7.3 Power Cone Optimization

Conic optimization is a generalization of linear optimization, allowing constraints of the type

\[x^t \in \K_t,\]

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

\[\begin{split}\begin{array}{lccccl} \mbox{minimize} & & & c^T x + c^f & & \\ \mbox{subject to} & l^c & \leq & A x & \leq & u^c, \\ & l^x & \leq & x & \leq & u^x, \\ & & & x \in \K, & & \end{array}\end{split}\]

where the domain restriction, \(x \in \K\), implies that all variables are partitioned into convex cones

\[x = (x^0, x^1, \ldots , x^{p-1}),\quad \mbox{with } x^t \in \K_t \subseteq \real^{n_t}.\]

In this tutorial we describe how to use the power cone. The primal power cone of dimension \(n\) with parameter \(0<\alpha<1\) is defined as:

\[\POW_n^{\alpha,1-\alpha} = \left\lbrace x \in \real^n: x_0^\alpha x_1^{1-\alpha}\geq\sqrt{\sum_{i=2}^{n-1}x_i^2},\ x_0,x_1\geq 0 \right\rbrace.\]

In particular, the most important special case is the three-dimensional power cone family:

\[\POW_3^{\alpha,1-\alpha} = \left\lbrace x \in \real^3: x_0^\alpha x_1^{1-\alpha}\geq |x_2|,\ x_0,x_1\geq 0 \right\rbrace.\]

For example, the conic constraint \((x,y,z)\in\POW_3^{0.25,0.75}\) is equivalent to \(x^{0.25}y^{0.75}\geq |z|\), or simply \(xy^3\geq z^4\) with \(x,y\geq 0\).

MOSEK also supports the dual power cone:

\[\left(\POW_n^{\alpha,1-\alpha}\right)^{*} = \left\lbrace x \in \real^n: \left(\frac{x_0}{\alpha}\right)^\alpha \left(\frac{x_1}{1-\alpha}\right)^{1-\alpha}\geq\sqrt{\sum_{i=2}^{n-1}x_i^2},\ x_0,x_1\geq 0 \right\rbrace.\]

For other types of cones supported by MOSEK see Sec. 7.2 (Conic Quadratic Optimization), Sec. 7.4 (Conic Exponential 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.

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.

7.3.1 Example POW1

Consider the following optimization problem which involves powers of variables:

(7.3)\[\begin{split}\begin{array} {lrcl} \mbox{maximize} & x^{0.2}y^{0.8} + z^{0.4} - x & & \\ \mbox{subject to} & x+y+\frac12 z & = & 2, \\ & x,y,z & \geq & 0. \end{array}\end{split}\]

With \((x,y,z)=(x_0,x_1,x_2)\) we convert it into conic form using auxiliary variables as bounds for the power expressions:

(7.4)\[\begin{split}\begin{array} {lrcl} \mbox{maximize} & x_3 + x_4 - x_0 & & \\ \mbox{subject to} & x_0+x_1+\frac12 x_2 & = & 2, \\ & (x_0,x_1,x_3) & \in & \POW_3^{0.2,0.8}, \\ & (x_2,x_5,x_4) & \in & \POW_3^{0.4,0.6}, \\ & x_5 & = & 1. \end{array}\end{split}\]

We start by creating the optimization model:

  Model::t M = new Model("pow1"); auto _M = finally([&]() { M->dispose(); });

We then define the variable x corresponding to the original problem (7.3), and auxiliary variables appearing in the conic reformulation (7.4).

  Variable::t x  = M->variable("x", 3, Domain::unbounded());
  Variable::t x3 = M->variable();
  Variable::t x4 = M->variable();

The linear constraint is defined using the dot product operator Expr.dot:

  // Create the linear constraint
  auto aval = new_array_ptr<double, 1>({1.0, 1.0, 0.5});
  M->constraint(Expr::dot(x, aval), Domain::equalsTo(2.0));

The primal power cone is referred to via Domain.inPPowerCone with an appropriate list of variables or expressions in each case.

  // Create the conic constraints
  M->constraint(Var::vstack(x->slice(0,2), x3), Domain::inPPowerCone(0.2));
  M->constraint(Expr::vstack(x->index(2), 1.0, x4), Domain::inPPowerCone(0.4));     

We only need the objective function:

  auto cval = new_array_ptr<double, 1>({1.0, 1.0, -1.0});
  M->objective(ObjectiveSense::Maximize, Expr::dot(cval, Var::vstack(x3, x4, x->index(0))));

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. Here we just display the primal solution

  ndarray<double, 1> xlvl   = *(x->level());
  std::cout << "x,y,z = " << xlvl << std::endl;

which is

[ 0.06389298  0.78308564  2.30604283 ]
Listing 7.3 Fusion implementation of model (7.3). Click here to download.
#include <iostream>
#include "fusion.h"

using namespace mosek::fusion;
using namespace monty;

int main(int argc, char ** argv)
{
  Model::t M = new Model("pow1"); auto _M = finally([&]() { M->dispose(); });

  Variable::t x  = M->variable("x", 3, Domain::unbounded());
  Variable::t x3 = M->variable();
  Variable::t x4 = M->variable();

  // Create the linear constraint
  auto aval = new_array_ptr<double, 1>({1.0, 1.0, 0.5});
  M->constraint(Expr::dot(x, aval), Domain::equalsTo(2.0));

  // Create the conic constraints
  M->constraint(Var::vstack(x->slice(0,2), x3), Domain::inPPowerCone(0.2));
  M->constraint(Expr::vstack(x->index(2), 1.0, x4), Domain::inPPowerCone(0.4));     

  auto cval = new_array_ptr<double, 1>({1.0, 1.0, -1.0});
  M->objective(ObjectiveSense::Maximize, Expr::dot(cval, Var::vstack(x3, x4, x->index(0))));

  // Solve the problem
  M->solve();

  // Get the linear solution values
  ndarray<double, 1> xlvl   = *(x->level());
  std::cout << "x,y,z = " << xlvl << std::endl;
}