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### gc_pwl_func_cs.cs

/* Copyright 2020, Gurobi Optimization, LLC

This example considers the following nonconvex nonlinear problem

maximize    2 x    + y
subject to  exp(x) + 4 sqrt(y) <= 9
x, y >= 0

We show you two approaches to solve this:

1) Use a piecewise-linear approach to handle general function
constraints (such as exp and sqrt).
u = exp(x)
v = sqrt(y)
b) Compute points (x, u) of u = exp(x) for some step length (e.g., x
= 0, 1e-3, 2e-3, ..., xmax) and points (y, v) of v = sqrt(y) for
some step length (e.g., y = 0, 1e-3, 2e-3, ..., ymax). We need to
compute xmax and ymax (which is easy for this example, but this
does not hold in general).
c) Use the points to add two general constraints of type
piecewise-linear.

2) Use the Gurobis built-in general function constraints directly (EXP
and POW). Here, we do not need to compute the points and the maximal
possible values, which will be done internally by Gurobi.  In this
approach, we show how to "zoom in" on the optimal solution and
tighten tolerances to improve the solution quality.
*/

using System;
using Gurobi;

class gc_pwl_func_cs {

private static double f(double u) { return Math.Exp(u); }
private static double g(double u) { return Math.Sqrt(u); }

private static void printsol(GRBModel m, GRBVar x, GRBVar y, GRBVar u, GRBVar v) {
Console.WriteLine("x = " + x.X + ", u = " + u.X);
Console.WriteLine("y = " + y.X + ", v = " + v.X);
Console.WriteLine("Obj = " + m.ObjVal);

// Calculate violation of exp(x) + 4 sqrt(y) <= 9
double vio = f(x.X) + 4 * g(y.X) - 9;
if (vio < 0.0) vio = 0.0;
Console.WriteLine("Vio = " + vio);
}

static void Main() {
try {

// Create environment

GRBEnv env = new GRBEnv();

// Create a new m

GRBModel m = new GRBModel(env);

double lb = 0.0, ub = GRB.INFINITY;

GRBVar x = m.AddVar(lb, ub, 0.0, GRB.CONTINUOUS, "x");
GRBVar y = m.AddVar(lb, ub, 0.0, GRB.CONTINUOUS, "y");
GRBVar u = m.AddVar(lb, ub, 0.0, GRB.CONTINUOUS, "u");
GRBVar v = m.AddVar(lb, ub, 0.0, GRB.CONTINUOUS, "v");

// Set objective

m.SetObjective(2*x + y, GRB.MAXIMIZE);

m.AddConstr(u + 4*v <= 9, "l1");

// Approach 1) PWL constraint approach

double intv = 1e-3;
double xmax = Math.Log(9.0);
int len = (int) Math.Ceiling(xmax/intv) + 1;
double[] xpts = new double[len];
double[] upts = new double[len];
for (int i = 0; i < len; i++) {
xpts[i] = i*intv;
upts[i] = f(i*intv);
}
GRBGenConstr gc1 = m.AddGenConstrPWL(x, u, xpts, upts, "gc1");

double ymax = (9.0/4.0)*(9.0/4.0);
len = (int) Math.Ceiling(ymax/intv) + 1;
double[] ypts = new double[len];
double[] vpts = new double[len];
for (int i = 0; i < len; i++) {
ypts[i] = i*intv;
vpts[i] = g(i*intv);
}
GRBGenConstr gc2  = m.AddGenConstrPWL(y, v, ypts, vpts, "gc2");

// Optimize the model and print solution

m.Optimize();
printsol(m, x, y, u, v);

// Approach 2) General function constraint approach with auto PWL
//             translation by Gurobi

// restore unsolved state and get rid of PWL constraints
m.Reset();
m.Remove(gc1);
m.Remove(gc2);
m.Update();

GRBGenConstr gcf1 = m.AddGenConstrExp(x, u, "gcf1", "");
GRBGenConstr gcf2 = m.AddGenConstrPow(y, v, 0.5, "gcf2", "");

m.Parameters.FuncPieceLength = 1e-3;

// Optimize the model and print solution

m.Optimize();
printsol(m, x, y, u, v);

// Zoom in, use optimal solution to reduce the ranges and use a smaller
// pclen=1e-5 to solve it

x.LB = Math.Max(x.LB, x.X-0.01);
x.UB = Math.Min(x.UB, x.X+0.01);
y.LB = Math.Max(y.LB, y.X-0.01);
y.UB = Math.Min(y.UB, y.X+0.01);
m.Update();
m.Reset();

m.Parameters.FuncPieceLength = 1e-5;

// Optimize the model and print solution

m.Optimize();
printsol(m, x, y, u, v);

// Dispose of model and environment

m.Dispose();
env.Dispose();
} catch (GRBException e) {
Console.WriteLine("Error code: " + e.ErrorCode + ". " + e.Message);
}
}
}