7.4 Conic Exponential Optimization¶
Conic optimization is a generalization of linear optimization, allowing constraints of the type
where \(x^t\) is a subset of the problem variables and \(\K_t\) is a convex cone. Since the set \(\real^n\) of real numbers is also a convex cone, we can simply write a compound conic constraint \(x\in \K\) where \(\K=\K_1\times\cdots\times\K_l\) is a product of smaller cones and \(x\) is the full problem variable.
MOSEK can solve conic optimization problems of the form
where the domain restriction, \(x \in \K\), implies that all variables are partitioned into convex cones
In this tutorial we describe how to use the primal exponential cone defined as:
MOSEK also supports the dual exponential cone:
For other types of cones supported by MOSEK see Sec. 7.2 (Conic Quadratic Optimization), Sec. 7.3 (Power Cone Optimization), Sec. 7.5 (Semidefinite Optimization). See Domain
for a list and definitions of available cone types. Different cone types can appear together in one optimization problem.
For example, the following constraint:
describes a convex cone in \(\real^3\) given by the inequalities:
In Fusion the coordinates of a cone are not restricted to single variables. They can be arbitrary linear expressions, and an auxiliary variable will be substituted by Fusion in a way transparent to the user.
7.4.1 Example CEO1¶
Consider the following basic conic exponential problem which involves some linear constraints and an exponential inequality:
The conic form of (7.5) is:
We start by creating the optimization model:
using (Model M = new Model("ceo1"))
{
We then define the variable x
.
Variable x = M.Variable("x", 3, Domain.Unbounded());
The linear constraint is defined using the sum operator Expr.Sum
:
// Create the constraint
// x[0] + x[1] + x[2] = 1.0
M.Constraint("lc", Expr.Sum(x), Domain.EqualsTo(1.0));
The conic exponential constraint in this case is very simple as it involves just the variable x
. The primal exponential cone is referred to via Domain.InPExpCone
, and it must be applied to a variable of length 3 or an array of such variables. Note that this is a basic way of defining conic constraints, and that in practice they would have more complicated structure.
// Create the exponential conic constraint
Constraint expc = M.Constraint("expc", x, Domain.InPExpCone());
We only need the objective function:
// Set the objective function to (x[0] + x[1])
M.Objective("obj", ObjectiveSense.Minimize, Expr.Sum(x.Slice(0,2)));
Calling the Model.Solve
method invokes the solver:
M.Solve();
The primal and dual solution values can be retrieved using Variable.Level
, Constraint.Level
and Variable.Dual
, Constraint.Dual
, respectively:
// Get the linear solution values
double[] solx = x.Level();
// Get conic solution of expc
double[] expclvl = expc.Level();
double[] expcsn = expc.Dual();
using System;
using mosek.fusion;
namespace mosek.fusion.example
{
public class ceo1
{
public static void Main(string[] args)
{
using (Model M = new Model("ceo1"))
{
Variable x = M.Variable("x", 3, Domain.Unbounded());
// Create the constraint
// x[0] + x[1] + x[2] = 1.0
M.Constraint("lc", Expr.Sum(x), Domain.EqualsTo(1.0));
// Create the exponential conic constraint
Constraint expc = M.Constraint("expc", x, Domain.InPExpCone());
// Set the objective function to (x[0] + x[1])
M.Objective("obj", ObjectiveSense.Minimize, Expr.Sum(x.Slice(0,2)));
// Solve the problem
M.Solve();
// Get the linear solution values
double[] solx = x.Level();
Console.WriteLine("x1,x2,x3 = {0}, {1}, {2}", solx[0], solx[1], solx[2]);
// Get conic solution of expc
double[] expclvl = expc.Level();
double[] expcsn = expc.Dual();
Console.Write("expc levels = {0}", expclvl[0]);
for (int i = 1; i < expclvl.Length; ++i)
Console.Write(", {0}", expclvl[i]);
Console.WriteLine();
Console.Write("expc dual conic var levels = {0}", expcsn[0]);
for (int i = 1; i < expcsn.Length; ++i)
Console.Write(", {0}", expcsn[i]);
Console.WriteLine();
}
}
}
}