solnl() R function from [NlcOptim]

Solve Optimization problem with Nonlinear Objective and Constraints

Sequential Quatratic Programming (SQP) method is implemented to find solution for general nonlinear optimization problem (with nonlinear objective and constraint functions). The SQP method can be find in detail in Chapter 18 of Jorge Nocedal and Stephen J. Wright's book. Linear or nonlinear equality and inequality constraints are allowed. It accepts the input parameters as a constrained matrix. The function solnl is to solve generalized nonlinear optimization problem: [REMOVE_ME] $min f(x) [REMOVE_ME_2]$

[REMOVE_ME] $s.t. ceq(x)=0 [REMOVE_ME_2]$

[REMOVE_ME] $c(x)\le 0 [REMOVE_ME_2]$

[REMOVE_ME] $Ax\le B [REMOVE_ME_2]$

[REMOVE_ME] $Aeq x \le Beq [REMOVE_ME_2]$

[REMOVE_ME] $lb\le x \le ub [REMOVE_ME_2]$


solnl(X = NULL, objfun = NULL, confun = NULL, A = NULL, B = NULL,
  Aeq = NULL, Beq = NULL, lb = NULL, ub = NULL, tolX = 1e-05,
  tolFun = 1e-06, tolCon = 1e-06, maxnFun = 1e+07, maxIter = 4000)

Arguments

X: Starting vector of parameter values.
objfun: Nonlinear objective function that is to be optimized.
confun: Nonlinear constraint function. Return a ceq vector and a c vector as nonlinear equality constraints and an inequality constraints.
A: A in the linear inequality constraints.
B: B in the linear inequality constraints.
Aeq: Aeq in the linear equality constraints.
Beq: Beq in the linear equality constraints.
lb: Lower bounds of parameters.
ub: Upper bounds of parameters.
tolX: The tolerance in X.
tolFun: The tolerance in the objective function.
tolCon: The tolenrance in the constraint function.
maxnFun: Maximum updates in the objective function.
maxIter: Maximum iteration.

Returns

Return a list with the following components: - par: The optimum solution.

fn: The value of the objective function at the optimal point.
counts: Number of function evaluations, and number of gradient evaluations.
lambda: Lagrangian multiplier.
grad: The gradient of the objective function at the optimal point.
hessian: Hessian of the objective function at the optimal point.

Description

min f(x)

s.t. ceq(x)=0

c(x)\le 0

Ax\le B

Aeq x \le Beq

lb\le x \le ub

Examples


library(MASS)
###ex1
objfun=function(x){
 return(exp(x[1]*x[2]*x[3]*x[4]*x[5]))  
}
#constraint function
confun=function(x){
 f=NULL
 f=rbind(f,x[1]^2+x[2]^2+x[3]^2+x[4]^2+x[5]^2-10)
 f=rbind(f,x[2]*x[3]-5*x[4]*x[5])
 f=rbind(f,x[1]^3+x[2]^3+1)
 return(list(ceq=f,c=NULL))
}

x0=c(-2,2,2,-1,-1)
solnl(x0,objfun=objfun,confun=confun)

####ex2
obj=function(x){
 return((x[1]-1)^2+(x[1]-x[2])^2+(x[2]-x[3])^3+(x[3]-x[4])^4+(x[4]-x[5])^4)  
}
#constraint function
con=function(x){
 f=NULL
 f=rbind(f,x[1]+x[2]^2+x[3]^3-2-3*sqrt(2))
 f=rbind(f,x[2]-x[3]^2+x[4]+2-2*sqrt(2))
 f=rbind(f,x[1]*x[5]-2)
 return(list(ceq=f,c=NULL))
}

x0=c(1,1,1,1,1)
solnl(x0,objfun=obj,confun=con)

##########ex3
obj=function(x){
 return((1-x[1])^2+(x[2]-x[1]^2)^2)  
}
#constraint function
con=function(x){
 f=NULL
 f=rbind(f,x[1]^2+x[2]^2-1.5)
 return(list(ceq=NULL,c=f))
}

x0=as.matrix(c(-1.9,2))
obj(x0)
con(x0)
solnl(x0,objfun=obj,confun=con)

##########ex4
objfun=function(x){
 return(x[1]^2+x[2]^2)  
}
#constraint function
confun=function(x){
 f=NULL
 f=rbind(f,-x[1] - x[2] + 1)
 f=rbind(f,-x[1]^2 - x[2]^2 + 1)
 f=rbind(f,-9*x[1]^2 - x[2]^2 + 9)
 f=rbind(f,-x[1]^2 + x[2])
 f=rbind(f,-x[2]^2 + x[1])
 return(list(ceq=NULL,c=f))
}

x0=as.matrix(c(3,1))
solnl(x0,objfun=objfun,confun=confun)

##############ex5
rosbkext.f <- function(x){
   n <- length(x)
   sum (100*(x[1:(n-1)]^2 - x[2:n])^2 + (x[1:(n-1)] - 1)^2)
}
n <- 2
set.seed(54321)
p0 <- rnorm(n)
Aeq <- matrix(rep(1, n), nrow=1)
Beq <- 1
lb <- c(rep(-Inf, n-1), 0)
solnl(X=p0,objfun=rosbkext.f, lb=lb, Aeq=Aeq, Beq=Beq)
ub <- rep(1, n)
solnl(X=p0,objfun=rosbkext.f, lb=lb, ub=ub, Aeq=Aeq, Beq=Beq)

##############ex6
nh <- vector("numeric", length = 5)

Nh <- c(6221,11738,4333,22809,5467)
ch <- c(120, 80, 80, 90, 150)

mh.rev <- c(85, 11, 23, 17, 126)
Sh.rev <- c(170.0, 8.8, 23.0, 25.5, 315.0)

mh.emp <- c(511, 21, 70, 32, 157)
Sh.emp <- c(255.50, 5.25, 35.00, 32.00, 471.00)

ph.rsch <- c(0.8, 0.2, 0.5, 0.3, 0.9)

ph.offsh <- c(0.06, 0.03, 0.03, 0.21, 0.77)

budget = 300000
n.min <- 100
relvar.rev <- function(nh){
 rv <- sum(Nh * (Nh/nh - 1)*Sh.rev^2)
 tot <- sum(Nh * mh.rev)
 rv/tot^2
}

relvar.emp <- function(nh){
 rv <- sum(Nh * (Nh/nh - 1)*Sh.emp^2)
 tot <- sum(Nh * mh.emp)
 rv/tot^2
}

relvar.rsch <- function(nh){
 rv <- sum( Nh * (Nh/nh - 1)*ph.rsch*(1-ph.rsch)*Nh/(Nh-1) )
 tot <- sum(Nh * ph.rsch)
 rv/tot^2
}

relvar.offsh <- function(nh){
 rv <- sum( Nh * (Nh/nh - 1)*ph.offsh*(1-ph.offsh)*Nh/(Nh-1) )
 tot <- sum(Nh * ph.offsh)
 rv/tot^2
}

nlc.constraints <- function(nh){
 h <- rep(NA, 13)
 h[1:length(nh)] <- (Nh + 0.01) - nh
 h[(length(nh)+1) : (2*length(nh)) ] <- (nh + 0.01) - n.min
 h[2*length(nh) + 1] <- 0.05^2 - relvar.emp(nh)
 h[2*length(nh) + 2] <- 0.03^2 - relvar.rsch(nh)
 h[2*length(nh) + 3] <- 0.03^2 - relvar.offsh(nh)
 return(list(ceq=NULL, c=-h))
}

nlc <- function(nh){
 h <- rep(NA, 3)
 h[ 1] <- 0.05^2 - relvar.emp(nh)
 h[ 2] <- 0.03^2 - relvar.rsch(nh)
 h[3] <- 0.03^2 - relvar.offsh(nh)
 return(list(ceq=NULL, c=-h))
}

Aeq <- matrix(ch/budget, nrow=1)
Beq <- 1

A=rbind(diag(-1,5,5),diag(1,5,5))
B=c(-Nh-0.01,rep(n.min-0.01,5))

solnl(X=rep(100,5),objfun=relvar.rev,confun=nlc.constraints, Aeq=Aeq, Beq=Beq)

solnl(X=rep(100,5),objfun=relvar.rev,confun=nlc, Aeq=Aeq, Beq=Beq, A=-A, B=-B)

References

Nocedal, Jorge, and Stephen Wright. Numerical optimization. Springer Science & Business Media, 2006.

Author(s)

Xianyan Chen, Xiangrong Yin

NlcOptim package Read PDF manual

Maintainer: Xianyan Chen
License: GPL-3
Last published: 2019-01-18

Useful links

solnl function