Description Usage Arguments Details Value Author(s) References Examples
View source: R/interior_point.R
This function solves a linear programming problem using a modification of Karmarkar's linear programming algorithm, developed by Vanderbei, Meketon and Freedman (1986). This algorithm uses a recentered projected gradient approach without a priori knowledge of the optimal objective function value.
1 2 |
t |
Type of problem. Put 1 if it is a maximization problem, and -1 if it is a minimization one. |
c |
The vector corresponding to the coefficients of the objective function |
bA |
The vector corresponding to the right hand side constants (with =) |
A |
The coefficients of the problem constraints matrix A (with =) |
bm |
The vector corresponding to the right hand side constants (with <=) |
m |
The coefficients of the problem constraints matrix A (with <=) |
bM |
The vector corresponding to the right hand side constants (with >=) |
M |
The coefficients of the problem constraints matrix A (with >=) |
e |
The value of ε. Default is 1e-04 |
a1 |
The value of α_1. Default is 1 |
a2 |
The value of α_2. Default is 0.97 |
The function is defined in the form that we only need to put the values used in the problem. For example, for a linear programming problem in the form:
Maximize Z=c_{1}x_{1}+c_{2}x_{2}+...+c_{n}x_{n} |
subject to |
a_{11}x_{1}+a_{12}x_{2}+...a_{1n}x_{n}≤q b_{1} |
a_{21}x_{1}+a_{22}x_{2}+...a_{2n}x_{n}≤q b_{2} |
... |
a_{m1}x_{1}+a_{m2}x_{2}+...a_{mn}x_{n}≤q b_{m} |
that is, all the restrictions are inequality in the form “≤q ”, we need only to specify the vector bm and the vector(s) corresponding to the row(s) of A.
a list with the values
Z |
The optimum value of the objective function |
xf |
The solution vector |
n |
The number of iterations to solve the problem |
Alejandro Quintela del Rio aquintela@udc.es
Gill, P.E., Murray, W. and Wright, M.H. (1991) Numerical Linear Algebra and Optimization vol. 1, Addison-Wesley.
Karmarkar, N. (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4, pp. 373-395.
Vanderbei, R.J., Meketon, M.S. and Freedman, B.A. (1986) A modification of Karmarkar's linear programming algorithm. Algorithmica 1, pp. 395-407.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 | ##
c<-c(1,3,-2)
bA<-c(25,30)
A<-array(4,c(2,3))
A[1,2]<-8;A[1,3]<-6;A[2,1]<-7;A[2,2]<-5;A[2,3]<-9
interior_point(-1,c,bA,A)
##
c<-c(1,2)
bm<-c(4)
bM<-c(1)
m<-array(1,c(1,2))
m[1,1]<-0
M<-array(1,c(1,2))
interior_point(1,c,bm=bm,m=m,bM=bM,M=M)
##
# This example is taken from Exercise 7.5 of Gill, Murray,
# and Wright (1991).
enj <- c(200, 6000, 3000, -200)
fat <- c(800, 6000, 1000, 400)
vitx <- c(50, 3, 150, 100)
vity <- c(10, 10, 75, 100)
vitz <- c(150, 35, 75, 5)
interior_point(1,c=enj, m=fat, bm=13800, M=rbind(vitx, vity, vitz),
bM = c(600, 300, 550))
|
$`The optimum value of Z is`
[1] -2.261901
$`The optimal solution X`
[,1]
[1,] 1.144358e-06
[2,] 1.071429e+00
[3,] 2.738094e+00
$`Number of iterations`
[1] 6
[1] "Unbounded problem"
$`The optimum value of Z is`
[1] 41399.91
$`The optimal solution X`
[,1]
[1,] 2.801898e-10
[2,] 3.313643e-11
[3,] 1.379997e+01
[4,] 2.349286e-09
$`Number of iterations`
[1] 51
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