Treatment of nonconvexities using homotopy¶
Using homotopy, a convex optimization problem can be continuously deformed into a nonconvex problem.

class
rtctools.optimization.homotopy_mixin.
HomotopyMixin
¶ Bases:
rtctools.optimization.optimization_problem.OptimizationProblem
Adds homotopy to your optimization problem. A homotopy is a continuous transformation between two optimization problems, parametrized by a single parameter \(\theta \in [0, 1]\).
Homotopy may be used to solve nonconvex optimization problems, by starting with a convex approximation at \(\theta = 0.0\) and ending with the nonconvex problem at \(\theta = 1.0\).
Note
It is advised to look for convex reformulations of your problem, before resorting to a use of the (potentially expensive) homotopy process.

homotopy_options
(self)¶ Returns a dictionary of options controlling the homotopy process.
Option Type Default value delta_theta_0
float
1.0
delta_theta_min
float
0.01
homotopy_parameter
string
theta
The homotopy process is controlled by the homotopy parameter in the model, specified by the option
homotopy_parameter
. The homotopy parameter is initialized to0.0
, and increases to a value of1.0
with a dynamically changing step size. This step size is initialized with the value of the optiondelta_theta_0
. If this step size is too large, i.e., if the problem with the increased homotopy parameter fails to converge, the step size is halved. The process of halving terminates when the step size falls below the minimum value specified by the optiondelta_theta_min
.Returns: A dictionary of homotopy options.
