# Copyright 2026, Gurobi Optimization, LLC## This example formulates and solves the following simple model# with PWL constraints:##  maximize#        sum c(j) * x(j)#  subject to#        sum A(i,j) * x(j) <= 0,  for i = 1, ..., m#        sum y(j) <= 3#        y(j) = pwl(x(j)),        for j = 1, ..., n#        x(j) free, y(j) >= 0,    for j = 1, ..., n##  where pwl(x) = 0,     if x  = 0#               = 1+|x|, if x != 0##  Note#   1. sum pwl(x(j)) <= b is to bound x vector and also to favor sparse x vector.#      Here b = 3 means that at most two x(j) can be nonzero and if two, then#      sum x(j) <= 1#   2. pwl(x) jumps from 1 to 0 and from 0 to 1, if x moves from negative 0 to 0,#      then to positive 0, so we need three points at x = 0. x has infinite bounds#      on both sides, the piece defined with two points (-1, 2) and (0, 1) can#      extend x to -infinite. Overall we can use five points (-1, 2), (0, 1),#      (0, 0), (0, 1) and (1, 2) to define y = pwl(x)library(gurobi)library(Matrix)n=5# A x <= 0A<-rbind(c(0,0,0,1,-1),c(0,0,1,1,-1),c(1,1,0,0,-1),c(1,0,1,0,-1),c(1,0,0,1,-1))# sum y(j) <= 3y<-rbind(c(1,1,1,1,1))# Initialize modelmodel<-list()# Constraint matrixmodel$A<-bdiag(A,y)# Right-hand-side coefficient vectormodel$rhs<-c(rep(0,n),3)# Objective function (x coefficients arbitrarily chosen)model$obj<-c(0.5,0.8,0.5,0.1,-1,rep(0,n))# It's a maximization modelmodel$modelsense<-"max"# Lower bounds for x and ymodel$lb<-c(rep(-Inf,n),rep(0,n))# PWL constraintsmodel$genconpwl<-list()for(kin1:n){model$genconpwl[[k]]<-list()model$genconpwl[[k]]$xvar<-kmodel$genconpwl[[k]]$yvar<-n+kmodel$genconpwl[[k]]$xpts<-c(-1,0,0,0,1)model$genconpwl[[k]]$ypts<-c(2,1,0,1,2)}# Solve the model and collect the resultsresult<-gurobi(model)# Display solution values for xfor(kin1:n)print(sprintf('x(%d) = %g',k,result$x[k]))print(sprintf('Objective value: %g',result$objval))# Clear spacerm(model,result)