Apps
~~*~~ 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. [[http://www.sciencedirect.com/science/article/pii/S036054421631814X|Article]]

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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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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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! 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:)

~~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

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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'

Objects

f = abs

Connections

f.x = x

f.y = y

Parameters

x = -2

Variables

y

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

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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

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

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## Mpec Examples

## Apps.MpecExamples History

Show minor edits - Show changes to output

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to:

!!!! 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. [[http://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. [[http://www.sciencedirect.com/science/article/pii/S036054421631814X|Article]]

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----

to:

----

* 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. [[http://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. [[http://www.sciencedirect.com/science/article/pii/S036054421631814X|Article]]

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! SIGN function as an Object

to:

! SIGN function MPEC as an Object

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! ABS function as an Object

to:

! 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

to:

! SIGN function as an Object

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

to:

! ABS function as an Object

Added lines 199-220:

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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:)

Added lines 143-164:

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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:)

Added lines 36-57:

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! 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

to:

! 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

Changed lines 68-72 from:

! 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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to:

End Objects

Changed lines 76-77 from:

to:

End Connections

Changed lines 80-81 from:

to:

End Parameters

Added line 84:

End Variables

Deleted line 85:

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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

Added lines 66-87:

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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'

Objects

f = abs

Connections

f.x = x

f.y = y

Parameters

x = -2

Variables

y

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

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to:

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to:

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to:

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to:

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----

to:

----

Changed line 3 from:

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.

Changed line 3 from:

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~~

to:

Model sign

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Model ~~signum~~

to:

Model abs

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y = x1 ~~+~~ s_a

to:

y = x1 - s_a

Added lines 75-134:

! 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

Added lines 41-72:

(: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]]

Changed lines 7-14 from:

!!! ~~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]]

Deleted line 35:

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!!! Absolute Value (ABS) Operator

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

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

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

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

----

Changed lines 9-13 from:

to:

----

!!!! SIGN Operator

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

!!!! SIGN Operator

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

Deleted lines 10-11:

Changed lines 12-13 from:

</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:)

----

! 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:)

----

Added lines 1-15:

!! 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]]

----

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

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