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piecewise.R
# Copyright 2023, Gurobi Optimization, LLC # # This example considers the following separable, convex problem: # # minimize # f(x) - y + g(z) # subject to # x + 2 y + 3 z <= 4 # x + y >= 1 # x, y, z <= 1 # # where f(u) = exp(-u) and g(u) = 2 u^2 - 4u, for all real u. It # formulates and solves a simpler LP model by approximating f and # g with piecewise-linear functions. Then it transforms the model # into a MIP by negating the approximation for f, which gives # a non-convex piecewise-linear function, and solves it again. library(gurobi) model <- list() model$A <- matrix(c(1,2,3,1,1,0), nrow=2, byrow=T) model$obj <- c(0,-1,0) model$ub <- c(1,1,1) model$rhs <- c(4,1) model$sense <- c('<', '>') # Uniformly spaced points in [0.0, 1.0] u <- seq(from=0, to=1, by=0.01) # First piecewise-linear function: f(x) = exp(-x) pwl1 <- list() pwl1$var <- 1 pwl1$x <- u pwl1$y <- sapply(u, function(x) exp(-x)) # Second piecewise-linear function: g(z) = 2 z^2 - 4 z pwl2 <- list() pwl2$var <- 3 pwl2$x <- u pwl2$y <- sapply(u, function(z) 2 * z * z - 4 * z) model$pwlobj <- list(pwl1, pwl2) result <- gurobi(model) print(result$objval) print(result$x) # Negate piecewise-linear function on x, making it non-convex model$pwlobj[[1]]$y <- sapply(u, function(x) -exp(-x)) result <- gurobi(model) gurobi_write(model, "pwl.lp") print(result$objval) print(result$x) # Clear space rm(model, pwl1, pwl2, result)
