objective.function: Computes the coefficient terms of the objective functions
In conroylau/lpinfer: An R Package for Inference in Linear Programs

objective.function

R Documentation

Computes the coefficient terms of the objective functions

Description

This function computes the matrices in the objective functions for linear programs. This function takes matrix \bm{A} and \bm{\beta} as input and computes the coefficients of the objective function.

Usage

objective.function(A, b, n, weight = NULL)

Arguments

`A`	The matrix `\bm{A}`.
`b`	The column vector `\bm{\beta}`.
`n`	The sample size `n`.
`weight`	The weighting matrix.

Details

Quadratic programs — Given inputs \bm{A} \in \mathbf{R}^{m\times m}, \bm{b} \in \mathbf{R}^m and \bm{W} \in \mathbf{R}^{m\times m}, the equation of the objective function of the quadratic program can be written as

n (\bm{A}\bm{x} -\bm{\beta})'\bm{W}(\bm{A}\bm{x}-\bm{\beta}) = n\bm{x}'\bm{A}'\bm{W}\bm{A}\bm{x} - 2n\bm{\beta}'\bm{W}\bm{A}\bm{x} + n\bm{\beta}'\bm{W}\bm{\beta}.

If the \bm{W} matrix is not specified, then it will be taken as an identity matrix of order m.
Linear programs — For all linear problems that are considered in this code, one of \bm{A} and \bm{b} is NULL or is a zero vector. The term that is nonzero and nonnull will be multiplied by n and used as obj1.

Value

Returns the following three quantities: obj2 is the coefficient of the quadratic term, obj1 is the coefficient of the linear term and obj0 is the constant term. More explicitly, their form are given as follows:

`obj2`	The coefficient for the second-order term. It is returned as `NULL` for linear programs and `n\bm{A}'\bm{W}\bm{A}` for quadratic programs.
`obj1`	The coefficient term of the linear term. For quadratic programs, it is returned as `-2n\bm{\beta}'\bm{W}\bm{A}`.
`obj0`	The constant term of the linear program. For quadratic programs, it is returned as `n\bm{\beta}'\bm{W}\bm{\beta}`.