- SOS2: use SOS2 variables to model the piecewise linear functions. This is an easy modeling exercise.
- BIGM_BIN: use binary variables and big-M constraints to enable and disable constraints. This is a more complex undertaking, especially if we want to use the smallest possible values for the big-M constants. Pyomo has bug in this version.
- BIGM_SOS1: a slight variation of the BIGM_BIN model. Pyomo has the same problem with this model.
Here I show a few more formulations:
Note that part 3 [2] will discuss some newer formulations: LOG and DLOG that only need a logarithmic number of binary variables.
For demonstration purposes we use the same small example with 4 breakpoints and 3 segments.
- DCC: Disaggregated Convex Combination formulation is a simulation of the SOS2 model by binary variables.
- CC: Convex Combination formulation is a similar SOS2 like approach using binary variables only.
- MC: Multiple Choice model uses a semi-continuous approach.
- INCR: Incremental formulation is using all previous segments.
Note that part 3 [2] will discuss some newer formulations: LOG and DLOG that only need a logarithmic number of binary variables.
For demonstration purposes we use the same small example with 4 breakpoints and 3 segments.
Method 4: DCC - Disaggregated Convex Combination Model
This is an impressive name for a model that is actually not very complicated.
| DCC - Disaggregated Convex Combination Formulation |
|---|
| \[\begin{align} & \color{DarkRed}x = \sum_{s,k|\color{DarkBlue}SK(s,k)} \color{DarkBlue}{\bar{x}}_k \color{DarkRed}\lambda_{s,k} \\ & \color{DarkRed}y = \sum_{s,k|\color{DarkBlue}SK(s,k)} \color{DarkBlue}{\bar{y}}_k \color{DarkRed}\lambda_{s,k}\\& \color{DarkRed}\delta_s = \sum_{k|\color{DarkBlue}SK(s,k)} \color{DarkRed} \lambda_{s,k} \\& \sum_s \color{DarkRed}\delta_s=1 \\ & \color{DarkRed}\lambda_{s,k} \ge 0 \\ & \color{DarkRed}\delta_s \in \{0,1\} \end{align}\] |
Here the mapping set \(SK(s,k)\) is TRUE for the two breakpoints \(k\) belonging to segment \(s\).
We can also write the model more directly, without the mapping \(SK(s,k)\):
| DCC - Disaggregated Convex Combination Formulation 2 |
|---|
| \[\begin{align} & \color{DarkRed}x = \sum_s \left( \color{DarkBlue}{\bar{x}}_s \color{DarkRed}\lambda_{s,s} + \color{DarkBlue}{\bar{x}}_{s+1} \color{DarkRed}\lambda_{s,s+1} \right) \\ & \color{DarkRed}y = \sum_s \left( \color{DarkBlue}{\bar{y}}_s \color{DarkRed}\lambda_{s,s}+\color{DarkBlue}{\bar{y}}_{s+1} \color{DarkRed}\lambda_{s,s+1}\right)\\& \color{DarkRed}\delta_s = \color{DarkRed} \lambda_{s,s}+ \color{DarkRed}\lambda_{s,s+1} \\& \sum_s \color{DarkRed}\delta_s=1 \\ & \color{DarkRed}\lambda_{s,k} \ge 0 \\ & \color{DarkRed}\delta_s \in \{0,1\} \end{align}\] |
Here is the GAMS version:
set
k 'breakpoints' /point1*point4/ s 'segments' /segment1*segment3/ sk(s,k) 'mapping' ; sk(s,k) = ord(s)=ord(k) or ord(s)=ord(k)-1; display sk; table data(k,*) x y point1 1 6 point2 3 2 point3 6 8 point4 10 7 ; positive variable lambda(s,k) 'interpolation'; binary variable delta(s) 'select segment'; variable x,y; equations xdef 'x' ydef 'y' link(s) 'link delta-lambda' sumdelta 'select segment' ; xdef.. x =e= sum(sk(s,k), lambda(s,k)*data(k,'x')); ydef.. y =e= sum(sk(s,k), lambda(s,k)*data(k,'y')); link(s).. delta(s) =e= sum(sk(s,k),lambda(s,k)); sumdelta.. sum(s, delta(s)) =e= 1; x.fx = 5; model m /all/; option optcr=0; solve m maximizing y using mip; display x.l,y.l,delta.l,lambda.l; |
In this model I made a strong distinction between segments \(s\) and breakpoints \(k\). This will help the GAMS model to perform domain checking (type checking), so we get a bit better protection against errors in the model. Of course, if you prefer you just can indicate segments by \(1,\dots,K-1\).
Note that the linking constraints link, perform two functions. First: they make sure that for unselected segments (with \(\delta_s=0\)), we have \(\lambda_{s,k}=0\). Second, for the selected segment, we automatically sum the \(\lambda\)'s to 1.
The output of the model looks like:
---- 13 SET sk mapping point1 point2 point3 point4 segment1 YES YES segment2 YES YES segment3 YES YES ---- 45 VARIABLE x.L = 5.000 VARIABLE y.L = 6.000 ---- 45 VARIABLE delta.L segment2 1.000 ---- 45 VARIABLE lambda.L point2 point3 segment2 0.333 0.667
The display of the set SK confirms we setup the topography correctly: segment \(k\) uses points \(k, k+1\).
We fixed \(x=5\) which yields \(y=6\). We see \(\delta_2=1\) so the second segment is selected, and this is also visible from the variables \(\lambda_{s,k}\) indicating we interpolate between points 2 and 3.
We can have a look at the generated equations:
---- xdef =E= xdef.. - lambda(segment1,point1) - 3*lambda(segment1,point2) - 3*lambda(segment2,point2) - 6*lambda(segment2,point3) - 6*lambda(segment3,point3) - 10*lambda(segment3,point4) + x =E= 0 ; (LHS = 5, INFES = 5 ****) ---- ydef =E= ydef.. - 6*lambda(segment1,point1) - 2*lambda(segment1,point2) - 2*lambda(segment2,point2) - 8*lambda(segment2,point3) - 8*lambda(segment3,point3) - 7*lambda(segment3,point4) + y =E= 0 ; (LHS = 0) ---- link =E= link(segment1).. - lambda(segment1,point1) - lambda(segment1,point2) + delta(segment1) =E= 0 ; (LHS = 0) link(segment2).. - lambda(segment2,point2) - lambda(segment2,point3) + delta(segment2) =E= 0 ; (LHS = 0) link(segment3).. - lambda(segment3,point3) - lambda(segment3,point4) + delta(segment3) =E= 0 ; (LHS = 0) ---- sumdelta =E= sumdelta.. delta(segment1) + delta(segment2) + delta(segment3) =E= 1 ; (LHS = 0, INFES = 1 ****)
In a sense we simulated the SOS2 model using binary variables. More visible here is the two-step approach:
- Select the segment \(s\) using the binary variables \(\delta_s\).
- Interpolate between the breakpoints of this segment using the positive variables \(\lambda_{s,k}\).
Of course in a MIP model we have simultaneous equations, so these things happen actually at the same time. The 2-step paradigm is more of a useful mental model.
For completeness, here is the GAMS model without set SK(s,k):
For completeness, here is the GAMS model without set SK(s,k):
* DCC2: Disaggregated Convex Combination formulation 2
* without mapping SK(s,k) set k 'breakpoints' /point1*point4/ s(k) 'segments' /point1*point3/ ; table data(k,*) x y point1 1 6 point2 3 2 point3 6 8 point4 10 7 ; positive variable lambda(s,k) 'interpolation'; binary variable delta(s) 'select segment'; variable x,y; equations xdef 'x' ydef 'y' link(s) 'link delta-lambda' sumdelta 'select segment' ; xdef.. x =e= sum(s(k), lambda(s,k)*data(k,'x')+lambda(s,k+1)*data(k+1,'x')); ydef.. y =e= sum(s(k), lambda(s,k)*data(k,'y')+lambda(s,k+1)*data(k+1,'y')); link(s(k)).. delta(s) =e= lambda(s,k)+lambda(s,k+1); sumdelta.. sum(s, delta(s)) =e= 1; x.fx = 5; model m /all/; option optcr=0; solve m maximizing y using mip; display x.l,y.l,delta.l,lambda.l; |
Method 5: CC - Convex Combination Model
This formulation is very similar to the previous one. The main difference is the structure of the \(\lambda\) variables. We index them by \(k\) only. This is more like the SOS2 model. The manner in which we link \(\lambda_k\) to \(\delta_s\) becomes somewhat different. Instead of an equality we use an inequality \[\lambda_k \le \sum_{s|SK(s,k)}\delta_s \] This make sure that only for a selected segment with \(\delta_s=1\), the corresponding \(\lambda_k\)'s can be nonzero. We need to add explicitly that \(\sum_k \lambda_k=1\) as this is no longer implied by the linking constraints. The model looks like:
| CC - Convex Combination Formulation |
|---|
| \[\begin{align} & \color{DarkRed}x = \sum_k \color{DarkBlue}{\bar{x}}_k \color{DarkRed}\lambda_{k} \\ & \color{DarkRed}y = \sum_k \color{DarkBlue}{\bar{y}}_k \color{DarkRed}\lambda_{k} \\ & \sum_k \color{DarkRed}\lambda_k = 1 \\& \color{DarkRed} \lambda_k \le \sum_{s|\color{DarkBlue}SK(s,k)} \color{DarkRed}\delta_s \\& \sum_s \color{DarkRed}\delta_s=1 \\ & \color{DarkRed}\lambda_k \ge 0 \\ & \color{DarkRed}\delta_s \in \{0,1\} \end{align}\] |
This model has fewer continuous variables than the DCC model. The GAMS version looks like:
set
k 'breakpoints' /point1*point4/ s 'segments' /segment1*segment3/ sk(s,k) 'mapping' ; sk(s,k) = ord(s)=ord(k) or ord(s)=ord(k)-1; display sk; table data(k,*) x y point1 1 6 point2 3 2 point3 6 8 point4 10 7 ; positive variable lambda(k) 'interpolation'; binary variable delta(s) 'select segment'; variable x,y; equations xdef 'x' ydef 'y' link(k) 'link delta-lambda' sumlambda 'interpolation' sumdelta 'select segment' ; xdef.. x =e= sum(k, lambda(k)*data(k,'x')); ydef.. y =e= sum(k, lambda(k)*data(k,'y')); link(k).. lambda(k) =l= sum(sk(s,k),delta(s)); sumlambda.. sum(k, lambda(k)) =e= 1; sumdelta.. sum(s, delta(s)) =e= 1; x.fx = 5; model m /all/; option optcr=0; solve m maximizing y using mip; display x.l,y.l,delta.l,lambda.l; |
The output (including the generated equations) is:
Generated equations
---- xdef =E= x xdef.. - lambda(point1) - 3*lambda(point2) - 6*lambda(point3) - 10*lambda(point4) + x =E= 0 ; (LHS = 5, INFES = 5 ****) ---- ydef =E= y ydef.. - 6*lambda(point1) - 2*lambda(point2) - 8*lambda(point3) - 7*lambda(point4) + y =E= 0 ; (LHS = 0) ---- link =L= link delta-lambda link(point1).. lambda(point1) - delta(segment1) =L= 0 ; (LHS = 0) link(point2).. lambda(point2) - delta(segment1) - delta(segment2) =L= 0 ; (LHS = 0) link(point3).. lambda(point3) - delta(segment2) - delta(segment3) =L= 0 ; (LHS = 0) link(point4).. lambda(point4) - delta(segment3) =L= 0 ; (LHS = 0)
Solution
---- sumlambda =E= interpolation sumlambda.. lambda(point1) + lambda(point2) + lambda(point3) + lambda(point4) =E= 1 ; (LHS = 0, INFES = 1 ****) ---- sumdelta =E= select segment sumdelta.. delta(segment1) + delta(segment2) + delta(segment3) =E= 1 ; (LHS = 0, INFES = 1 ****) ---- 49 VARIABLE x.L = 5.000 VARIABLE y.L = 6.000 ---- 49 VARIABLE delta.L select segment segment2 1.000 ---- 49 VARIABLE lambda.L interpolation point2 0.333, point3 0.667
The methods DCC and CC simulate our SOS2 model from [1] using binary variables. This means the model can handle step functions (we don't need to form a slope). The models are not too difficult to setup, as is illustrated by the GAMS models implementing them.
An early reference to this method is [4].
Method 6: MC - Multiple Choice formulation
This is a well-known method. We assume we can calculate slopes and intercepts for each segment. I.e. we have \[ y = a_s + b_s x\>\> \text{ if $\bar{x}_s \le x \le \bar{x}_{s+1}$} \] with \[\begin{align} & a_s = \frac{\bar{y}_{s+1}-\bar{y}_s}{\bar{x}_{s+1}-\bar{x}_s} \\ & b_s = \bar{y}_s - a_s \bar{x}_s \end{align}\]
For the model we introduce binary variables \(\delta_s\) to indicate which segment is selected, and so-called semi-continuous variables \(v_s \in {0} \cup [\bar{x}_s,\bar{x}_{s+1}] \). The variable \(v_s\) can be modeled as \[\bar{x}_s \delta_s \le v_s \le \bar{x}_{s+1} \delta_s \]
With this we can formulate the complete model:
| MC - Multiple Choice Formulation |
|---|
| \[\begin{align} & \color{DarkRed}x = \sum_s \color{DarkRed}v_s \\ & \color{DarkRed}y = \sum_s \left( \color{DarkBlue}a_s \color{DarkRed} v_s + \color{DarkBlue}b_s \color{DarkRed} \delta_s \right) \\ &\color{DarkBlue}{\bar{x}}_s \color{DarkRed}\delta_s \le \color{DarkRed}v_s \le \color{DarkBlue}{\bar{x}}_{s+1} \color{DarkRed}\delta_s \\& \sum_s \color{DarkRed}\delta_s=1 \\ & \color{DarkRed}\delta_s \in \{0,1\}\\ & \color{DarkBlue}a_s = \frac{\color{DarkBlue}{\bar{y}}_{s+1}-\color{DarkBlue}{\bar{y}}_s}{\color{DarkBlue}{\bar{x}}_{s+1}-\color{DarkBlue}{\bar{x}}_s} \\ &\color{DarkBlue}b_s = \color{DarkBlue}{\bar{y}}_s - \color{DarkBlue} a_s \color{DarkBlue}{\bar{x}}_s \end{align}\] |
- The sandwich equation \(\bar{x}_s \delta_s \le v_s \le \bar{x}_{s+1} \delta_s \) must likely be implemented as two inequalities.
- This construct says: \(v_s=0\) or \(v_s \in [\bar{x}_s, \bar{x}_{s+1}]\). \(v_s\) is sometimes called semi-continuous.
- The variables \(\delta_s\) and \(v_s\) are connected: \[\begin{align} &\delta_s = 0 \Rightarrow v_s = 0\\ & \delta_s = 1 \Rightarrow v_s = x\end{align}\] I.e. they operate in parallel.
- I have seen cases where this model outperformed the SOS2 formulation.
The GAMS model is simple:
set
k 'breakpoints' /k1*k4/ s(k) 'segments' /k1*k3/ ; table data(k,*) x y k1 1 6 k2 3 2 k3 6 8 k4 10 7 ; data(s(k),'dx') = data(k+1,'x')-data(k,'x'); data(s(k),'dy') = data(k+1,'y')-data(k,'y'); data(s(k),'slope') = data(k,'dy')/data(k,'dx'); data(s(k),'intercept') = data(k,'y')-data(k,'slope')*data(k,'x'); display data; variable v(s) 'equal to x or 0'; binary variable delta(s) 'select segment'; variable x,y; equations xdef 'x' ydef 'y' semicont1(k) semicont2(k) sumdelta 'select segment' ; xdef.. x =e= sum(s, v(s)); ydef.. y =e= sum(s, data(s,'slope')*v(s)+data(s,'intercept')*delta(s)); semicont1(s(k)).. v(s) =l= data(k+1,'x')*delta(s); semicont2(s).. v(s) =g= data(s,'y')*delta(s); sumdelta.. sum(s, delta(s)) =e= 1; x.fx = 5; model m /all/; option optcr=0; solve m maximizing y using mip; display data,x.l,y.l,delta.l,v.l; |
The results look like:
---- 43 PARAMETER data x y dx dy slope intercept k1 1.000 6.000 2.000 -4.000 -2.000 8.000 k2 3.000 2.000 3.000 6.000 2.000 -4.000 k3 6.000 8.000 4.000 -1.000 -0.250 9.500 k4 10.000 7.000 ---- 43 VARIABLE x.L = 5.000 VARIABLE y.L = 6.000 ---- 43 VARIABLE delta.L select segment k2 1.000 ---- 43 VARIABLE v.L equal to x or 0 k2 5.000
Segment 2 is selected. We can see that directly from the binary variable \(\delta_s\), but also from the semi-continuous variable \(v_s\).
Method 7: INC - Incremental Formulation
In the incremental or delta method we add up all contributions of "earlier" segments, to find our \(x\) and \(y\).
Let \(s'\) be the segment that contains our current \(x\). We define a binary variable \(\delta_{s}\) as \[\delta_{s} = \begin{cases} 1 & \text{for $s\lt s'$}\\ 0 & \text{for $s \ge s'$}\end{cases}\] In addition we use a continuous variable \(\lambda_s \in [0,1]\) indicating how much of each segment we "use up". The contribution for earlier segments is 1 and for the current segment we have a fractional value. So we have: \[\begin{cases} \lambda_{s} = 1 & \text{for $s\lt s'$} \\\lambda_{s} \in [0,1] & \text{for $s= s'$} \\ \lambda_{s} = 0 & \text{for $s\gt s'$}\end{cases}\] With these definitions we can write: \[\begin{align} x = \bar{x}_1 + \sum_s \lambda_s (\bar{x}_{s+1} - \bar{x}_s) \\ y = \bar{y}_1 + \sum_s \lambda_s (\bar{y}_{s+1} - \bar{y}_s) \end{align}\]
Now we need to formulate a structure that enforce these rules on \(\delta\) and \(\lambda\). The following will do that: \[\lambda_{s+1} \le \delta_s \le \lambda_s\] Typically you will need to implement this as two separate constraints.
The complete model looks like:
| INC - Incremental Formulation |
|---|
| \[\begin{align} & \color{DarkRed}x = \color{DarkBlue}{\bar{x}}_1 + \sum_s \color{DarkRed}\lambda_s (\color{DarkBlue}{\bar{x}}_{s+1} - \color{DarkBlue}{\bar{x}}_s) \\ & \color{DarkRed}y = \color{DarkBlue}{\bar{y}}_1 + \sum_s \color{DarkRed}\lambda_s (\color{DarkBlue}{\bar{y}}_{s+1} - \color{DarkBlue}{\bar{y}}_s) \\ & \color{DarkRed} \lambda_{s+1} \le \color{DarkRed}\delta_s \le \color{DarkRed}\lambda_s \\ & \color{DarkRed} \delta_s \in \{0,1\} \\ & \color{DarkRed} \lambda_s \in [0,1] \end{align}\] |
We can see that we can fix the last \(\delta_s=0\). The model will behave correctly without this, but we may help the presolver a bit with this.
The GAMS model can look like:
set
k 'breakpoints' /k1*k4/ s(k) 'segments' /k1*k3/ ; table data(k,*) x y k1 1 6 k2 3 2 k3 6 8 k4 10 7 ; positive variable lambda(s) 'contribution factor of segment'; lambda.up(s) = 1; binary variable delta(s) 'previously contributing segments'; variable x,y; equations xdef 'x' ydef 'y' link1(s) 'links lambda delta' link2(s) 'links lambda delta' ; * fix last delta(s) to 0. * this is not strictly needed delta.fx(s)$(ord(s)=card(s)) = 0; xdef.. x =e= data('k1','x')+ sum(s(k), lambda(s)*(data(k+1,'x')-data(k,'x'))); ydef.. y =e= data('k1','y')+ sum(s(k), lambda(s)*(data(k+1,'y')-data(k,'y'))); link1(s).. lambda(s+1) =l= delta(s); link2(s).. delta(s) =l= lambda(s); x.fx = 5; model m /all/; option optcr=0; solve m maximizing y using mip; display x.l,y.l,delta.l,lambda.l; |
Note that in equation link1, we go one position too far when addressing \(\lambda_{s+1}\) for the last \(s\). GAMS will make that reference zero, and that is correct for this case. We see in the equation listing:
---- link1 =L= links lambda delta link1(k1).. lambda(k2) - delta(k1) =L= 0 ; (LHS = 0) link1(k2).. lambda(k3) - delta(k2) =L= 0 ; (LHS = 0) link1(k3).. - delta(k3) =L= 0 ; (LHS = 0)
The solution is
---- 36 VARIABLE x.L = 5.000 VARIABLE y.L = 6.000 ---- 36 VARIABLE delta.L previously contributing segments k1 1.000 ---- 36 VARIABLE lambda.L contribution factor of segment k1 1.000, k2 0.667
Notes:
- A slightly different formulation is the following. Use the definition: \[\delta_{s} = \begin{cases} 1 & \text{for $s\le s'$}\\ 0 & \text{for $s \gt s'$}\end{cases}\] Now we use the sandwich equation: \[\delta_{s+1} \le \lambda_s \le \delta_s\] With this formulation we can fix: \(\delta_1=1\).
- This approach goes back to [5].
References
- Piecewise linear functions and formulations for interpolation (part 1). This post discusses formulations SOS2, BIGM_BIN and BIGM_SOS1. The SOS2 formulation is the most important. Also illustrated how a Pyomo implementation is not implementing BIGM_BIN and BIGM_SOS1 correctly. http://yetanothermathprogrammingconsultant.blogspot.com/2019/02/piecewise-linear-functions-and.html
- Piecewise linear functions and formulations for interpolation (part 3). This post describes the LOG and DLOG formulations, which have a logarithmic number of binary variables. We see also some issues with the Pyomo implementation. http://yetanothermathprogrammingconsultant.blogspot.com/2019/03/piecewise-linear-functions-and.html
- Keely L. Croxton, Bernard Gendron, Thomas L. Magnanti, A Comparison of Mixed-Integer Programming Models for Non-Convex Piecewise Linear Cost Minimization Problems, Management Science, Volume 49, Issue 9, 2003, pages 1121-1273
- G. Dantzig, On the significance of solving linear programming problems with some integer variables, Econometrica 28, 30-44, 1960
- H. Markowitz, A. Manne, On the solution of discrete programming problems, Econometrica 25, 84-110, 1957
