phase1: Function finding an initial feasible solution (provided it...

In cmottet/GLP: GLP

Description

Finds a feasible solution (provided it exists) to programs such as (24). It uses the generalized linear programming approach described in Algorithm 2. We slightly generalize this algorithm so that inequalities in (24) can be replaced by lower inequalities or equalities if desired.

Usage

phase1(constFun, constRHS, constDir, constLambda = rep(0, length(constRHS)),
  x = 0, C = 10000, IterMax = 100)

Arguments

constFun	List containing the functions G_j
constRHS	Vector containing the bounds gamma_j
constDir	Vector containing the direction of the constraints (e.g. '=', '<=', ">=" )
constLambda	Vector containing the the values of the parameters λ_{j,M} (use default vector of 0's to solve classic generalized linear programs)
x	Non-negative scalar representing the first point support to enter the procedure (default 0)
C	Upper bound of the support of feasible distribution functions (default 1e4)
IterMax	Maximum number of iterations for the procedure (default = 100)

Value

A list containing

p	Vector of point masses
x	Vector of point supports
s	Scalar
r	Optimal value of the variable r when the program is feasible and an optimal solution exists
feasible	Boolean indicating wether (p, x, s) is a feasible solution to (24)
status	Integer describing the final status of the procedure (0 => Reached a solution, 1 => the algorithm terminated by reaching the maximum number of iterations, 2 => the algorithm entered a cycle)
nIter	Number of iterations reached when the procedure terminated
eps	Opposite value of the inner optimization program when the procedure terminated

Examples

####
#### Finding a basic feasible solution (p, x, s) such that
####
#### sum(p)  = 1
#### sum(px) = 1
#### sum(px^2) = 2 + s
####
####

# Function for the integrals of the constraints inequality
constFun = rep(list(
function(x) 1,
function(x) x,
function(x) x^2 + s
),2)

# Direction of the inequality constraints
constDir <- rep(c("<=", ">="), each = 3)

# Values on the RHS of each inequality
mu0 <- 1
mu1 <- 1
mu2 <- 2

constRHS <- rep(c(mu0,mu1,mu2), 2)

# Lambdas for the objective function and the constraints functions
constLambda <- rep(c(0,0,1),2)

# Get a basic feasible solution
initBFS  <-  phase1(constFun, constRHS,constDir, constLambda)

# Check feasibility
with(initBFS, c(sum(p), sum(p*x), sum(p*x^2) + s))

cmottet/GLP documentation built on May 6, 2019, 12:05 a.m.