]

Mojica, J.L., Petersen, D.J., Hansen, B., Powell, K.M., Hedengren, J.D., Optimal Combined Long-Term Facility Design and Short-Term Operational Strategy for CHP Capacity Investments, Energy, Vol 118, 1 January 2017, pp. 97–115. [[https://www.sciencedirect.com/science/article/pii/S036054421631814X|Article]]

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----

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* Mojica, J.L., Petersen, D.J., Hansen, B., Powell, K.M., Hedengren, J.D., Optimal Combined Long-Term Facility Design and Short-Term Operational Strategy for CHP Capacity Investments, Energy, Vol 118, 1 January 2017, pp. 97–115. [[https://www.sciencedirect.com/science/article/pii/S036054421631814X|Article]]

* Mojica, J.L., Petersen, D.J., Hansen, B., Powell, K.M., Hedengren, J.D., Optimal Combined Long-Term Facility Design and Short-Term Operational Strategy for CHP Capacity Investments, Energy, Vol 118, 1 January 2017, pp. 97–115. [[https://www.sciencedirect.com/science/article/pii/S036054421631814X|Article]]

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! SIGN function as an Object

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! SIGN function MPEC as an Object

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! ABS function as an Object

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! ABS function MPEC as an Object

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! MIN function MPEC as an Object

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! MAX function MPEC as an Object

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! SIGN function ~~with~~ Object

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! SIGN function as an Object

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! ~~MPEC formulation for ABS function using an object~~

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! ABS function as an Object

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(:html:)<font size=2><pre>

Objects

f = max

End Objects

Connections

f.x[1] = x1

f.x[2] = x2

f.y = y

End Connections

Parameters

x1 = -2

x2 = 4

End Parameters

Variables

y

End Variables

</pre></font>(:htmlend:)

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(:html:)<font size=2><pre>

Objects

f = min

End Objects

Connections

f.x[1] = x1

f.x[2] = x2

f.y = y

End Connections

Parameters

x1 = -2

x2 = -1

End Parameters

Variables

y

End Variables

</pre></font>(:htmlend:)

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(:html:)<font size=2><pre>

! SIGN function with Object

Objects

f = sign

End Objects

Connections

f.x = x

f.y = y

End Connections

Parameters

x = -2

End Parameters

Variables

y

End Variables

</pre></font>(:htmlend:)

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! y = x if the corresponding element of X is ~~greater~~ than zero

! y = -x if the corresponding element of X is~~less~~ than zero

! y = -x if the corresponding element of X is

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! y = x if the corresponding element of X is > than zero

! y = -x if the corresponding element of X is < than zero

! y = -x if the corresponding element of X is < than zero

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! 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

! this uses the APMonitor object 'abs'

! y = x if the corresponding element of X is greater than zero

! y = -x if the corresponding element of X is less than zero

! this uses the APMonitor object 'abs'

to:

! MPEC formulation for ABS function using an object

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End Objects

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to:

End Connections

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End Parameters

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End Variables

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Parameters

Variables

s_a >

Equations

x = s_b - s_a

y = s_a + s_b

minimize s_a*s_b

End Equations

End Model

to:

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

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

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(:html:)<font size=2><pre>

! 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

! this uses the APMonitor object 'abs'

Objects

f = abs

Connections

f.x = x

f.y = y

Parameters

x = -2

Variables

y

</pre></font>(:htmlend:)

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----

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----

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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.

to:

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.

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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 problems.

to:

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.

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Model ~~signum~~

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Model sign

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Model ~~signum~~

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Model abs

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y = x1 ~~+~~ s_a

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y = x1 - s_a

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! 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

</pre></font>(:htmlend:)

----

!!! Maximum Selector (MAX) Operator

* %list list-page% [[Attach:max.apm | MAX Operator Example]]

(:html:)<font size=1><pre>

! 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

! 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

</pre></font>(:htmlend:)

----

!!! Maximum Selector (MAX) Operator

* %list list-page% [[Attach:max.apm | MAX Operator Example]]

(:html:)<font size=1><pre>

! 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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(:html:)<font size=1><pre>

! 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 signum

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

</pre></font>(:htmlend:)

----

!!! Minimum Selector (MIN) Operator

* %list list-page% [[Attach:min.apm | MIN Operator Example]]

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!!! ~~MPEC examples~~

-~~---~~

!!!! SIGN Operator

* %list list-page% [[Attach:sign.apm | Download SIGN Operator Example]]

!!!! SIGN Operator

* %list list-page% [[Attach:sign.apm | Download

to:

!!! SIGN Operator

* %list list-page% [[Attach:sign.apm | SIGN Operator Example]]

* %list list-page% [[Attach:sign.apm | SIGN Operator Example]]

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!!! Absolute Value (ABS) Operator

* %list list-page% [[Attach:abs.apm | ABS Operator Example]]

(:html:)<font size=1><pre>

</pre></font>(:htmlend:)

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* %list list-page% [[Attach:abs.apm | ABS Operator Example]]

(:html:)<font size=1><pre>

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!!!! SIGN Operator

* %list list-page% [[Attach:sign.apm | Download SIGN Operator Example]]

!!!! SIGN Operator

* %list list-page% [[Attach:sign.apm | Download SIGN Operator Example]]

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</pre></font>(:htmlend:)

to:

! 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 signum

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

</pre></font>(:htmlend:)

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! 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 signum

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

</pre></font>(:htmlend:)

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!! 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 problems.

----

!!! MPEC examples

* %list list-page% [[Attach:sign.apm | SIGN Operator]]

----

(:html:)<font size=1><pre>

</pre></font>(:htmlend:)

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 problems.

----

!!! MPEC examples

* %list list-page% [[Attach:sign.apm | SIGN Operator]]

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(:html:)<font size=1><pre>

</pre></font>(:htmlend:)