absmip.gms : Discontinous functions abs() min() max() sign() as MIPs
This illustrates the MIP formulations of some difficult functions.
Reference:
- GAMS Development Corporation, Formulation and Language Example.
Small Model of Type: MIP
<![CDATA[
$title Discontinous functions abs, min, max, and sign as MIPs (ABSMIP,SEQ=208)
$Ontext
This illustrates the MIP formulations of some difficult functions.
GAMS Development Corporation, Formulation and Language Example.
$Offtext
Variable x Argument to the various functions;
Positive Variable xp Positive part of x
xn Negative part of x;
Binary Variable b decision if we have positive or negative x;
Equations e1 split x in positive and negative part
e2 restrict positive part of x
e3 restrict negative part of x;
e1.. x =e= xp - xn;
e2.. xp =l= abs(x.up)*b;
e3.. xn =l= abs(x.lo)*(1-b);
Variable y result of function evaluation;
Equation defabs "definition of y = abs(x)"
defzmax "definition of y = zmax(x) = max(x,0)"
defzmin "definition of y = zmin(x) = min(x,0)";
defabs.. y =e= xp + xn;
defzmax.. y =e= xp;
defzmin.. y =e= - xn;
model absM / e1,e2,e3,defabs /
zmaxM / e1,e2,e3,defzmax /
zminM / e1,e2,e3,defzmin /;
Scalar tol tolerance for being zero / 1e-04 /;
Binary Variables b1 decides if xp is positive
b2 decides if xn is positive;
Equations e2up upper restriction for xp
e2lo lower restriction for xp
e3up upper restriction for xn
e3lo lower restriction for xn
one either b1 or b2
defsign definition of y = sign(x);
e2up.. xp =l= abs(x.up)*b1;
e2lo.. xp =g= tol*b1;
e3up.. xn =l= abs(x.lo)*b2;
e3lo.. xn =g= tol*b2;
one.. b1 + b2 =l= 1;
defsign.. y =e= b1 - b2;
Model signM / e1,e2up,e2lo,e3up,e3lo,one,defsign /;
Option limcol=0,limrow=0,optcr=1e-6;
Set runs number of test runs / 1*5 /
nos / "v.lo", "v.up", "w.lo", "w.up" /
mm min and max / max, min /;
Parameter bnds_unary(runs,*) /
1.lo -5, 1.up 5
2.lo -5, 2.up -2
3.lo 2, 3.up 5
4.lo 0, 4.up 5
5.lo -5, 5.up 0
/;
Parameter rep_unary;
loop(runs,
x.lo = bnds_unary(runs,"lo"); rep_unary(mm,runs,"x.lo") = x.lo;
x.up = bnds_unary(runs,"up"); rep_unary(mm,runs,"x.up") = x.up;
solve absM max y using mip; rep_unary("max",runs,"abs" ) = y.l;
solve zmaxM max y using mip; rep_unary("max",runs,"zmax") = y.l;
solve zminM max y using mip; rep_unary("max",runs,"zmin") = y.l;
solve signM max y using mip; rep_unary("max",runs,"sign") = y.l;
solve absM min y using mip; rep_unary("min",runs,"abs" ) = y.l;
solve zmaxM min y using mip; rep_unary("min",runs,"zmax") = y.l;
solve zminM min y using mip; rep_unary("min",runs,"zmin") = y.l;
solve signM min y using mip; rep_unary("min",runs,"sign") = y.l;
);
Variables z "result of function evaluation"
v "argument to max/min"
w "argument to max/min"
Equations defvwx "substitution x = v - w"
defmXX "definition of z = max/min(v,w) = max/min(x,0) + w";
defvwx.. x =e= v - w;
defmXX.. z =e= y + w;
Model maxM / e1,e2,e3,defzmax,defvwx,defmXX /
minM / e1,e2,e3,defzmin,defvwx,defmXX /;
Parameter bnds_binary(runs,*) /
1.vlo -5, 1.vup 5, 1.wlo -2, 1.wup 2
2.vlo -2, 2.vup 2, 2.wlo -5, 2.wup 5
3.vlo -5, 3.vup -2, 3.wlo 2, 3.wup 5
4.vlo 2, 4.vup 5, 4.wlo -5, 4.wup -2
5.vlo -5, 5.vup 2, 5.wlo -2, 5.wup 5
/;
Parameter rep_binary;
loop(runs,
v.lo = bnds_binary(runs,"vlo"); rep_binary(mm,runs,"v.lo") = v.lo;
v.up = bnds_binary(runs,"vup"); rep_binary(mm,runs,"v.up") = v.up;
w.lo = bnds_binary(runs,"wlo"); rep_binary(mm,runs,"w.lo") = w.lo;
w.up = bnds_binary(runs,"wup"); rep_binary(mm,runs,"w.up") = w.up;
x.lo = -max(abs(w.up-v.lo), abs(v.up-w.lo)); x.up = -x.lo;
solve maxM max z using mip; rep_binary("max",runs,"max" ) = z.l;
solve minM max z using mip; rep_binary("max",runs,"min" ) = z.l;
solve maxM min z using mip; rep_binary("min",runs,"max" ) = z.l;
solve minM min z using mip; rep_binary("min",runs,"min" ) = z.l;
);
display rep_unary, rep_binary;
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