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RGA - algorithm for black box global optimization
Protein Folding: Lennard-Jones Clusters
The problem is taken from physical chemistry: find such a structure of a protein, a cluster of N atoms, interacting via the Lennard-Jones (dimensionless non-quantizied pair ) potential : V ( r ) = r- 12 - 2 r- 6 , that its energy E is (globally) minimal. There are fairly smart approaches to solving the problem of LJ- clusters, which exploit the known analytical form of the objective function (OF : potential energy of the microcluster conformation).
Here we would like just to illustrate the capabilities of the RGA-algorithm, i.e. to solve the problem with no use a priori information on the OF by as few OF-calls as possible. Therefore we use the OF as a black box and define the search region as a (3 N - 6)-dimensional box composed with Cartesian coordinates (x, y, z) of N atoms/molecules. The intrinsic dimension (the number of independent variables) of the optimization problem is reduced - when compared with 3N-dimensional internal configurational space - due to the evident symmetry of the objective function: the energy of the LJ-cluster depends just on its geometry and therefore is invariant with respect to translation (3 parameters) and rotation (3 parameters) of the cluster as a rigid body ( see related Hessians ). Because of permutation symmetry (the atoms are assumed identical spherical) the OF has about (N-2) ! global extrema. Besides the nonconvex OF is known to have O (eN'**2) local extrema.
Thus "our" OF is the black box which returns the OF-value provided all the values of ( 3 N - 6 ) "controlling" parameters from intervals [ -2.0 : 2.0 ] are given (except three parameters: the intervals for those are chosen [ 0.0 : 2.0 ] , in our case it is valid for x2, x3 , y3 ) . The analytical form of the OF is utilized by us for analyzing the results exclusively.
Case: N= 8
(18-dimensional optimization problem)
A few metastable states
Fig. 9. The 1st example of a cluster configuration which corresponds to one of local minima: the energy of the cluster is E = -18.8286715. Some of two-bodies forces are shown with lines: red and blue lines correspond to tensional ( r < 1 ) and attractional ( r >1 ) forces respectively; a) and b) are the images of the same structure from two standpoints.
Table A9. Cartesian coordinates of the cluster.
X Y Z 1 -0.79212E+00 0.38599E+00 0.37373E+00 2 0.74053E+00 0.00000E+00 0.00000E+00 3 0.63796E-01 0.73243E+00 0.00000E+00 4 -0.67481E+00 -0.57931E+00 0.61308E+00 5 -0.22312E+00 -0.20538E+00 -0.18497E+00 6 0.78930E-01 0.57610E-01 0.73405E+00 7 0.49073E+00 -0.68082E+00 -0.68007E+00 8 0.31606E+00 0.28948E+00 -0.85582E+00
Table B9. Balnce of acting forces & Binding energies
for the cluster atoms/molecules.
1 3.97341625E-06 -4.23076338 2 4.84256563E-06 -5.19762808 3 8.24184313E-06 -5.20910929 4 1.53265139E-06 -3.30113289 5 5.75156584E-07 -6.98914127 6 8.96049811E-06 -5.19765774 7 4.79571129E-07 -3.30114546 8 8.39391996E-06 -4.23076486 a)b)
Fig. 10. Another example of a cluster (equilibrium) configuration which corresponds to the local minima: E = -18. 8568262 . Just r < 1 are shown on b.
Table A10. Cartesian coordinates of the cluster.
X Y Z 1 -0.47329E+00 -0.46362E+00 0.83369E+00 2 0.87991E+00 0.00000E+00 0.00000E+00 3 -0.47354E+00 0.95384E+00 0.00000E+00 4 0.73348E-01 0.29247E+00 0.49721E+00 5 0.60052E+00 -0.48633E+00 -0.82711E+00 6 - 0.74316E+00 -0.49481E-02 -0.82610E-02 7 0.68073E-01 0.28668E+00 -0.50204E+00 8 0.68137E-01 -0.57809E+00 0.65074E-02
Table B10. Balance & Energies.
1 1.54630091E-07 -3.32797226 2 3.27740205E-07 -4.29217003 3 6.10375403E-07 -3.32797749 4 5.67574885E-07 -6.09744147 5 1.40071978E-07 -3.26687473 6 1.41823356E-07 -5.19765774 7 3.23987277E-07 -3.30114546 8 5.09238628E-07 -4.23076486
The stable configuration
Fig. 11. The deepest equilibrium cluster structure found with the RGA-algorithm: E = -19. 8214892.
It took in total 23 162 calls of OF to find this cluster.
Note that "b" looks similar to that of Fig. 9 b, but is more compact (more red links).
Table A11. Cartesian coordinates of the cluster.
X Y Z 1 -0.14563E+00 -0.12550E+00 0.37908E+00 2 0.76767E+00 0.00000E+00 0.00000E+00 3 0.11327E+00 0.75927E+00 0.00000E+00 4 0.51290E+00 0.44205E+00 0.85575E+00 5 -0.31102E-02 -0.26808E-02 -0.62338E+00 6 -0.70804E+00 -0.61026E+00 -0.27821E+00 7 0.26047E+00 -0.84521E+00 -0.16662E+00 8 -0.79753E+00 0.38233E+00 -0.16663E+00
Table B11. Balance & Energies.
1 3.01558811E-06 -6.97399361 2 7.01741062E-07 -5.21090377 3 5.00437642E-07 -5.21090300 4 1.36031693E-07 -3.31896741 5 2.36973478E-07 -6.08124997 6 6.17511574E-07 -4.24293398 7 6.65146213E-08 -4.30200814 8 4.18525834E-07 -4.30201848 Fig. 12. The energy landscape: 2-D section of the objective function , which involves all of 3 structures displayed above. V1 and V2 represent distances from the global minimum point in the 18-dimensional box along two basic vectors V1 and V2 respectively. Points of discontinuity of the OF are not shown: the OF-values, which are greater than 8.5, are cut off.
More about stability of the structures:Hessians in configurational space.
2D section of the OF (conformation energy)
.to be continued .