Nothing
swf <- function(x) {
# Schwefel problem
#
# -500 <= xi <= 500, i = {1, 2, ..., n}
# The number of local minima for a given n is not known, but the global
# minimum f(x*) = -418.9829n is located at x* = (s, s, ..., s), s = 420.97.
#
# Source:
# Ali, M. Montaz, Khompatraporn, Charoenchai, and Zabinsky, Zelda B. (2005).
# A numerical evaluation of several stochastic algorithms on selected
# continuous global optimization test problems.
# Journal of Global Optimization 31, 635-672.
-crossprod( x, sin(sqrt(abs(x))) )
}
sf1 <- function(x) {
# Schaffer 1 problem
#
# -100 <= x1, x2 <= 100
# The number of local minima is not known but the global minimum
# is located at x* = (0, 0) with f(x*) = 0.
#
# Source:
# Ali, M. Montaz, Khompatraporn, Charoenchai, and Zabinsky, Zelda B. (2005).
# A numerical evaluation of several stochastic algorithms on selected
# continuous global optimization test problems.
# Journal of Global Optimization 31, 635-672.
temp <- x[1]^2 + x[2]^2
0.5 + (sin(sqrt(temp))^2 - 0.5)/(1 + 0.001*temp)^2
}
g11 <- list(
obj = function(x) {
# -1 <= xi <= 1 (i = 1, 2)
# The optimal solution is x* = (+-1/sqrt(2), 1/2)
# and the optimal value f(x*) = 0.75.
#
# Source:
# Runarsson, Thomas P., and Yao, Xin (2000).
# Stochastic ranking for constrained evolutionary optimization.
# IEEE Transactions on Evolutionary Computation 4, 284-294.
x[1]^2 + (x[2] - 1)^2
},
eq = 1,
con = function(x) x[2] - x[1]^2
)
RND <- list(
obj = function(x) {
# Reactor network design
#
# 1e-5 <= x5, x6 <= 16
# It possesses two local solutions at x = (16, 0) with f = -0.37461
# and at x = (0, 16) with f = -0.38808.
# The global optimum is (x5, x6; f) = (3.036504, 5.096052; -0.388812).
#
# Source:
# Babu, B. V., and Angira, Rakesh (2006).
# Modified differential evolution (MDE) for optimization of nonlinear
# chemical processes.
# Computers and Chemical Engineering 30, 989-1002.
x5 <- x[1]; x6 <- x[2]
k1 <- 0.09755988; k2 <- 0.99*k1; k3 <- 0.0391908; k4 <- 0.9*k3
-( k2*x6*(1 + k3*x5) +
k1*x5*(1 + k2*x6) ) /
( (1 + k1*x5)*(1 + k2*x6)*
(1 + k3*x5)*(1 + k4*x6) )
},
con = function(x) sqrt(x[1]) + sqrt(x[2]) - 4
)
HEND <- list(
obj = function(x) {
# Heat exchanger network design
#
# 100 <= x1 <= 10000, 1000 <= x2, x3 <= 10000, 10 <= x4, x5 <= 1000
# The global optimum is (x1, x2, x3, x4, x5; f) = (579.19, 1360.13,
# 5109.92, 182.01, 295.60; 7049.25).
#
# Source:
# Babu, B. V., and Angira, Rakesh (2006).
# Modified differential evolution (MDE) for optimization of nonlinear
# chemical processes.
# Computers and Chemical Engineering 30, 989-1002.
x[1] + x[2] + x[3]
},
con = function(x) {
x1 <- x[1]; x2 <- x[2]; x3 <- x[3]; x4 <- x[4]; x5 <- x[5]
c(100*x1 - x1*(400 - x4) + 833.33252*x4 - 83333.333,
x2*x4 - x2*(400 - x5 + x4) - 1250*x4 + 1250*x5,
x3*x5 - x3*(100 + x5) - 2500*x5 + 1250000)
}
)
alkylation <- list(
obj = function(x) {
# Optimal operation of alkylation unit
#
# Variable Lower Bound Upper Bound
# ------------------------------------
# x1 1500 2000
# x2 1 120
# x3 3000 3500
# x4 85 93
# x5 90 95
# x6 3 12
# x7 145 162
# ------------------------------------
# The maximum profit is $1766.36 per day, and the optimal
# variable values are x1 = 1698.256922, x2 = 54.274463, x3 = 3031.357313,
# x4 = 90.190233, x5 = 95.0, x6 = 10.504119, x7 = 153.535355.
#
# Source:
# Babu, B. V., and Angira, Rakesh (2006).
# Modified differential evolution (MDE) for optimization of nonlinear
# chemical processes.
# Computers and Chemical Engineering 30, 989-1002.
x1 <- x[1]; x3 <- x[3]
1.715*x1 + 0.035*x1*x[6] + 4.0565*x3 + 10.0*x[2] - 0.063*x3*x[5]
},
con = function(x) {
x1 <- x[1]; x2 <- x[2]; x3 <- x[3]; x4 <- x[4]
x5 <- x[5]; x6 <- x[6]; x7 <- x[7]
c(0.0059553571*x6^2*x1 + 0.88392857*x3 - 0.1175625*x6*x1 - x1,
1.1088*x1 + 0.1303533*x1*x6 - 0.0066033*x1*x6^2 - x3,
6.66173269*x6^2 + 172.39878*x5 - 56.596669*x4 - 191.20592*x6 - 10000,
1.08702*x6 + 0.32175*x4 - 0.03762*x6^2 - x5 + 56.85075,
0.006198*x7*x4*x3 + 2462.3121*x2 - 25.125634*x2*x4 - x3*x4,
161.18996*x3*x4 + 5000.0*x2*x4 - 489510.0*x2 - x3*x4*x7,
0.33*x7 - x5 + 44.333333,
0.022556*x5 - 0.007595*x7 - 1.0,
0.00061*x3 - 0.0005*x1 - 1.0,
0.819672*x1 - x3 + 0.819672,
24500.0*x2 - 250.0*x2*x4 - x3*x4,
1020.4082*x4*x2 - 1.2244898*x3*x4 - 100000*x2,
6.25*x1*x6 + 6.25*x1 - 7.625*x3 - 100000,
1.22*x3 - x6*x1 - x1 + 1.0)
}
)
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