# 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]$$ lb\le x \le ub [REMOVE_ME_2]$$ ```r 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 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: $$ min f(x) $$ $$ s.t. ceq(x)=0 $$ $$ c(x)\le 0 $$ $$ Ax\le B $$ $$ Aeq x \le Beq $$ $$ lb\le x \le ub $$ ## Examples ```r 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

solnl function