performWeightedBisectionOptimization: Performs a weighted bi-section optimization.

In kerschke/mogsa: A Multi-Objective Optimization Algorithm Based on Multi-Objective Gradients

Description

Weighted version of the bisection optimization method. Given two points x1 and x2 on opposite sides of the optimum, this optimizer iteratively splits the interval [x1, x2] into two parts [x1, x.new] and [x.new, x2] and proceeds with the interval, whose boundaries are still located on opposite sides of the optimum. Instead to the classical bisection method, where x.new is the arithmetic mean of x1 and x2, this version uses the lengths of the bi-objective gradients in x1 and x2 to compute a more promising cut-point x.new.

Usage

performWeightedBisectionOptimization(x1, x2, fn1, fn2, g1 = NULL,
  g2 = NULL, prec.grad = 1e-06, prec.norm = 1e-06,
  max.steps = 1000L, lower, upper)

Arguments

x1	[numeric(d)] d-dimensional individual located on one side of the (bi-objective) optimum.
x2	[numeric(d)] d-dimensional individual located on the opposite side (w.r.t. x1) of the (bi-objective) optimum.
fn1	[function] The first objective used for computing the multi-objective gradient.
fn2	[function] The second objective used for computing the multi-objective gradient.
g1	[function] The gradient of the first objective in ind. If missing, it will be approximated using estimateGradientBothDirections.
g2	[function] The gradient of the second objective in ind. If missing, it will be approximated using estimateGradientBothDirections.
prec.grad	[numeric(1L)] Precision value (= step size) used for approximating the gradient. The default is 1e-6.
prec.norm	[numeric(1L)] Precision threshold when normalizing a vector. That is, every element of the vector, whose absolute value is below this threshold, will be replaced by 0. The default is 1e-6.
max.steps	[integer(1L)] Maximum number of allowed bi-section steps to reach an optimum. The default is 1000L.
lower	[numeric(d)] Vector of lower bounds.
upper	[numeric(d)] Vector of upper bounds.

Value

[list(4L)]
List containing a matrix (opt.path) with the individuals along the optimization path, the corresponding number of function evaluations (fn.evals), the single-objective gradients of the last individual (gradient.list) and a flag, indicating whether the optimizer found a local optimum.

Examples

# Define two single-objective test problems:
fn1 = function(x) sum((x - c(2, 0))^2)
fn2 = function(x) sum((x - c(0, 1))^2)

# Visualize locally efficient set, i.e., the "area" where we ideally want to find a point:
plot(c(2, 0), c(0, 1), type = "o", pch = 19,
  xlab = expression(x[1]), ylab = expression(x[2]), las = 1, asp = 1)
text(2, 0, "Optimum of fn1", pos = 2, offset = 1.5)
text(0, 1, "Optimum of fn2", pos = 4, offset = 1.5)

# Place two points x1 and x2 on opposite sides of the bi-objective optimum:
x1 = c(1, 1)
x2 = c(0.5, 0)
points(rbind(x1, x2), pch = 19, type = "o", lty = "dotted")
text(rbind(x1, x2), labels = c("x1", "x2"), pos = 4)

# Optimize using weighted bisection optimization:
opt.path = performWeightedBisectionOptimization(x1 = x1, x2 = x2, fn1 = fn1, fn2 = fn2)$opt.path

# Visualize the optimization path:
points(opt.path)

# Highlight the found local efficient point (= local optimum w.r.t. both objectives):
n = nrow(opt.path)
points(opt.path[n, 1], opt.path[n, 2], pch = 4, col = "red", cex = 2)
text(opt.path[n, 1], opt.path[n, 2], "Found Local Efficient Point", pos = 4, offset = 1.5)

kerschke/mogsa documentation built on July 11, 2019, 11:52 p.m.