# shekel.mod	OUR2-AN-4-8
# Original AMPL coding by Elena Bobrovnikova (summer 1996 at Bell Labs).

# Ref.: C. Jansson and O. Knueppel, "A Global Minimization Method:
# the Multi-Dimensional Case", Technische Informatik III,
# TU Hamburg-Hamburg, Jan. 1992, p. 39 (problem "S5").

# Shekel function

# Number of variables:  4
# Number of constraints:  8

# The global minimum is Fsh = -10.1532,
# x = (4.00004, 4.00013, 4.00004, 4.00013).

# There are 5 local minima with f ~ -1/c[i] and x approximately at a[i,*].

# Objective nonseparable nonconvex
# Simple bound constraints

set I := 1 .. 5;
set J := 1 .. 4;
param a {i in I, j in J};
param c {I};
var x{j in J} := j, >= 0, <= 10;

minimize Fsh:
  -sum{i in I} 1 / (sum {j in J} (x[j] - a[i,j])^2 + c[i]);

data;

param    a:
          1   2   3   4  :=
    1     4   4   4   4
    2     1   1   1   1
    3     8   8   8   8
    4     6   6   6   6
    5     3   7   3   7   ;

param c  :=
    1    0.1
    2    0.2
    3    0.2
    4    0.4
    5    0.4  ;

.