Changes

The size and ''terrain'' of a particular search space are two concepts that one should include when speaking of the difficulty of a search space.  The computing time drastically increases with the size (or dimensionality) of  search space [1].  However, it is still possible to have higher dimensional problems that are unimodal or have very few local minima.  Thus, one must also take into account the shape of search space when contemplating the difficulty.

The size and ''terrain'' of a particular search space are two concepts that one should include when speaking of the difficulty of a search space.  The computing time drastically increases with the size (or dimensionality) of  search space [1].  However, it is still possible to have higher dimensional problems that are unimodal or have very few local minima.  Thus, one must also take into account the shape of search space when contemplating the difficulty.

In ''the diamond problem'', a solution consists of values given to two amplitudes and three phases.  The amplitudes represent the intensity of the light from the reference mirror and the diamond {the front and the back only differ by a constant).  The phases represent each of the three surfaces: the reference mirror and the front and back of the diamond.  Each of these quantities is represented by a certain order Legendre polynomial, which is represented in the form of a matrix with the number of rows and columns to match the order with an offset of one.  Since the order Legendre Polynomial for each of the amplitudes and phases is set by the user, the dimensionality of the problem varies depending on the desired order.  Thus, the lowest dimensionality that ''the diamond problem'' could take on would be 11 and the highest would be (cutting the Legendre polynomial order off at five) 125.  However, it is most likely somewhere in between most likely greater than 50 since,

In ''the diamond problem'', a solution consists of values given to two amplitudes and three phases.  The amplitudes represent the intensity of the light from the reference mirror and the diamond {the front and the back only differ by a constant).  The phases represent each of the three surfaces: the reference mirror and the front and back of the diamond.  Each of these quantities is represented by a certain order Legendre polynomial, which is represented in the form of a matrix with the number of rows and columns to match the order with an offset of one.  Since the order Legendre Polynomial for each of the amplitudes and phases is set by the user, the dimensionality of the problem varies depending on the desired order.  Thus, the lowest dimensionality that ''the diamond problem'' could take on would be 11 and the highest would be (cutting the Legendre polynomial order off at five) 125.  However, it is most likely somewhere in between most likely slightly greater than 50, due to wanting to fully exploit the range of the Legendre polynomials to give the greatest accuracy in mapping the diamond surface.

 

A solution in our search space corresponds to an ordered set of numbers that are determined by the order of the legendre polynomial for the phase and amplitude of the surface of the diamond in the simulated annealing run.  We have three phases and two amplitudes (make a picture of the setup, with the diamond and reference labelled).

[1] Esin Onbasoglu and Ozdamar Yeditepe University, Istanbul, Turkey Journal of Global Optimization 2001

[1] Esin Onbasoglu and Ozdamar Yeditepe University, Istanbul, Turkey Journal of Global Optimization 2001