Given electrons, find the equilibrium state distribution (of minimal Coulomb potential) of the electrons positioned on a conducting sphere.

This problem, known as the Thomson problem of finding the lowest energy configuration of point charges on a conducting sphere, originated with Thomson's plum pudding model of the atomic nucleus. This problem is representative of an important class of problems in physics and chemistry that determine a structure with respect to atomic positions.

The potential energy for points
is defined by

and the constraints on the points are

Data for this problem appears in Table 2.1.

This problem has many local minima at which the objective
value is relatively close to the objective value at the global
minimum. Experimental and theoretical results [18,20]
show that

Also, the number of local minima grows exponentially with . Thus, determining the global minimum is computationally difficult, and solvers are usually expected to find only a local minimum.

Variables | |

Constraints | |

Bounds | 0 |

Linear equality constraints | 0 |

Linear inequality constraints | 0 |

Nonlinear equality constraints | |

Nonlinear inequality constraints | 0 |

Nonzeros in | |

Nonzeros in |

Results for the AMPL implementation are summarized in Table 2.2. The starting point is a quasi-uniform distribution of the points on a unit sphere. The best solution for is shown in Figure 2.1.

Solver | ||||

LANCELOT | 3.98 s | 8.08 s | 53.36 s | 371.6 s |

2.43812e+02 | 1.05518e+03 | 4.44841e+03 | 1.84389e+04 | |

violation | 2.34380e-06 | 2.58920e-08 | 2.32410e-07 | 1.94010e-06 |

iterations | 50 | 46 | 67 | 127 |

LOQO | 0.84 s | 7.94 s | 179.06 s | 2437.78 s |

2.43812e+02 | 1.05518e+03 | 4.44835e+03 | 1.84389e+04 | |

violation | 2.8e-09 | 4.9e-09 | 3.0e-09 | 1.9e-09 |

iterations | 27 | 46 | 130 | 264 |

MINOS | 6.22 s | 36.85 s | 794.08 s | |

2.43812e+02 | 1.05518e+03 | 1.25964e+04 | ||

violation | 7.7e-08 | 1.2e-12 | 6.6e+09 | |

iterations | 45 | 59 | 38 | |

SNOPT | 9.65 s | 10.68 s | 73.66 s | 1600.48 s |

2.43812e+02 | 1.05518e+03 | 4.44841e+03 | 1.84390e+04 | |

violation | 2.2e-09 | 1.8e-10 | 5.2e-10 | 9.0e-10 |

iterations | 168 | 102 | 146 | 561 |

Errors or warnings. Timed out. |

MINOS cannot solve the problem for . For it gives the error message
*unbounded (or badly scaled) problem.*

2001-01-02