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 } 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) })