Univariate global minimization on a segment

### 16.3.1 Univariate global minimization on a segment

The find_minimum command is used for global minimization of a continuous function on a segment.

- find_minimum takes between three and five arguments:
  - f, a continuous univariate function or expression.
  - a, a real number.
  - b, a real number such that a<b.
  - Optionally, a sequence of one or two numbers, each of which being either:
    - ε∈⟨ 0,1⟩, the optimality tolerance (by default, ε=epsilon()2/3), or
    - n∈ℤ, the maximum number of iterations (n≥ 1 is required) for Brent’s method subroutine (by default, n=100).
- find_minimum(f,a,b ⟨,ε,n ⟩) returns the location of the smallest local minimum found.
- find_minimum applies a modification of Brent’s algorithm which usually finds a global minimum, even when f has several local extrema. The original Brent’s method, which searches for a local minimum, is used as a subroutine which is applied recursively by partitioning the search interval. Each instance is allowed the maximum of n iterations.

#### Examples

We minimize a function which has several local extrema, one of which is global.

f(x):=Airy_Ai(x+sin(x))+cos(x^2):; x0:=find_minimum(f(x),-5,5)

−3.96344565209 |  |  |  |  |  |  |  |  |  |

plot(f(x),x=-5..5); point(x0,f(x0),display=point_width_3+point_point+red)

The objective function in the example below is unimodal, with a single local minimum which is also the global minimum.

f(x):=besselJ(x,2)/Gamma(x+1)+(x+1)^sin(x):; x0:=find_minimum(f(x),0,10)

4.82843934915 |  |  |  |  |  |  |  |  |  |

plot(f(x),x=0..10); point(x0,f(x0),display=point_width_3+point_point+red)

The function f defined below is continuous but not differentiable. It has three local and only one global minimum at x0=−1. Note that we are passing the function itself as the first argument. Also, the parameters ε and n are set to 10−5 and 30, respectively.

f(x):=min(sqrt(abs(x+4))-1,sqrt(abs(x+1))-1005/1000,sqrt(abs(x-3))+1/2):; x0:=find_minimum(f,-5,5,1e-5,30)

−0.999998230068 |  |  |  |  |  |  |  |  |  |

plot(f(x),x=-5..5,xstep=5e-4); point(x0,f(x0),display=point_width_3+point_point+red)