6.5 Conic Exponential Optimization

The structure of a typical conic optimization problem is

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

(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:

  \[\EXP = \left\lbrace x \in \real^3: x_0 \geq x_1 \exp(x_2/x_1),\ x_0,x_1\geq 0 \right\rbrace.\]

• Dual exponential cone:

  \[\EXP^* = \left\lbrace s \in \real^3: s_0 \geq -s_2 e^{-1} \exp(s_1/s_2),\ s_2\leq 0,s_0\geq 0 \right\rbrace.\]

For example, consider the following constraint:

\[(x_4, x_0, x_2) \in \EXP\]

which describes a convex cone in \(\real^3\) given by the inequalities:

\[x_4 \geq x_0\exp(x_2/x_0),\ x_0,x_4\geq 0.\]

For other types of cones supported by MOSEK, see Sec. 15.8 (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:

(6.12)\[\begin{split}\begin{array} {lrcl} \mbox{minimize} & x_0 + x_1 & & \\ \mbox{subject to} & x_0+x_1+x_2 & = & 1, \\ & x_0 & \geq & x_1\exp(x_2/x_1), \\ & x_0, x_1 & \geq & 0. \end{array}\end{split}\]

The affine conic form of (6.12) is:

(6.13)\[\begin{split}\begin{array} {lrcl} \mbox{minimize} & x_0 + x_1 & & \\ \mbox{subject to} & x_0+x_1+x_2 & = & 1, \\ & Ix & \in & \EXP, \\ & x & \in & \real^3. \end{array}\end{split}\]

where \(I\) is the \(3\times 3\) identity matrix.

Setting up the linear part

The linear parts (constraints, variables, objective) are set up exactly the same way 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 problem, retrieving the solution and so on.

Setting up the conic constraints

To define the conic constraint, we set the prob.f equal to the sparse identity matrix, as required in (6.13). Since g is zero it can be omitted. The domains and dimensions of affine conic constraints are specified using the structure accs (the exponential cone always has dimension 3 but the value is still required).

Listing 6.8 demonstrates how to solve the example (6.12) using MOSEK.

Listing 6.8 Script implementing problem (6.12). Click here to download.

function ceo1()

clear prob;

[r, res] = mosekopt('symbcon');
% Specify the non-conic part of the problem.

prob.c   = [1 1 0];
prob.a   = sparse([1 1 1]);
prob.blc = 1;
prob.buc = 1;
prob.blx = [-inf -inf -inf];
prob.bux = [ inf  inf  inf];

% Specify the affine conic constraint with one exponential cone.

prob.accs   = [res.symbcon.MSK_DOMAIN_PRIMAL_EXP_CONE 3];
prob.f = speye(3);

% prob.accs the domain types, in this case a single exponential cone
% The matrix f is the ientity, meaning that 
%
% I * x \in EXP
%
% which is exactly
%
%   x(1) >= x(2)*exp(x(3)/x(2))

% Optimize the problem. 

[r,res]=mosekopt('minimize',prob);

% Display the primal solution.

res.sol.itr.xx'

For a step by step introduction to formulating problems with affine conic constraints (ACC) see also Sec. 6.2 (From Linear to Conic Optimization).