Shared Concepts
Folded Concave Penalized Selection
This section applies to the glm action in the regression action set.
The folded concave penalized (FCP) selection coefficients 
are the solution to the regularized least squares
where 
is a folded concave penalty.
A variety of nonconvex penalties have been proposed. Two of the earliest and most influential nonconvex penalties are the smoothly clipped absolute deviation (SCAD) and the minimax concave penalty (MCP).
SCAD
When the method subparameter value is 'scad', the model selection action performs the SCAD selection method, which minimizes ordinary least squares plus the smoothly clipped absolute deviation (SCAD) function 
:
The quadratic program (QP) reformulation of the SCAD problem is given by
where 
and 
.
Liu, Yao, and Li (2016) show that the preceding QP (1) is equivalent to the following mixed integer linear program (MILP):
where 
is a properly large constant.
MCP
When the method subparameter value is 'mcp', the model selection action performs the MCP selection method, which minimizes ordinary least squares plus the minimax concave penalty (MCP) function 
:
The QP reformulation of the MCP problem is given by
Liu, Yao, and Li (2016) show that the preceding QP (3) is equivalent to the following MILP:
where is a properly large constant.
SOLVER
When the solver subparameter value is 'MILP' in the fcpSelectionOptions subparameter, the coefficients 
are obtained by solving the minimization problem (2) or (4). Furthermore, you can also specify the bigM subparameter in the fcpSelectionOptions subparameter for the 
value and the intTol subparameter in the fcpSelectionOptions subparameter for the tolerance of integer variables 
. By default, 
and intTol=1E–7.
When the solver subparameter value is 'NLP' in the fcpSelectionOptions subparameter, the coefficients are obtained by solving the minimization problem (1) or (3).
LAMBDAGRID
When you search for the optimal model, the default values of 
and 
are, respectively,
where 
is the standard deviation of 
, m is the length of , and 
is the natural logarithm.
Suppose that the number of steps in the search for a suitable 
value is 
. You can define the value by specifying the maxIterLambda subparameter. When you specify 'LINSPACE' for the lambdaGrid subparameter, the values of series are generated as follows:
where 
.
When the lambdaGrid subparameter value is 'LOGSPACE' in the fcpSelectionOptions subparameter, the values of series are generated as follows:
where 
.