Qcp Examples#
This section includes source code for all of the Gurobi qcp examples.
The same source code can be found in the examples directory of the
Gurobi distribution.
/* Copyright 2025, Gurobi Optimization, LLC */
/* This example formulates and solves the following simple QCP model:
maximize x
subject to x + y + z = 1
x^2 + y^2 <= z^2 (second-order cone)
x^2 <= yz (rotated second-order cone)
x, y, z non-negative
*/
#include <stdlib.h>
#include <stdio.h>
#include "gurobi_c.h"
int
main(int argc,
char *argv[])
{
GRBenv *env = NULL;
GRBmodel *model = NULL;
int error = 0;
double sol[3];
int ind[3];
double val[3];
double obj[] = {1, 0, 0};
int qrow[3];
int qcol[3];
double qval[3];
int optimstatus;
double objval;
/* Create environment */
error = GRBloadenv(&env, "qcp.log");
if (error) goto QUIT;
/* Create an empty model */
error = GRBnewmodel(env, &model, "qcp", 0, NULL, NULL, NULL, NULL, NULL);
if (error) goto QUIT;
/* Add variables */
error = GRBaddvars(model, 3, 0, NULL, NULL, NULL, obj, NULL, NULL, NULL,
NULL);
if (error) goto QUIT;
/* Change sense to maximization */
error = GRBsetintattr(model, GRB_INT_ATTR_MODELSENSE, GRB_MAXIMIZE);
if (error) goto QUIT;
/* Linear constraint: x + y + z = 1 */
ind[0] = 0; ind[1] = 1; ind[2] = 2;
val[0] = 1; val[1] = 1; val[2] = 1;
error = GRBaddconstr(model, 3, ind, val, GRB_EQUAL, 1.0, "c0");
if (error) goto QUIT;
/* Cone: x^2 + y^2 <= z^2 */
qrow[0] = 0; qcol[0] = 0; qval[0] = 1.0;
qrow[1] = 1; qcol[1] = 1; qval[1] = 1.0;
qrow[2] = 2; qcol[2] = 2; qval[2] = -1.0;
error = GRBaddqconstr(model, 0, NULL, NULL, 3, qrow, qcol, qval,
GRB_LESS_EQUAL, 0.0, "qc0");
if (error) goto QUIT;
/* Rotated cone: x^2 <= yz */
qrow[0] = 0; qcol[0] = 0; qval[0] = 1.0;
qrow[1] = 1; qcol[1] = 2; qval[1] = -1.0;
error = GRBaddqconstr(model, 0, NULL, NULL, 2, qrow, qcol, qval,
GRB_LESS_EQUAL, 0.0, "qc1");
if (error) goto QUIT;
/* Optimize model */
error = GRBoptimize(model);
if (error) goto QUIT;
/* Write model to 'qcp.lp' */
error = GRBwrite(model, "qcp.lp");
if (error) goto QUIT;
/* Capture solution information */
error = GRBgetintattr(model, GRB_INT_ATTR_STATUS, &optimstatus);
if (error) goto QUIT;
error = GRBgetdblattr(model, GRB_DBL_ATTR_OBJVAL, &objval);
if (error) goto QUIT;
error = GRBgetdblattrarray(model, GRB_DBL_ATTR_X, 0, 3, sol);
if (error) goto QUIT;
printf("\nOptimization complete\n");
if (optimstatus == GRB_OPTIMAL) {
printf("Optimal objective: %.4e\n", objval);
printf(" x=%.2f, y=%.2f, z=%.2f\n", sol[0], sol[1], sol[2]);
} else if (optimstatus == GRB_INF_OR_UNBD) {
printf("Model is infeasible or unbounded\n");
} else {
printf("Optimization was stopped early\n");
}
QUIT:
/* Error reporting */
if (error) {
printf("ERROR: %s\n", GRBgeterrormsg(env));
exit(1);
}
/* Free model */
GRBfreemodel(model);
/* Free environment */
GRBfreeenv(env);
return 0;
}
/* Copyright 2025, Gurobi Optimization, LLC */
/* This example formulates and solves the following simple QCP model:
maximize x
subject to x + y + z = 1
x^2 + y^2 <= z^2 (second-order cone)
x^2 <= yz (rotated second-order cone)
x, y, z non-negative
*/
#include "gurobi_c++.h"
using namespace std;
int
main(int argc,
char *argv[])
{
try {
GRBEnv env = GRBEnv();
GRBModel model = GRBModel(env);
// Create variables
GRBVar x = model.addVar(0.0, GRB_INFINITY, 0.0, GRB_CONTINUOUS, "x");
GRBVar y = model.addVar(0.0, GRB_INFINITY, 0.0, GRB_CONTINUOUS, "y");
GRBVar z = model.addVar(0.0, GRB_INFINITY, 0.0, GRB_CONTINUOUS, "z");
// Set objective
GRBLinExpr obj = x;
model.setObjective(obj, GRB_MAXIMIZE);
// Add linear constraint: x + y + z = 1
model.addConstr(x + y + z == 1, "c0");
// Add second-order cone: x^2 + y^2 <= z^2
model.addQConstr(x*x + y*y <= z*z, "qc0");
// Add rotated cone: x^2 <= yz
model.addQConstr(x*x <= y*z, "qc1");
// Optimize model
model.optimize();
cout << x.get(GRB_StringAttr_VarName) << " "
<< x.get(GRB_DoubleAttr_X) << endl;
cout << y.get(GRB_StringAttr_VarName) << " "
<< y.get(GRB_DoubleAttr_X) << endl;
cout << z.get(GRB_StringAttr_VarName) << " "
<< z.get(GRB_DoubleAttr_X) << endl;
cout << "Obj: " << model.get(GRB_DoubleAttr_ObjVal) << endl;
} catch(GRBException e) {
cout << "Error code = " << e.getErrorCode() << endl;
cout << e.getMessage() << endl;
} catch(...) {
cout << "Exception during optimization" << endl;
}
return 0;
}
/* Copyright 2025, Gurobi Optimization, LLC */
/* This example formulates and solves the following simple QCP model:
maximize x
subject to x + y + z = 1
x^2 + y^2 <= z^2 (second-order cone)
x^2 <= yz (rotated second-order cone)
x, y, z non-negative
*/
using System;
using Gurobi;
class qcp_cs
{
static void Main()
{
try {
GRBEnv env = new GRBEnv("qcp.log");
GRBModel model = new GRBModel(env);
// Create variables
GRBVar x = model.AddVar(0.0, GRB.INFINITY, 0.0, GRB.CONTINUOUS, "x");
GRBVar y = model.AddVar(0.0, GRB.INFINITY, 0.0, GRB.CONTINUOUS, "y");
GRBVar z = model.AddVar(0.0, GRB.INFINITY, 0.0, GRB.CONTINUOUS, "z");
// Set objective
GRBLinExpr obj = x;
model.SetObjective(obj, GRB.MAXIMIZE);
// Add linear constraint: x + y + z = 1
model.AddConstr(x + y + z == 1.0, "c0");
// Add second-order cone: x^2 + y^2 <= z^2
model.AddQConstr(x*x + y*y <= z*z, "qc0");
// Add rotated cone: x^2 <= yz
model.AddQConstr(x*x <= y*z, "qc1");
// Optimize model
model.Optimize();
Console.WriteLine(x.VarName + " " + x.X);
Console.WriteLine(y.VarName + " " + y.X);
Console.WriteLine(z.VarName + " " + z.X);
Console.WriteLine("Obj: " + model.ObjVal + " " + obj.Value);
// Dispose of model and env
model.Dispose();
env.Dispose();
} catch (GRBException e) {
Console.WriteLine("Error code: " + e.ErrorCode + ". " + e.Message);
}
}
}
/* Copyright 2025, Gurobi Optimization, LLC */
/* This example formulates and solves the following simple QCP model:
maximize x
subject to x + y + z = 1
x^2 + y^2 <= z^2 (second-order cone)
x^2 <= yz (rotated second-order cone)
x, y, z non-negative
*/
import com.gurobi.gurobi.*;
public class Qcp {
public static void main(String[] args) {
try {
GRBEnv env = new GRBEnv("qcp.log");
GRBModel model = new GRBModel(env);
// Create variables
GRBVar x = model.addVar(0.0, GRB.INFINITY, 0.0, GRB.CONTINUOUS, "x");
GRBVar y = model.addVar(0.0, GRB.INFINITY, 0.0, GRB.CONTINUOUS, "y");
GRBVar z = model.addVar(0.0, GRB.INFINITY, 0.0, GRB.CONTINUOUS, "z");
// Set objective
GRBLinExpr obj = new GRBLinExpr();
obj.addTerm(1.0, x);
model.setObjective(obj, GRB.MAXIMIZE);
// Add linear constraint: x + y + z = 1
GRBLinExpr expr = new GRBLinExpr();
expr.addTerm(1.0, x); expr.addTerm(1.0, y); expr.addTerm(1.0, z);
model.addConstr(expr, GRB.EQUAL, 1.0, "c0");
// Add second-order cone: x^2 + y^2 <= z^2
GRBQuadExpr qexpr = new GRBQuadExpr();
qexpr.addTerm(1.0, x, x);
qexpr.addTerm(1.0, y, y);
qexpr.addTerm(-1.0, z, z);
model.addQConstr(qexpr, GRB.LESS_EQUAL, 0.0, "qc0");
// Add rotated cone: x^2 <= yz
qexpr = new GRBQuadExpr();
qexpr.addTerm(1.0, x, x);
qexpr.addTerm(-1.0, y, z);
model.addQConstr(qexpr, GRB.LESS_EQUAL, 0.0, "qc1");
// Optimize model
model.optimize();
System.out.println(x.get(GRB.StringAttr.VarName)
+ " " +x.get(GRB.DoubleAttr.X));
System.out.println(y.get(GRB.StringAttr.VarName)
+ " " +y.get(GRB.DoubleAttr.X));
System.out.println(z.get(GRB.StringAttr.VarName)
+ " " +z.get(GRB.DoubleAttr.X));
System.out.println("Obj: " + model.get(GRB.DoubleAttr.ObjVal) + " " +
obj.getValue());
System.out.println();
// Dispose of model and environment
model.dispose();
env.dispose();
} catch (GRBException e) {
System.out.println("Error code: " + e.getErrorCode() + ". " +
e.getMessage());
}
}
}
function qcp()
% Copyright 2025, Gurobi Optimization, LLC
%
% This example formulates and solves the following simple QCP model:
% maximize
% x
% subject to
% x + y + z = 1
% x^2 + y^2 <= z^2 (second-order cone)
% x^2 <= yz (rotated second-order cone)
% x, y, z non-negative
names = {'x', 'y', 'z'};
model.varnames = names;
% Set objective: x
model.obj = [ 1 0 0 ];
model.modelsense = 'max';
% Add constraint: x + y + z = 1
model.A = sparse([1 1 1]);
model.rhs = 1;
model.sense = '=';
% Add second-order cone: x^2 + y^2 <= z^2 using a sparse matrix
model.quadcon(1).Qc = sparse([
1 0 0;
0 1 0;
0 0 -1]);
model.quadcon(1).q = zeros(3,1);
model.quadcon(1).rhs = 0.0;
model.quadcon(1).name = 'std_cone';
% Add rotated cone: x^2 <= yz using sparse triplet representation
% Equivalent sparse matrix data:
%model.quadcon(2).Qc = sparse([
% 1 0 0;
% 0 0 -1;
% 0 0 0]);
model.quadcon(2).Qrow = [1, 2]
model.quadcon(2).Qcol = [1, 3]
model.quadcon(2).Qval = [1, -1]
% All-zero sparse 3-by-1 vector
model.quadcon(2).q = sparse(3,1);
model.quadcon(2).rhs = 0.0;
model.quadcon(2).name = 'rot_cone';
gurobi_write(model, 'qcp.lp');
result = gurobi(model);
for j=1:3
fprintf('%s %e\n', names{j}, result.x(j))
end
fprintf('Obj: %e\n', result.objval);
end
#!/usr/bin/env python3.11
# Copyright 2025, Gurobi Optimization, LLC
# This example formulates and solves the following simple QCP model:
# maximize x
# subject to x + y + z = 1
# x^2 + y^2 <= z^2 (second-order cone)
# x^2 <= yz (rotated second-order cone)
# x, y, z non-negative
import gurobipy as gp
from gurobipy import GRB
# Create a new model
m = gp.Model("qcp")
# Create variables
x = m.addVar(name="x")
y = m.addVar(name="y")
z = m.addVar(name="z")
# Set objective: x
obj = 1.0 * x
m.setObjective(obj, GRB.MAXIMIZE)
# Add constraint: x + y + z = 1
m.addConstr(x + y + z == 1, "c0")
# Add second-order cone: x^2 + y^2 <= z^2
m.addConstr(x**2 + y**2 <= z**2, "qc0")
# Add rotated cone: x^2 <= yz
m.addConstr(x**2 <= y * z, "qc1")
m.optimize()
for v in m.getVars():
print(f"{v.VarName} {v.X:g}")
print(f"Obj: {m.ObjVal:g}")
# Copyright 2025, Gurobi Optimization, LLC
#
# This example formulates and solves the following simple QCP model:
# maximize
# x
# subject to
# x + y + z = 1
# x^2 + y^2 <= z^2 (second-order cone)
# x^2 <= yz (rotated second-order cone)
# x, y, z non-negative
library(gurobi)
library(Matrix)
model <- list()
model$A <- matrix(c(1,1,1), nrow=1, byrow=T)
model$modelsense <- 'max'
model$obj <- c(1,0,0)
model$rhs <- c(1)
model$sense <- c('=')
# First quadratic constraint: x^2 + y^2 - z^2 <= 0
qc1 <- list()
qc1$Qc <- spMatrix(3, 3, c(1, 2, 3), c(1, 2, 3), c(1.0, 1.0, -1.0))
qc1$rhs <- 0.0
# Second quadratic constraint: x^2 - yz <= 0
qc2 <- list()
qc2$Qc <- spMatrix(3, 3, c(1, 2), c(1, 3), c(1.0, -1.0))
qc2$rhs <- 0.0
model$quadcon <- list(qc1, qc2)
result <- gurobi(model)
print(result$objval)
print(result$x)
# Clear space
rm(model, result)
' Copyright 2025, Gurobi Optimization, LLC
' This example formulates and solves the following simple QCP model:
'
' maximize x
' subject to x + y + z = 1
' x^2 + y^2 <= z^2 (second-order cone)
' x^2 <= yz (rotated second-order cone)
' x, y, z non-negative
Imports Gurobi
Class qcp_vb
Shared Sub Main()
Try
Dim env As New GRBEnv("qcp.log")
Dim model As New GRBModel(env)
' Create variables
Dim x As GRBVar = model.AddVar(0.0, GRB.INFINITY, 0.0, GRB.CONTINUOUS, "x")
Dim y As GRBVar = model.AddVar(0.0, GRB.INFINITY, 0.0, GRB.CONTINUOUS, "y")
Dim z As GRBVar = model.AddVar(0.0, GRB.INFINITY, 0.0, GRB.CONTINUOUS, "z")
' Set objective
Dim obj As GRBLinExpr = x
model.SetObjective(obj, GRB.MAXIMIZE)
' Add linear constraint: x + y + z = 1
model.AddConstr(x + y + z = 1.0, "c0")
' Add second-order cone: x^2 + y^2 <= z^2
model.AddQConstr(x * x + y * y <= z * z, "qc0")
' Add rotated cone: x^2 <= yz
model.AddQConstr(x * x <= y * z, "qc1")
' Optimize model
model.Optimize()
Console.WriteLine(x.VarName & " " & x.X)
Console.WriteLine(y.VarName & " " & y.X)
Console.WriteLine(z.VarName & " " & z.X)
Console.WriteLine("Obj: " & model.ObjVal & " " & obj.Value)
' Dispose of model and env
model.Dispose()
env.Dispose()
Catch e As GRBException
Console.WriteLine("Error code: " & e.ErrorCode & ". " & e.Message)
End Try
End Sub
End Class