MEIGOR (version 1.6.0)

essR: Global optimization algorithm for MINLPs based on Scatter Search

Description

essR attempts to solve problems of the form:

$min f(x,p_1,p_2,...,p_n)$

subject to:

$c_e=0$ $c_L \le c(x) \le c_U$ $x_L \le x \le x_U$

Usage

essR(problem, opts = list(maxeval = NULL, maxtime = NULL), ...)

Arguments

problem

List containing problem definition.

opts

List containing options (if set as opts <- numeric(0) default options will be loaded).

...

Additional variables passed to the objective function

Value

fbest

Best objective function value found after the optimization

xbest

Vector providing the best function value

cpu_time

Time in seconds consumed in the optimization

f

Vector containing the best objective function value after each iteration

x

Matrix containing the best vector after each iteration

time

Vector containing the cpu time consumed after each iteration

neval

Vector containing the number of function evaluations after each iteration

numeval

Number of function evaluations

local_solutions

Local solutions found by the local solver (in rows)

local_solutions_values

Function values of the local solutions

end_crit

Criterion to finish the optimization:

1

Maximal number of function evaluations achieved

2

Maximum allowed CPU Time achieved

3

Value to reach achieved

Details

Problem definition:

problem$f: Name of the file containing the objective function (String). problem$x_L: Lower bounds of decision variables (vector). problem$x_U: Upper bounds of decision variables (vector). problem$x_0: Initial point(s) (optional; vector or matrix). problem$f_0: Function values of initial point(s) (optional). These values MUST correspond to feasible points.

NOTE: The dimension of f_0 and x_0 may be different. For example, if we want to introduce 5 initial points but we only know the values for 3 of them, x_0 would have 5 rows whereas f_0 would have only 3 elements. In this example, it is mandatory that the first 3 rows of x_0 correspond to the values of f_0.

Fill the following fields if your problem has non-linear constraints:

problem$neq: Number of equality constraints (Integer; do not define it if there are no equality constraints). problem$c_L: Lower bounds of nonlinear inequality constraints (vector). problem$c_U: Upper bounds of nonlinear inequality constraints (vector). problem$int_var: Number of integer variables (Integer). problem$bin_var: Number of binary variables (Integer). problem$vtr: Objective function value to be reached (optional).

User options:

opts$maxeval: Maximum number of function evaluations (Default 1000). opts$maxtime: Maximum CPU time in seconds (Default 60). opts$iterprint: Print each iteration on screen: 0-Deactivated; 1-Activated (Default 1). opts$plot: Plots convergence curves: 0-Deactivated; 1-Plot curves on line; 2-Plot final results (Default 0). opts$weight: Weight that multiplies the penalty term added to the objective function in constrained problems (Default 1e6). opts$log_var: Indexes of the variables which will be used to generate diverse solutions in different orders of magnitude (vector). opts$tolc: Maximum absolute violation of the constraints (Default 1e-5). opts$prob_bound: Probability (0-1) of biasing the search towards the bounds (Default 0.5). opts$inter_save: Saves results in a mat file in intermediate iterations. Useful for very long runs (Binary; Default = 0). Global options:

opts$dim_refset: Number of elements in Refset (Integer; automatically calculated). opts$ndiverse: Number of solutions generated by the diversificator (Default 10*nvar). opts$combination: Type of combination of Refset elements (Default 1);1: hyper-rectangles; 2: linear combinations.

Local options:

opts$local_solver: Choose local solver 0: Local search deactivated (Default), "NM", "BFGS", "CG", "LBFGSB","SA","SOLNP". opts$local_tol: Level of tolerance in local search. opts$local_iterprint: Print each iteration of local solver on screen (Binary; default = 0). opts$local_n1: Number of iterations before applying local search for the 1st time (Default 1). opts$local_n2: Minimum number of iterations in the global phase between 2 local calls (Default 10). opts$local_balance: Balances between quality (=0) and diversity (=1) for choosing initial points for the local search (default 0.5). opts$local_finish: Applies local search to the best solution found once the optimization if finished (same values as opts.local.solver). opts$local_bestx: When activated (i.e. =1) only applies local search to the best solution found to date,ignoring filters (Default=0).

References

If you use essR and publish the results, please cite the following papers:

Egea, J.A., Maria, R., Banga, J.R. (2010) An evolutionary method for complex-process optimization. Computers & Operations Research 37(2): 315-324. Egea, J.A., Balsa-Canto, E., Garcia, M.S.G., Banga, J.R. (2009) Dynamic optimization of nonlinear processes with an enhanced scatter search method. Industrial & Engineering Chemistry Research 49(9): 4388-4401.

Examples

#1 Unconstrained problem

ex1 <- function(x){
	y<-4*x[1]*x[1]-2.1*x[1]^4+1/3*x[1]^6+x[1]*x[2]-4*x[2]*x[2]+4*x[2]^4;
	return(y)
}

#global optimum
#x*=[0.0898, -0.7127];
# or    
#x*=[-0.0898, 0.7127];
#
#f(x*)= -1.03163;

#========================= PROBLEM SPECIFICATIONS ===========================
problem<-list(f="ex1",x_L=rep(-1,2),x_U=rep(1,2))
opts<-list(maxeval=500, ndiverse=10, dim_refset=4, local_solver="solnp", local_n2=1)
#========================= END OF PROBLEM SPECIFICATIONS =====================

Results<-essR(problem,opts);




#2 Constrained problem

ex2<-function(x){
	F=-x[1]-x[2];
	g<-rep(0,2);
	g[1]<-x[2]-2*x[1]^4+8*x[1]^3-8*x[1]^2;
	g[2]<-x[2]-4*x[1]^4+32*x[1]^3-88*x[1]^2+96*x[1];
	return(list(F=F,g=g))
}

# global optimum
#x*=[2.32952, 3.17849];    
#f(x*)=-5.50801

#========================= PROBLEM SPECIFICATIONS ===========================
problem<-list(f="ex2",x_L=rep(0,2),x_U=c(3,4), c_L=rep(-Inf,2), c_U=c(2,36))
opts<-list(maxeval=750, local_solver="solnp", local_n2=1)
#========================= END OF PROBLEM SPECIFICATIONS =====================

Results<-essR(problem,opts);




#3 Constrained problem with equality constraints

ex3<-function(x,k1,k2,k3,k4){
	f=-x[4];
#Equality constraints
	g<-rep(0,5);
	g[1]=x[4]-x[3]+x[2]-x[1]+k4*x[4]*x[6];
	g[2]=x[1]-1+k1*x[1]*x[5];
	g[3]=x[2]-x[1]+k2*x[2]*x[6];
	g[4]=x[3]+x[1]-1+k3*x[3]*x[5];

#Inequality constraint
	g[5]=x[5]^0.5+x[6]^0.5;
	return(list(f=f,g=g));
}

#global optimum
#x*=[0.77152
# 	0.516994
# 	0.204189
# 	0.388811
# 	3.0355
# 	5.0973];
# 
#f(x*)= -0.388811;

#========================= PROBLEM SPECIFICATIONS ===========================
problem<-list(f="ex3",x_L=rep(0,6),x_U=c(rep(1,4),16,16), neq=4, c_L=-Inf, c_U=4)
opts<-list(maxtime=7, local_solver="solnp", local_n2=10)
#========================= END OF PROBLEM SPECIFICATIONS =====================

k1=0.09755988;
k3=0.0391908;
k2=0.99*k1;
k4=0.9*k3;
Results<-essR(problem,opts,k1,k2,k3,k4);




#4 Mixed integer problem

ex4<-function(x){
	F = x[2]^2 + x[3]^2 + 2.0*x[1]^2 + x[4]^2 - 5.0*x[2] - 5.0*x[3] - 21.0*x[1] + 7.0*x[4];
	g<-rep(0,3);
	g[1] = x[2]^2 + x[3]^2 + x[1]^2 + x[4]^2 + x[2] - x[3] + x[1] - x[4];
	g[2] = x[2]^2 + 2.0*x[3]^2 + x[1]^2 + 2.0*x[4]^2 - x[2] - x[4];
	g[3] = 2.0*x[2]^2 + x[3]^2 + x[1]^2 + 2.0*x[2] - x[3] - x[4];
	return(list(F=F, g=g));
}

# global optimum
#x*=[2.23607, 0, 1, 0];
#f(x*)=-40.9575;

#========================= PROBLEM SPECIFICATIONS ===========================
problem<-list(f="ex4", x_L=rep(0,4), x_U=rep(10,4), x_0=c(3,4,5,1),int_var=3, c_L=rep(-Inf,3), c_U=c(8,10,5))
opts<-list(maxtime=2)
#========================= END OF PROBLEM SPECIFICATIONS =====================

Results<-essR(problem,opts);