public static void Main(string[] args)
{
using (Model M = new Model("sdo1"))
{
// Setting up the variables
Variable X = M.Variable("X", Domain.InPSDCone(3));
Variable x = M.Variable("x", Domain.InQCone(3));
DenseMatrix C = new DenseMatrix ( new double[][] { new double[] {2,1,0}, new double[] {1,2,1}, new double[] {0,1,2}} );
DenseMatrix A1 = new DenseMatrix ( new double[][] { new double[] {1,0,0}, new double[] {0,1,0}, new double[] {0,0,1}} );
DenseMatrix A2 = new DenseMatrix ( new double[][] { new double[] {1,1,1}, new double[] {1,1,1}, new double[] {1,1,1}} );
// Objective
M.Objective(ObjectiveSense.Minimize, Expr.Add(Expr.Dot(C, X), x.Index(0)));
// Constraints
M.Constraint("c1", Expr.Add(Expr.Dot(A1, X), x.Index(0)), Domain.EqualsTo(1.0));
M.Constraint("c2", Expr.Add(Expr.Dot(A2, X), Expr.Sum(x.Slice(1,3))), Domain.EqualsTo(0.5));
M.Solve();
Console.WriteLine("[{0}]", (new Utils.StringBuffer()).A(X.Level()).ToString());
Console.WriteLine("[{0}]", (new Utils.StringBuffer()).A(x.Level()).ToString());
}
}
public static void Main(string[] args)
{
double[][] A =
{ new double[] { 3.0, 2.0, 0.0, 1.0 },
new double[] { 2.0, 3.0, 1.0, 1.0 },
new double[] { 0.0, 0.0, 3.0, 2.0 } };
double[] c = { 3.0, 5.0, 1.0, 1.0 };
// Create a model with the name 'lo1'
Model M = new Model("lo1");
// Create variable 'x' of length 3
Variable x = M.Variable("x", 3, Domain.GreaterThan(0.0));
// Create variable 'y' of length 1
Variable y = M.Variable("y", 1, Domain.InRange(0.0, 10.0));
// Create a variable vector consisting of x and y
Variable z = Variable.Vstack(x,y);
// Create three constraints
M.Constraint("c1", Expr.Dot(A[0], z), Domain.EqualsTo(30.0));
M.Constraint("c2", Expr.Dot(A[1], z), Domain.GreaterThan(15.0));
M.Constraint("c3", Expr.Dot(A[2], z), Domain.LessThan(25.0));
// Set the objective function to (c^t * x)
M.Objective("obj", ObjectiveSense.Maximize, Expr.Dot(c, z));
// Solve the problem
M.Solve();
// Get the solution values
double[] sol = z.Level();
Console.WriteLine("x1,x2,x3,y = {0}, {1}, {2}, {3}",sol[0],sol[1],sol[2],sol[3]);
}
public static void Main(string[] args)
{
double[][] A =
{ new double[] { 50.0, 31.0 },
new double[] { 3.0, -2.0 } };
double[] c = { 1.0, 0.64 };
using (Model M = new Model("milo1"))
{
Variable x = M.Variable("x", 2, Domain.GreaterThan(0.0), Domain.IsInteger());
// Create the constraints
// 50.0 x[0] + 31.0 x[1] <= 250.0
// 3.0 x[0] - 2.0 x[1] >= -4.0
M.Constraint("c1", Expr.Dot(A[0], x), Domain.LessThan(250.0));
M.Constraint("c2", Expr.Dot(A[1], x), Domain.GreaterThan(-4.0));
// Set max solution time
M.SetSolverParam("mioMaxTime", 60.0);
// Set max relative gap (to its default value)
M.SetSolverParam("mioTolRelGap", 1e-4);
// Set max absolute gap (to its default value)
M.SetSolverParam("mioTolAbsGap", 0.0);
// Set the objective function to (c^T * x)
M.Objective("obj", ObjectiveSense.Maximize, Expr.Dot(c, x));
// Solve the problem
M.Solve();
// Get the solution values
double[] sol = x.Level();
Console.WriteLine("x1,x2 = {0}, {1}", sol[0], sol[1]);
Console.WriteLine("MIP rel gap = {0} ({0})",M.GetSolverDoubleInfo("mioObjRelGap"),M.GetSolverDoubleInfo("mioObjAbsGap"));
}
}
public static void Main(String[] args)
{
using (Model M = new Model("alan"))
{
Variable x = M.Variable("x", numsec, Domain.GreaterThan(0.0));
Variable t = M.Variable("variance", Domain.GreaterThan(0.0));
M.Objective("minvar", ObjectiveSense.Minimize, t.AsExpr());
// sum securities to 1.0
M.Constraint("wealth", Expr.Sum(x), Domain.EqualsTo(1.0));
// define target expected return
M.Constraint("dmean", Expr.Dot(mean, x), Domain.GreaterThan(target));
M.Constraint(Expr.Vstack(Expr.ConstTerm(1,0.5),
t.AsExpr(),
Expr.Mul(U,x)),
Domain.InRotatedQCone());
Console.WriteLine("Solve...");
M.Solve();
Console.WriteLine("... Solved.");
double[] solx = x.Level();
Console.WriteLine("Primal solution = {0}", solx[0]);
for (int i = 1; i < numsec; ++i)
Console.Write(", {0}", solx[i]);
Console.WriteLine("");
}
}
public static void Main(string[] args)
{
Matrix recipe = new DenseMatrix(recipe_data);
using (Model M = new Model("Recipe"))
{
// "production" defines the amount of each product to bake.
Variable production = M.Variable("production",
new StringSet(productnames),
Domain.GreaterThan(0.0));
// The objective is to maximize the total revenue.
M.Objective("revenue",
ObjectiveSense.Maximize,
Expr.Dot(revenue, production));
// The prodoction is constrained by stock:
M.Constraint(Expr.Mul(recipe, production), Domain.LessThan(stock));
M.SetLogHandler(Console.Out);
// We solve and fetch the solution:
M.Solve();
double[] res = production.Level();
Console.WriteLine("Solution:");
for (int i = 0; i < res.Length; ++i)
{
Console.WriteLine(" Number of {0} : {1}", productnames[i], res[i]);
}
Console.WriteLine(" Revenue : ${0}",
res[0] * revenue[0] + res[1] * revenue[1]);
}
}
public static void Main(string[] argv)
{
int N = 5;
var A = new double[][]
{ new double[] { 0.0, 0.5, -0.1, -0.2, 0.5},
new double[] { 0.5, 1.25, -0.05, -0.1, 0.25},
new double[] {-0.1, -0.05, 0.51, 0.02, -0.05},
new double[] {-0.2, -0.1, 0.02, 0.54, -0.1},
new double[] { 0.5, 0.25, -0.05, -0.1, 1.25} };
// Create a model with the name 'NearestCorrelation
using (var M = new Model("NearestCorrelation"))
{
// Setting up the variables
var X = M.Variable("X", Domain.InPSDCone(N));
var t = M.Variable("t", 1, Domain.Unbounded());
// (t, vec (A-X)) \in Q
M.Constraint( Expr.Vstack(t, Vec(Expr.Sub(new DenseMatrix(A),X))), Domain.InQCone() );
// diag(X) = e
M.Constraint(X.Diag(), Domain.EqualsTo(1.0));
// Objective: Minimize t
M.Objective(ObjectiveSense.Minimize, t);
// Solve the problem
M.Solve();
// Get the solution values
Console.WriteLine("X = {0}",arrtostr(X.Level()));
Console.WriteLine("t = {0}", arrtostr(t.Level()));
}
}
public static void Main(string[] args)
{
using (Model M = new Model("cqo1"))
{
Variable x = M.Variable("x", 3, Domain.GreaterThan(0.0));
Variable y = M.Variable("y", 3, Domain.Unbounded());
// Create the aliases
// z1 = [ y[0],x[0],x[1] ]
// and z2 = [ y[1],y[2],x[2] ]
Variable z1 = Variable.Vstack(y.Index(0), x.Slice(0,2));
Variable z2 = Variable.Vstack(y.Slice(1,3),x.Index(2));
// Create the constraint
// x[0] + x[1] + 2.0 x[2] = 1.0
double[] aval = new double[] {1.0, 1.0, 2.0};
M.Constraint("lc", Expr.Dot(aval, x), Domain.EqualsTo(1.0));
// Create the constraints
// z1 belongs to C_3
// z2 belongs to K_3
// where C_3 and K_3 are respectively the quadratic and
// rotated quadratic cone of size 3, i.e.
// z1[0] > sqrt(z1[1]^2 + z1[2]^2)
// and 2.0 z2[0] z2[1] > z2[2]^2
Constraint qc1 = M.Constraint("qc1", z1.AsExpr(), Domain.InQCone());
Constraint qc2 = M.Constraint("qc2", z2.AsExpr(), Domain.InRotatedQCone());
// Set the objective function to (y[0] + y[1] + y[2])
M.Objective("obj", ObjectiveSense.Minimize, Expr.Sum(y));
// Solve the problem
M.Solve();
// Get the linearsolution values
double[] solx = x.Level();
double[] soly = y.Level();
Console.WriteLine("x1,x2,x3 = {0}, {1}, {2}",solx[0],solx[1],solx[2]);
Console.WriteLine("y1,y2,y3 = {0}, {1}, {2}",soly[0],soly[1],soly[2]);
// Get conic solution of qc1
double[] qc1lvl = qc1.Level();
double[] qc1sn = qc1.Dual();
Console.Write("qc1 levels = {0}", qc1lvl[0]);
for (int i = 1; i < qc1lvl.Length; ++i)
Console.Write(", {0}", qc1lvl[i]);
Console.WriteLine();
Console.Write("qc1 dual conic var levels = {0}", qc1sn[0]);
for (int i = 1; i < qc1sn.Length; ++i)
Console.Write(", {0}", qc1sn[i]);
Console.WriteLine();
}
}
public static void Main(string[] args)
{
double[][] A = new double[][] { new double[] { -0.5, 1.0 } };
double[] b = new double[] { 1.0 };
double[] c = new double[] { 1.0, 1.0 };
using (Model M = new Model("duality"))
{
Variable x = M.Variable("x", 2, Domain.GreaterThan(0.0));
Constraint con = M.Constraint(Expr.Sub(b, Expr.Mul(new DenseMatrix(A), x)), Domain.EqualsTo(0.0));
M.Objective("obj", ObjectiveSense.Minimize, Expr.Dot(c, x));
M.Solve();
double[] xsol = x.Level();
double[] ysol = con.Dual();
Console.WriteLine("x1,x2,y = {0}, {1}, {2}\n",xsol[0],xsol[1],ysol[0]);
}
}
public static void Main(string[] args)
{
// Create a model object
using (Model M = new Model("simple"))
{
// Add two variables
Variable x = M.Variable("x",1,Domain.InRange(0.0,1.0));
Variable y = M.Variable("y",1,Domain.Unbounded());
// Add a constraint on the variables
M.Constraint("bound y", Expr.Sub(y,x), Domain.InRange(-1.0, 2.0));
// Define the objective
M.Objective("obj", ObjectiveSense.Maximize, Expr.Add(x,y));
// Solve the problem
M.Solve();
// Print the solution
Console.WriteLine("Solution. x = {0}, y = {1}",x.Level()[0],y.Level()[0]);
}
}
public static void Main(string[] args)
{
using (Model M = new Model("FacilityLocation"))
{
// Variable holding the facility location
Variable f = M.Variable("facility",new NDSet(1,2),Domain.Unbounded());
// Variable defining the euclidian distances to each customer
Variable d = M.Variable("dist", new NDSet(N,1), Domain.GreaterThan(0.0));
// Variable defining the x and y differences to each customer;
Variable t = M.Variable("t",new NDSet(N,2), Domain.Unbounded());
M.Constraint("dist measure",
Variable.Stack(new Variable[][] { new Variable[] { d, t }}),
Domain.InQCone(N,3));
Variable fxy = Variable.Repeat(f, N);
M.Constraint("xy diff", Expr.Add(t, fxy), Domain.EqualsTo(customerloc));
M.Objective("total_dist", ObjectiveSense.Minimize, Expr.Sum(d));
M.Solve();
double[] floc = f.Level();
Console.WriteLine("Facility location = {0},{1}",floc[0],floc[1]);
}
}
public static double[] fitpoly(double[,] data, int n)
{
using (var M = new Model("smooth poly"))
{
int m = data.GetLength(0);
double[] Adata = new double[m * (n+1)];
for (int j = 0, k = 0; j < m; ++j)
for (int i = 0; i < n+1; ++i, ++k)
Adata[k] = Math.Pow(data[j,0], i);
Matrix A = new DenseMatrix(m,n+1,Adata);
double[] b = new double[m];
for (int j = 0; j < m; ++j)
b[j] = data[j,1];
Variable x = M.Variable("x", n+1, Domain.Unbounded());
Variable z = M.Variable("z", 1, Domain.Unbounded());
Variable dx = diff(M, x);
M.Constraint(Expr.Mul(A, x), Domain.EqualsTo(b));
// z - f'(t) >= 0, for all t \in [a, b]
Variable ub = M.Variable(n, Domain.Unbounded());
M.Constraint(Expr.Sub(ub,
Expr.Vstack(Expr.Sub(z,dx.Index(0)), Expr.Neg(dx.Slice(1,n)))),
Domain.EqualsTo(0.0));
nn_finite(M, ub, data[0,0], data[m-1,0]);
// f'(t) + z >= 0, for all t \in [a, b]
Variable lb = M.Variable(n, Domain.Unbounded());
M.Constraint(Expr.Sub(lb,
Expr.Vstack(Expr.Add(z, dx.Index(0)), dx.Slice(1,n).AsExpr())),
Domain.EqualsTo(0.0));
nn_finite(M, lb, data[0,0], data[m-1,0]);
M.Objective(ObjectiveSense.Minimize, z);
M.Solve();
return x.Level();
}
}
/*
Description:
Extends the basic Markowitz model with a market cost term.
Input:
n: Number of assets
mu: An n dimmensional vector of expected returns
GT: A matrix with n columns so (GT")*GT = covariance matrix"
x0: Initial holdings
w: Initial cash holding
gamma: Maximum risk (=std. dev) accepted
f: If asset j is traded then a fixed cost f_j must be paid
g: If asset j is traded then a cost g_j must be paid for each unit traded
Output:
Optimal expected return and the optimal portfolio
*/
public static double[] MarkowitzWithTransactionsCost
( int n,
double[] mu,
DenseMatrix GT,
double[] x0,
double w,
double gamma,
double[] f,
double[] g)
{
// Upper bound on the traded amount
double[] u = new double[n];
{
double v = w+sum(x0);
for (int i = 0; i < n; ++i) u[i] = v;
}
using( Model M = new Model("Markowitz portfolio with transaction costs") )
{
Console.WriteLine("\n-------------------------------------------------------------------------");
Console.WriteLine("Markowitz portfolio optimization with transaction cost\n");
Console.WriteLine("------------------------------------------------------------------------\n");
//M.SetLogHandler(Console.Out);
// Defines the variables. No shortselling is allowed.
Variable x = M.Variable("x", n, Domain.GreaterThan(0.0));
// Addtional "helper" variables
Variable z = M.Variable("z", n, Domain.Unbounded());
// Binary varables
Variable y = M.Variable("y", n, Domain.InRange(0.0,1.0), Domain.IsInteger());
// Maximize expected return
M.Objective("obj", ObjectiveSense.Maximize, Expr.Dot(mu,x));
// Invest amount + transactions costs = initial wealth
M.Constraint("budget", Expr.Add(Expr.Add(Expr.Sum(x),Expr.Dot(f,y)),Expr.Dot(g,z)),
Domain.EqualsTo(w+sum(x0)));
// Imposes a bound on the risk
M.Constraint("risk", Expr.Vstack(Expr.ConstTerm(new double[] {gamma}),Expr.Mul(GT,x)), Domain.InQCone());
// z >= |x-x0|
M.Constraint("buy", Expr.Sub(z,Expr.Sub(x,x0)),Domain.GreaterThan(0.0));
M.Constraint("sell", Expr.Sub(z,Expr.Sub(x0,x)),Domain.GreaterThan(0.0));
//M.constraint("trade", Expr.hstack(z.asExpr(),Expr.sub(x,x0)), Domain.inQcone())"
// Consraints for turning y off and on. z-diag(u)*y<=0 i.e. z_j <= u_j*y_j
M.Constraint("y_on_off", Expr.Sub(z,Expr.Mul(Matrix.Diag(u),y)), Domain.LessThan(0.0));
// Integer optimization problems can be very hard to solve so limiting the
// maximum amount of time is a valuable safe guard
M.SetSolverParam("mioMaxTime", 180.0);
M.Solve();
Console.WriteLine("Expected return: {0:e4} Std. deviation: {1:e4} Transactions cost: {2:e4}",
dot(mu, x.Level()), gamma, dot(f, y.Level()) + dot(g, z.Level()));
return x.Level();
}
}
public static double[][] lownerjohn_outer(double[][] x)
{
using( Model M = new Model("lownerjohn_outer") )
{
// Direct log output to the terminal for debugging.
M.SetLogHandler(Console.Out);
int m = x.Length;
int n = x[0].Length;
// Setup variables
Variable t = M.Variable("t", 1, Domain.GreaterThan(0.0));
Variable P = M.Variable("P", new NDSet(n,n), Domain.Unbounded());
Variable c = M.Variable("c", n, Domain.Unbounded());
// (1, P(*xi+c)) \in Q
for (int i = 0; i < m; ++i)
M.Constraint( "qc" + i, Expr.Vstack(Expr.Ones(1), Expr.Sub(Expr.Mul(P,x[i]), c)), Domain.InQCone() );
// t <= det(P)^{1/n}
det_rootn(M, P, t);
// Objective: Maximize t
M.Objective(ObjectiveSense.Maximize, t);
M.Solve();
double[] Plvl = P.Level();
double[] clvl = c.Level();
double[][] Pc = new double[n+1][];
for (int i = 0; i < n; ++i)
{
Pc[i] = new double[n];
System.Array.Copy(Plvl,i*n, Pc[i],0, n);
}
Pc[n] = clvl;
return Pc;
}
}
public static double[][] lownerjohn_inner(double[][] A, double[] b)
{
using( Model M = new Model("lownerjohn_inner"))
{
// Direct log output to the terminal for debugging.
M.SetLogHandler(Console.Out);
int m = A.Length;
int n = A[0].Length;
// Setup variables
Variable t = M.Variable("t", 1, Domain.GreaterThan(0.0));
Variable C = M.Variable("C", new NDSet(n,n), Domain.Unbounded());
Variable d = M.Variable("d", n, Domain.Unbounded());
// (bi - ai^T*d, C*ai) \in Q
for (int i = 0; i < m; ++i)
M.Constraint( "qc" + i, Expr.Vstack(Expr.Sub(b[i],Expr.Dot(A[i],d)), Expr.Mul(C,A[i])), Domain.InQCone() );
// t <= det(C)^{1/n}
det_rootn(M, C, t);
// Objective: Maximize t
M.Objective(ObjectiveSense.Maximize, t);
M.Solve();
double[] Clvl = C.Level();
double[] dlvl = d.Level();
double[][] Cres_d = new double[n+1][];
for (int i = 0; i < n; ++i)
{
Cres_d[i] = new double[n];
System.Array.Copy(Clvl,i*n,Cres_d[i],0, n);
}
Cres_d[n] = dlvl;
return Cres_d;
}
}
/*
Purpose:
Computes several portfolios on the optimal portfolios by
for alpha in alphas:
maximize expected return - alpha * standard deviation
subject to the constraints
Input:
n: Number of assets
mu: An n dimmensional vector of expected returns
GT: A matrix with n columns so (GT")*GT = covariance matrix"
x0: Initial holdings
w: Initial cash holding
alphas: List of the alphas
Output:
The efficient frontier as list of tuples (alpha,expected return,risk)
*/
public static void EfficientFrontier
( int n,
double[] mu,
DenseMatrix GT,
double[] x0,
double w,
double[] alphas,
double[] frontier_mux,
double[] frontier_s)
{
using(Model M = new Model("Efficient frontier"))
{
//M.SetLogHandler(Console.Out);
// Defines the variables (holdings). Shortselling is not allowed.
Variable x = M.Variable("x", n, Domain.GreaterThan(0.0)); // Portfolio variables
Variable s = M.Variable("s", 1, Domain.Unbounded()); // Risk variable
M.Constraint("budget", Expr.Sum(x), Domain.EqualsTo(w+sum(x0)));
// Computes the risk
M.Constraint("risk", Expr.Vstack(s.AsExpr(),Expr.Mul(GT,x)),Domain.InQCone());
for (int i = 0; i < alphas.Length; ++i)
{
// Define objective as a weighted combination of return and risk
M.Objective("obj", ObjectiveSense.Maximize, Expr.Sub(Expr.Dot(mu,x),Expr.Mul(alphas[i],s)));
M.Solve();
frontier_mux[i] = dot(mu,x.Level());
frontier_s[i] = s.Level()[0];
}
}
}
/*
Description:
Extends the basic Markowitz model with a market cost term.
Input:
n: Number of assets
mu: An n dimmensional vector of expected returns
GT: A matrix with n columns so (GT')*GT = covariance matrix'
x0: Initial holdings
w: Initial cash holding
gamma: Maximum risk (=std. dev) accepted
m: It is assumed that market impact cost for the j'th asset is
m_j|x_j-x0_j|^3/2
Output:
Optimal expected return and the optimal portfolio
*/
public static void MarkowitzWithMarketImpact
( int n,
double[] mu,
DenseMatrix GT,
double[] x0,
double w,
double gamma,
double[] m,
double[] xsol,
double[] tsol)
{
using(Model M = new Model("Markowitz portfolio with market impact"))
{
//M.SetLogHandler(Console.Out);
// Defines the variables. No shortselling is allowed.
Variable x = M.Variable("x", n, Domain.GreaterThan(0.0));
// Addtional "helper" variables
Variable t = M.Variable("t", n, Domain.Unbounded());
Variable z = M.Variable("z", n, Domain.Unbounded());
Variable v = M.Variable("v", n, Domain.Unbounded());
// Maximize expected return
M.Objective("obj", ObjectiveSense.Maximize, Expr.Dot(mu,x));
// Invested amount + slippage cost = initial wealth
M.Constraint("budget", Expr.Add(Expr.Sum(x),Expr.Dot(m,t)), Domain.EqualsTo(w+sum(x0)));
// Imposes a bound on the risk
M.Constraint("risk", Expr.Vstack(Expr.ConstTerm(new double[] {gamma}),Expr.Mul(GT,x)), Domain.InQCone());
// z >= |x-x0|
M.Constraint("buy", Expr.Sub(z,Expr.Sub(x,x0)),Domain.GreaterThan(0.0));
M.Constraint("sell", Expr.Sub(z,Expr.Sub(x0,x)),Domain.GreaterThan(0.0));
// t >= z^1.5, z >= 0.0. Needs two rotated quadratic cones to model this term
M.Constraint("ta", Expr.Hstack(v.AsExpr(),t.AsExpr(),z.AsExpr()),Domain.InRotatedQCone());
M.Constraint("tb", Expr.Hstack(z.AsExpr(),Expr.ConstTerm(n,1.0/8.0),v.AsExpr()),
Domain.InRotatedQCone());
M.Solve();
if (xsol != null)
Array.Copy(x.Level(),xsol,n);
if (tsol != null)
Array.Copy(t.Level(),tsol,n);
}
}
/*
Purpose:
Computes the optimal portfolio for a given risk
Input:
n: Number of assets
mu: An n dimmensional vector of expected returns
GT: A matrix with n columns so (GT")*GT = covariance matrix"
x0: Initial holdings
w: Initial cash holding
gamma: Maximum risk (=std. dev) accepted
Output:
Optimal expected return and the optimal portfolio
*/
public static double BasicMarkowitz
( int n,
double[] mu,
DenseMatrix GT,
double[] x0,
double w,
double gamma)
{
using( Model M = new Model("Basic Markowitz"))
{
// Redirect log output from the solver to stdout for debugging.
// ifuncommented.
//M.SetLogHandler(Console.Out);
// Defines the variables (holdings). Shortselling is not allowed.
Variable x = M.Variable("x", n, Domain.GreaterThan(0.0));
// Maximize expected return
M.Objective("obj", ObjectiveSense.Maximize, Expr.Dot(mu,x));
// The amount invested must be identical to intial wealth
M.Constraint("budget", Expr.Sum(x), Domain.EqualsTo(w+sum(x0)));
// Imposes a bound on the risk
M.Constraint("risk", Expr.Vstack(Expr.ConstTerm(new double[] {gamma}),Expr.Mul(GT,x)), Domain.InQCone());
// Solves the model.
M.Solve();
return dot(mu,x.Level());
}
}
