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Check out http://www.gromacs.org. 34 */ 35 /*! \internal \file 36 * 37 * \brief Declare function optimization routines. 38 * 39 * \author Christian Blau <blau@kth.se> 40 * \ingroup module_math 41 */ 42 43 #ifndef GMX_MATH_OPTIMIZATION_H 44 #define GMX_MATH_OPTIMIZATION_H 45 46 #include <functional> 47 #include <vector> 48 49 #include "gromacs/utility/arrayref.h" 50 #include "gromacs/utility/real.h" 51 52 namespace gmx 53 { 54 55 /*! \internal 56 * \brief Compiles results of an a function optimisation. 57 */ 58 struct OptimisationResult 59 { 60 //! The coordinates at which the optimal function value has been found 61 std::vector<real> coordinates_; 62 //! The value of the function at the optimal coordinates 63 real functionValue_; 64 }; 65 66 /*! \brief Derivative-free downhill simplex optimisation. 67 * 68 * Find a local minimum of an N-dimensional mapping 69 * \f$\mathbb{R}^N\to\mathbb{R}\f$ using the downhill simplex algorithm by 70 * Nelder and Mead as described in 71 * 72 * Saša Singer and John Nelder (2009), Scholarpedia, 4(7):2928. 73 * doi:10.4249/scholarpedia.2928 74 * 75 * Stops when the oriented simplex length is less than a constant factor times the 76 * initial lengths or when a maximum step size is reached. 77 * 78 * For best performance pre-condition problem magnitudes to 1. 79 * 80 * The following algorithm is implemented in this function 81 * 1 Define the N+1 vertices of the initial simplex 82 * The inital simplex is constructed from the initial guess and N 83 * additional vertices by adding 0.05 to the initial guess (or 0.0025 if 84 * the initial guess is the null vector) from the initial vertex (in line 85 * with usual implementations). 86 * 87 * 1a Sort vertices according to function value with the lowest function value 88 * first in order to minimize the function. 89 * 90 * 2 Calculate the centroid of the simplex as arithmetic mean of all vertices 91 * except the worst, \f$ x_c = \frac1n \sum_{i=1}{N} x_i\f$. 92 * 93 * 3 Reflect the worst simplex vertex (the one with the highest function value) 94 * at the centroid to obtain a reflection point 95 * \f$ x_r = x_c + \alpha (x_c - x_{N+1}) \f$ which lies outside the vertex. 96 * 97 * 3a Replace worst point with reflection point if reflection point function 98 * value is better than second worst point, but not better than best and go 99 * to 1a. 100 * 101 * 4 If the reflection point is better than all other points so far, attempt 102 * an expansion by calculating the expansion point at 103 * \f$ x_e = x_c + \gamma (x_r - x_c) \f$. Swap out the worst point in the 104 * vertex with the expansion point if better than reflection point, otherwise 105 * use the reflection point and go to 1a. 106 * 107 * 5 Attempt contraction, because reflection was not successful; 108 * \f$ x_t = x_c + \rho (x_{N+1} - x_c) \f$. If the contraction point is 109 * better than the worst point, swap out worst point with contracted point 110 * and go to 1a. 111 * 112 * 6 Shrink the vertex. Replace all points except the best one with 113 * \f$ x_i = x_1 + \sigma (x_i - x_1) \f$ and go to 1a. 114 * 115 * \param[in] functionToMinimize function to be minimized 116 * \param[in] initialGuess of coordinates 117 * \param[in] minimumRelativeSimplexLength minimal oriented simplex length with 118 * respect to initial simplex 119 * \param[in] maxSteps to run algorithm for 120 * 121 * \returns the lowest found function value and corresponding coordinates. 122 */ 123 OptimisationResult nelderMead(const std::function<real(ArrayRef<const real>)>& functionToMinimize, 124 ArrayRef<const real> initialGuess, 125 real minimumRelativeSimplexLength = 1e-8, 126 int maxSteps = 10'000); 127 128 } // namespace gmx 129 #endif // GMX_MATH_OPTIMIZATION_H 130