6.5 Conic Exponential 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 primal exponential cone defined as:

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

MOSEK also supports the 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 other types of cones supported by MOSEK see Sec. 6.3 (Conic Quadratic Optimization), Sec. 6.4 (Power Cone Optimization), Sec. 6.6 (Semidefinite Optimization). Different cone types can appear together in one optimization problem.

For example, the following constraint:

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

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

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

Furthermore, each variable may belong to one cone at most. The constraint \(x_i - x_j = 0\) would however allow \(x_i\) and \(x_j\) to belong to different cones with same effect.

6.5.1 Example CEO1

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

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

The conic form of (6.9) is:

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

The linear constraints are specified as if the problem was a linear problem whereas the cones are specified using two index lists cones.subptr and cones.sub and list of cone-type identifiers cones.type. The elements of all the cones are listed in cones.sub, and cones.subptr specifies the index of the first element in cones.sub for each cone.

Listing 6.9 demonstrates how to solve the example (6.9) using MOSEK.

Listing 6.9 Script implementing problem (6.9). 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 cones.

prob.cones.type   = [res.symbcon.MSK_CT_PEXP];
prob.cones.sub    = [1, 2, 3];
prob.cones.subptr = [1];
% The field 'type' specifies the cone types, in this case an exponential
% cone with key MSK_CT_PEXP.
%
% The fields 'sub' and 'subptr' specify the members of the cones,
% i.e., the above definitions imply that 
%   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'