6.5 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. 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:
describes a convex cone in \(\real^3\) given by the inequalities:
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:
The conic form of (6.9) is:
The linear constraints are specified as if the problem was a linear problem whereas the cones are specified using two index lists
cones.sub and list of cone-type identifiers
cones.type. The elements of all the cones are listed in
cones.subptr specifies the index of the first element in
cones.sub for each cone.
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 = ; % 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'