6.5 Conic Exponential Optimization

The structure of a typical conic optimization problem is

$$\begin{array}{r}\begin{array}{lccccl}\text{minimize}& & & c^{T}x+c^f& & \\ \text{subject to}& l^c& \le & Ax& \le & u^c,\\ & l^x& \le & x& \le & u^x,\\ & & & Fx+g& \in & \mathcal{D},\end{array}\end{array}$$

(see Sec. 12 (Problem Formulation and Solutions) for detailed formulations). We recommend Sec. 6.2 (From Linear to Conic Optimization) for a tutorial on how problems of that form are represented in MOSEK and what data structures are relevant. Here we discuss how to set-up problems with the primal/dual exponential cones.

MOSEK supports two exponential cones, namely:

• Primal exponential cone:

  $$K_{\exp }=\{x\in {\mathbb{R}}^3:x_0\ge x_1\exp (x_2/x_1),x_0,x_1\ge 0\}.$$

• Dual exponential cone:

  $$K_{\exp }^{\ast }=\{s\in {\mathbb{R}}^3:s_0\ge -s_2{e}^{-1}\exp (s_1/s_2),s_2\le 0,s_0\ge 0\}.$$

For example, consider the following constraint:

$$(x_4,x_0,x_2)\in K_{\exp }$$

which describes a convex cone in ${\mathbb{R}}^3$ given by the inequalities:

$$x_4\ge x_0\exp (x_2/x_0),x_0,x_4\ge 0.$$

For other types of cones supported by MOSEK, see Sec. 15.11 (Supported domains) and the other tutorials in this chapter. Different cone types can appear together in one optimization problem.

6.5.1 Example CEO1

Consider the following basic conic exponential problem which involves some linear constraints and an exponential inequality:

(6.15)$$\begin{array}{r}\begin{array}{lrcl}\text{minimize}& x_0+x_1& & \\ \text{subject to}& x_0+x_1+x_2& =& 1,\\ & x_0& \ge & x_1\exp (x_2/x_1),\\ & x_0,x_1& \ge & 0.\end{array}\end{array}$$

The affine conic form of (6.15) is:

(6.16)$$\begin{array}{r}\begin{array}{lrcl}\text{minimize}& x_0+x_1& & \\ \text{subject to}& x_0+x_1+x_2& =& 1,\\ & Ix& \in & K_{\exp },\\ & x& \in & {\mathbb{R}}^3.\end{array}\end{array}$$

where $I$ is the $3\times 3$ identity matrix.

Setting up the linear part

The linear parts (constraints, variables, objective) are set up using exactly the same methods as for linear problems, and we refer to Sec. 6.1 (Linear Optimization) for all the details. The same applies to technical aspects such as defining an optimization task, retrieving the solution and so on.

Setting up the conic constraints

In order to append the conic constraints we first input the sparse identity matrix $\mathbf{F}$ as indicated by (6.16).

The affine conic constraint is then appended using the function Task.appendacc, with the primal exponential domain and the list of $\mathbf{F}$ rows, in this case consisting of all rows in their natural order.

        # Create a 3x3 identity matrix F
        task.appendafes(3)
        task.putafefentrylist([0, 1, 2],         # Rows 
                              [0, 1, 2],         # Columns 
                              [1.0] * 3)

        # Exponential cone (x(0),x(1),x(2)) \in EXP 
        expdomain  = task.appendprimalexpconedomain()
        task.appendacc(expdomain,               # Domain
                       [0, 1, 2],               # Rows from F 
                       None)                    # Unused 

The first argument selects the domain, which must be appended before being used, and must have the dimension matching the number of affine expressions appearing in the constraint. Variants of this method are available to append multiple ACCs at a time. It is also possible to define the matrix $\mathbf{F}$ using a variety of methods (row after row, column by column, individual entries, etc.) similarly as for the linear constraint matrix $A$.

For a more thorough exposition of the affine expression storage (AFE) matrix $\mathbf{F}$ and vector $\mathbf{g}$ see Sec. 6.2 (From Linear to Conic Optimization).

Source code

Listing 6.6 Source code solving problem (6.15). Click here to download.

import sys
import mosek

# Define a stream printer to grab output from MOSEK
def streamprinter(text):
    sys.stdout.write(text)
    sys.stdout.flush()

def main():

    # Only a symbolic constant
    inf = 0.0

    # Create a task
    with mosek.Task() as task:
        # Attach a printer to the task
        task.set_Stream(mosek.streamtype.log, streamprinter)

        c = [1.0, 1.0, 0.0]
        a = [1.0, 1.0, 1.0]
        numvar, numcon = 3, 1

        # Append 'numcon' empty constraints.
        # The constraints will initially have no bounds.
        task.appendcons(numcon)

        # Append 'numvar' variables.
        # The variables will initially be fixed at zero (x=0).
        task.appendvars(numvar)

        # Set up the linear part of the problem
        task.putcslice(0, numvar, c)
        task.putarow(0, [0, 1, 2], a)
        task.putvarboundslice(0, numvar, [mosek.boundkey.fr] * numvar, [inf] * numvar, [inf] * numvar)
        task.putconbound(0, mosek.boundkey.fx, 1.0, 1.0)

        # Add a conic constraint
        # Create a 3x3 identity matrix F
        task.appendafes(3)
        task.putafefentrylist([0, 1, 2],         # Rows 
                              [0, 1, 2],         # Columns 
                              [1.0] * 3)

        # Exponential cone (x(0),x(1),x(2)) \in EXP 
        expdomain  = task.appendprimalexpconedomain()
        task.appendacc(expdomain,               # Domain
                       [0, 1, 2],               # Rows from F 
                       None)                    # Unused 

        # Input the objective sense (minimize/maximize)
        task.putobjsense(mosek.objsense.minimize)
        task.writedata("ceo1-py.ptf")

        # Optimize the task
        task.optimize()
        # Print a summary containing information
        # about the solution for debugging purposes
        task.solutionsummary(mosek.streamtype.msg)
        prosta = task.getprosta(mosek.soltype.itr)
        solsta = task.getsolsta(mosek.soltype.itr)

        # Output a solution
        xx = task.getxx(mosek.soltype.itr)

        if solsta == mosek.solsta.optimal:
            print("Optimal solution: %s" % xx)
        elif solsta == mosek.solsta.dual_infeas_cer:
            print("Primal or dual infeasibility.\n")
        elif solsta == mosek.solsta.prim_infeas_cer:
            print("Primal or dual infeasibility.\n")
        elif mosek.solsta.unknown:
            print("Unknown solution status")
        else:
            print("Other solution status")


# call the main function
try:
    main()
except mosek.MosekException as e:
    print("ERROR: %s" % str(e.code))
    if msg is not None:
        print("\t%s" % e.msg)
        sys.exit(1)
except:
    import traceback
    traceback.print_exc()
    sys.exit(1)