在 CPLEX 中对函数进行积分

Question

我尝试在 CPLEX 中为 MIP 模型建模。我有一个包含决策变量的函数，我需要对该函数进行积分来计算其预期值。有没有办法在 CPLEX 中对函数进行积分？谢谢！

Answer 1

使用分段线性，您可以逼近任何函数并依赖数学规划。

见Interpolate any functionTips and Tricks in OPL

// linearization of f(x)=1/x through a piecewise linear function

int sampleSize=10000;
float s=1;
float e=10;
float x[i in 0..sampleSize]=s+(e-s)*i/sampleSize;
int nbSegments=5;
float x2[i in 0..nbSegments]=(s)+(e-s)*i/nbSegments;
float y2[i in 0..nbSegments]=1/x2[i];  // y=f(x)
float firstSlope=0;
 float lastSlope=0;
 
 tuple breakpoint // y=f(x)
 {
  key float x;
  float y;
 }
 
 sorted { breakpoint } breakpoints={<x2[i],y2[i]> | i in 0..nbSegments};
 
 float slopesBeforeBreakpoint[b in breakpoints]=
 (b.x==first(breakpoints).x)
 ?firstSlope
 :(b.y-prev(breakpoints,b).y)/(b.x-prev(breakpoints,b).x);
 
 pwlFunction f=piecewise(b in breakpoints)
 { slopesBeforeBreakpoint[b]->b.x; lastSlope } (first(breakpoints).x, first(breakpoints).y);
 
 assert forall(b in breakpoints) abs(f(b.x)-b.y)<=0.001;

通过 CPLEX 中的约束编程，您可以使用任何函数（甚至不是线性函数）甚至黑盒函数。

Example of not linear function:

using CP;

// CPOptimizer allows all kind of non linearities

int nbKids=300;
float costBus40=500;
float costBus30=400;
 
dvar int+ nbBus40;
dvar int+ nbBus30;
 
// Non linear objective (exponential) 
minimize
 costBus40*exp(nbBus40) +exp(nbBus30)*costBus30;
 
subject to
{
 40*nbBus40+nbBus30*30>=nbKids;
}