├── LICENSE ├── README.md ├── example.py └── python_lbfgsb ├── __init__.py ├── __pycache__ ├── __init__.cpython-36.pyc └── lbfgsb.cpython-36.pyc └── lbfgsb.py /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. By contrast, 15 | the GNU General Public License is intended to guarantee your freedom to 16 | share and change all versions of a program--to make sure it remains free 17 | software for all its users. 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But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # python_lbfgsb 2 | Pure Python-based L-BFGS-B implementation 3 | -------------------------------------------------------------------------------- /example.py: -------------------------------------------------------------------------------- 1 | #!/usr/bin/python3 2 | 3 | #python lbfgsb works with numpy arrays 4 | import numpy as np 5 | import python_lbfgsb as lbfgsb 6 | 7 | #First example: min x^Tx such that x>=1. Optimal solution hits the boundary. 8 | def quad(x): 9 | return x.dot(x) 10 | 11 | def grad_quad(x): 12 | return x 13 | 14 | l = np.array([1,1]) 15 | u = np.array([np.inf, np.inf]) 16 | x0 = np.array([5, 5]) 17 | 18 | print("====================== Quadratic example ======================") 19 | opt_quad = lbfgsb.L_BFGS_B(x0, quad, grad_quad, l, u) 20 | print("") 21 | print("") 22 | xOpt = np.array([1, 1]) 23 | theoOpt = {'x': xOpt, 'f': quad(xOpt), 'df': grad_quad(xOpt)} 24 | print("Theoretical optimal value: ") 25 | print(theoOpt) 26 | print("Optimal value found: ") 27 | print(opt_quad) 28 | 29 | 30 | print("") 31 | print("") 32 | print("") 33 | print("") 34 | #Second example : min Rosenbrock function 35 | print("====================== Rosenbrock example ======================") 36 | def rosenbrock(x): 37 | return 100*(x[1]-x[0]**2)**2+(1-x[0])**2 38 | 39 | def grad_rosenbrock(x): 40 | g = np.empty(x.size) 41 | g[0] = 400*x[0]*(x[0]**2-x[1]) + 2*(x[0]-1) 42 | g[1]= 200*(-x[0]**2 + x[1]) 43 | return g 44 | 45 | l = np.array([-2,-2]) 46 | u = np.array([2,2]) 47 | #x0 = np.array([0.12, 0.12]) 48 | x0 = np.array([-1, -1]) 49 | 50 | opt_rosenbrock = lbfgsb.L_BFGS_B(x0, rosenbrock, grad_rosenbrock, l, u) 51 | print("") 52 | print("") 53 | theoOpt = {'x': np.array([1, 1]), 'f': 0, 'df': grad_rosenbrock(np.array([1,1]))} 54 | print("Theoretical optimal value: ") 55 | print(theoOpt) 56 | print("Optimal value found: ") 57 | print(opt_rosenbrock) 58 | 59 | 60 | print("") 61 | print("") 62 | print("") 63 | print("") 64 | #Third example : min Beale function 65 | print("====================== Beale example ======================") 66 | def beale(x): 67 | return (1.5-x[0]+x[0]*x[1])**2+(2.25-x[0]+x[0]*x[1]**2)**2+(2.625-x[0]+x[0]*x[1]**3)**2 68 | 69 | def grad_beale(x): 70 | y1 = x[1] 71 | y2 = y1*y1 72 | y3 = y2*y1 73 | f1 = 1.5 - x[0] + x[0] * y1 74 | f2 = 2.25 - x[0] + x[0] * y2 75 | f3 = 2.625 - x[0] + x[0] * y3 76 | 77 | return np.array([2*(y1 - 1)*f1 + 2*(y2 - 1)*f2 + 2*(y3 - 1)*f3,\ 78 | 2*x[0]*f1+ 4*x[0]*y1*f2 + 6*x[0]*y2*f3]) 79 | 80 | l = -4.5*np.ones(2) 81 | u = -l 82 | x0 = np.array([2.5,-1.3]) 83 | 84 | opt_beale = lbfgsb.L_BFGS_B(x0, beale, grad_beale, l, u, epsf = 1e1, epsg = 1e-10) 85 | print("") 86 | print("") 87 | theoOpt = {'x': np.array([3, 0.5]), 'f': beale(np.array([3, 0.5])), 'df': grad_beale(np.array([3,0.5]))} 88 | print("Theoretical optimal value: ") 89 | print(theoOpt) 90 | print("Optimal value found: ") 91 | print(opt_beale) 92 | -------------------------------------------------------------------------------- /python_lbfgsb/__init__.py: -------------------------------------------------------------------------------- 1 | from python_lbfgsb.lbfgsb import compute_Cauchy_point, minimize_model,\ 2 | max_allowed_steplength, line_search,\ 3 | update_SY, L_BFGS_B -------------------------------------------------------------------------------- /python_lbfgsb/__pycache__/__init__.cpython-36.pyc: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/avieira/python_lbfgsb/21231a01633a5e70c5d20cb59ed0ec50a49d17c6/python_lbfgsb/__pycache__/__init__.cpython-36.pyc -------------------------------------------------------------------------------- /python_lbfgsb/__pycache__/lbfgsb.cpython-36.pyc: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/avieira/python_lbfgsb/21231a01633a5e70c5d20cb59ed0ec50a49d17c6/python_lbfgsb/__pycache__/lbfgsb.cpython-36.pyc -------------------------------------------------------------------------------- /python_lbfgsb/lbfgsb.py: -------------------------------------------------------------------------------- 1 | import numpy as np 2 | from scipy.optimize import minpack2 3 | from collections import deque 4 | 5 | def compute_Cauchy_point(x, g, l, u, W, M, theta): 6 | """ 7 | Computes the generalized Cauchy point (GCP), defined as the first 8 | local minimizer of the quadratic 9 | 10 | .. math:: 11 | :nowrap: 12 | \[\langle g,s\rangle + \frac{1}{2} \langle s, (\theta I + WMW^\intercal)s\rangle\] 13 | 14 | along the projected gradient direction 15 | .. math:: $P_[l,u](x-\theta g).$ 16 | 17 | 18 | :param x: starting point for the GCP computation 19 | :type x: np.array 20 | 21 | :param g: gradient of f(x). g must be a nonzero vector 22 | :type g: np.array 23 | 24 | :param l: the lower bound of x 25 | :type l: np.array 26 | 27 | :param u: the upper bound of x 28 | :type u: np.array 29 | 30 | :param W: part of limited memory BFGS Hessian approximation 31 | :type W: np.array 32 | 33 | :param M: part of limited memory BFGS Hessian approximation 34 | :type M: np.array 35 | 36 | :param theta: part of limited memory BFGS Hessian approximation 37 | :type theta: float 38 | 39 | :return: dict containing a computed value of: 40 | - 'xc' the GCP 41 | - 'c' = W^(T)(xc-x), used for the subspace minimization 42 | - 'F' set of free variables 43 | :rtype: dict 44 | ..todo check F 45 | 46 | .. seealso:: 47 | 48 | [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited 49 | memory algorithm for bound constrained optimization'', 50 | SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. 51 | 52 | [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN 53 | Subroutines for Large Scale Bound Constrained Optimization'' 54 | Tech. Report, NAM-11, EECS Department, Northwestern University, 55 | 1994. 56 | """ 57 | eps_f_sec = 1e-30 58 | t = np.empty(x.size) 59 | d = np.empty(x.size) 60 | x_cp = x.copy() 61 | for i in range(x.size): 62 | if g[i]<0: 63 | t[i] = (x[i]-u[i])/g[i] 64 | elif g[i]>0: 65 | t[i] = (x[i]-l[i])/g[i] 66 | else: 67 | t[i]=np.inf 68 | if t[i]==0: 69 | d[i]=0 70 | else: 71 | d[i]=-g[i] 72 | 73 | F = np.argsort(t) 74 | F = [i for i in F if t[i] >0] 75 | t_old = 0 76 | F_i = 0 77 | b=F[0] 78 | t_min = t[b] 79 | Dt = t_min 80 | 81 | p = np.transpose(W).dot(d) 82 | c = np.zeros(p.size) 83 | f_prime = -d.dot(d) 84 | f_second = -theta*f_prime-p.dot(M.dot(p)) 85 | f_sec0 = f_second 86 | Dt_min = -f_prime/f_second 87 | 88 | while Dt_min>=Dt and F_i0: 90 | x_cp[b] = u[b] 91 | elif d[b]<0: 92 | x_cp[b] = l[b] 93 | x_bcp = x_cp[b] 94 | 95 | zb = x_bcp - x[b] 96 | c += Dt*p 97 | W_b = W[b,:] 98 | g_b = g[b] 99 | 100 | f_prime += Dt*f_second+ g_b*(g_b+theta*zb-W_b.dot(M.dot(c))) 101 | f_second -= g_b*(g_b*theta+W_b.dot(M.dot(2*p+g_b*W_b))) 102 | f_second = min(f_second, eps_f_sec*f_sec0) 103 | 104 | Dt_min = -f_prime/f_second 105 | 106 | p += g_b*W_b 107 | d[b] = 0 108 | t_old = t_min 109 | F_i+=1 110 | 111 | if F_i=t_min: 123 | x_cp[i] = x[i] + t_old*d[i] 124 | 125 | F = [i for i in F if t[i]!=t_min] 126 | 127 | c += Dt_min*p 128 | return {'xc':x_cp, 'c':c, 'F':F} 129 | 130 | def minimize_model(x, xc, c, g, l, u, W, M, theta): 131 | """ 132 | Computes an approximate solution of the subspace problem 133 | .. math:: 134 | :nowrap: 135 | 136 | \[\begin{aligned} 137 | \min& &\langle r, (x-xcp)\rangle + 1/2 \langle x-xcp, B (x-xcp)\rangle\\ 138 | \text{s.t.}& &l<=x<=u\\ 139 | & & x_i=xcp_i \text{for all} i \in A(xcp) 140 | \] 141 | 142 | along the subspace unconstrained Newton direction 143 | .. math:: $d = -(Z'BZ)^(-1) r.$ 144 | 145 | 146 | :param x: starting point for the GCP computation 147 | :type x: np.array 148 | 149 | :param xc: Cauchy point 150 | :type xc: np.array 151 | 152 | :param c: W^T(xc-x), computed with the Cauchy point 153 | :type c: np.array 154 | 155 | :param g: gradient of f(x). g must be a nonzero vector 156 | :type g: np.array 157 | 158 | :param l: the lower bound of x 159 | :type l: np.array 160 | 161 | :param u: the upper bound of x u 162 | :type u: np.array 163 | 164 | :param W: part of limited memory BFGS Hessian approximation 165 | :type W: np.array 166 | 167 | :param M: part of limited memory BFGS Hessian approximation 168 | :type M: np.array 169 | 170 | :param theta: part of limited memory BFGS Hessian approximation 171 | :type theta: float 172 | 173 | :return: dict containing a computed value of: 174 | - 'xbar' the minimizer 175 | :rtype: dict 176 | 177 | ..todo Normaly, free_vars is already defined in compute_Cauchy_point in F. 178 | 179 | 180 | .. seealso:: 181 | 182 | [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited 183 | memory algorithm for bound constrained optimization'', 184 | SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. 185 | 186 | [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN 187 | Subroutines for Large Scale Bound Constrained Optimization'' 188 | Tech. Report, NAM-11, EECS Department, Northwestern University, 189 | 1994. 190 | """ 191 | invThet = 1.0/theta 192 | 193 | Z = list() 194 | free_vars = list() 195 | n = xc.size 196 | unit = np.zeros(n) 197 | for i in range(n): 198 | unit[i] = 1 199 | if ((xc[i] != u[i]) and (xc[i] != l[i])): 200 | free_vars.append(i) 201 | Z.append(unit.copy()) 202 | unit[i]=0 203 | 204 | if len(free_vars) == 0: 205 | return {'xbar':xc} 206 | 207 | Z = np.asarray(Z).T 208 | WTZ = W.T.dot(Z) 209 | 210 | rHat = [(g + theta*(xc-x) - W.dot(M.dot(c)))[ind] for ind in free_vars] 211 | v = WTZ.dot(rHat) 212 | v = M.dot(v) 213 | 214 | N = invThet*WTZ.dot(np.transpose(WTZ)) 215 | N = np.eye(N.shape[0])-M.dot(N) 216 | v = np.linalg.solve(N, v) 217 | 218 | dHat = -invThet * (rHat + invThet * np.transpose(WTZ).dot(v)) 219 | 220 | #Find alpha 221 | alpha_star = 1 222 | for i in range(len(free_vars)): 223 | idx = free_vars[i] 224 | if dHat[i] > 0: 225 | alpha_star = min(alpha_star, (u[idx]-xc[idx])/dHat[i]) 226 | elif dHat[i] < 0: 227 | alpha_star = min(alpha_star, (l[idx]-xc[idx])/dHat[i]) 228 | 229 | d_star = alpha_star*dHat; 230 | xbar = xc; 231 | for i in range(len(free_vars)): 232 | idx = free_vars[i]; 233 | xbar[idx] += d_star[i]; 234 | 235 | return {'xbar':xbar} 236 | 237 | 238 | def max_allowed_steplength(x, d, l, u, max_steplength): 239 | """ 240 | Computes the biggest 0<=k<=max_steplength such that: 241 | l<= x+kd <= u 242 | 243 | :param x: starting point 244 | :type x: np.array 245 | 246 | :param d: direction 247 | :type d: np.array 248 | 249 | :param l: the lower bound of x 250 | :type l: np.array 251 | 252 | :param u: the upper bound of x 253 | :type u: np.array 254 | 255 | :param max_steplength: maximum steplength allowed 256 | :type max_steplength: float 257 | 258 | :return: maximum steplength allowed 259 | :rtype: float 260 | 261 | 262 | .. seealso:: 263 | 264 | [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited 265 | memory algorithm for bound constrained optimization'', 266 | SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. 267 | 268 | [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN 269 | Subroutines for Large Scale Bound Constrained Optimization'' 270 | Tech. Report, NAM-11, EECS Department, Northwestern University, 271 | 1994. 272 | """ 273 | max_stpl = max_steplength 274 | for i in range(x.size): 275 | if d[i] > 0: 276 | max_stpl = min(max_stpl, (u[i] - x[i]) / d[i]) 277 | elif d[i] < 0 : 278 | max_stpl = min(max_stpl, (l[i] - x[i]) / d[i]) 279 | 280 | return max_stpl 281 | 282 | def line_search(x0, f0, g0, d, above_iter, max_steplength,\ 283 | fct_f, fct_grad,\ 284 | alpha = 1e-4, beta = 0.9,\ 285 | xtol_minpack = 1e-5, max_iter = 30): 286 | """ 287 | Finds a step that satisfies a sufficient decrease condition and a curvature condition. 288 | 289 | The algorithm is designed to find a step that satisfies the sufficient decrease condition 290 | 291 | f(x0+stp*d) <= f(x0) + alpha*stp*\langle f'(x0),d\rangle, 292 | 293 | and the curvature condition 294 | 295 | abs(f'(x0+stp*d)) <= beta*abs(\langle f'(x0),d\rangle). 296 | 297 | If alpha is less than beta and if, for example, the functionis bounded below, then 298 | there is always a step which satisfies both conditions. 299 | 300 | :param x0: starting point 301 | :type x0: np.array 302 | 303 | :param f0: f(x0) 304 | :type f0: float 305 | 306 | :param g0: f'(x0), gradient 307 | :type g0: np.array 308 | 309 | :param d: search direction 310 | :type d: np.array 311 | 312 | :param above_iter: current iteration in optimization process 313 | :type above_iter: integer 314 | 315 | :param max_steplength: maximum steplength allowed 316 | :type max_steplength: float 317 | 318 | :param fct_f: callable, function f(x) 319 | :type fct_f: function returning float 320 | 321 | :param fct_grad: callable, function f'(x) 322 | :type fct_grad: function returning np.array 323 | 324 | :param alpha, beta: parameters of the decrease and curvature conditions 325 | :type alpha, beta: floats 326 | 327 | :param xtol_minpack: tolerance used in minpack2.dcsrch 328 | :type xtol_minpack: float 329 | 330 | :param max_iter: number of iteration allowed for finding a steplength 331 | :type max_iter: integer 332 | 333 | :return: optimal steplength meeting both decrease and curvature condition 334 | :rtype: float 335 | 336 | 337 | 338 | 339 | .. seealso:: 340 | 341 | [minpack] scipy.optimize.minpack2.dcsrch 342 | 343 | [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited 344 | memory algorithm for bound constrained optimization'', 345 | SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. 346 | 347 | [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN 348 | Subroutines for Large Scale Bound Constrained Optimization'' 349 | Tech. Report, NAM-11, EECS Department, Northwestern University, 350 | 1994. 351 | """ 352 | 353 | steplength_0 = 1 if max_steplength > 1 else 0.5*max_steplength 354 | f_m1 = f0 355 | dphi = g0.dot(d) 356 | dphi_m1 = dphi 357 | i = 0 358 | 359 | if(above_iter == 0): 360 | max_steplength = 1.0 361 | steplength_0 = min(1.0/np.sqrt(d.dot(d)), 1.0) 362 | 363 | isave = np.zeros((2,), np.intc) 364 | dsave = np.zeros((13,), float) 365 | task = b'START' 366 | 367 | while i0 416 | 417 | :return: updated [W, M, thet] 418 | :rtype: tuple 419 | 420 | .. seealso:: 421 | 422 | [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited 423 | memory algorithm for bound constrained optimization'', 424 | SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. 425 | 426 | [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN 427 | Subroutines for Large Scale Bound Constrained Optimization'' 428 | Tech. Report, NAM-11, EECS Department, Northwestern University, 429 | 1994. 430 | """ 431 | sTy = sk.dot(yk) 432 | yTy = yk.dot(yk) 433 | if (sTy > eps*yTy): 434 | S.append(sk) 435 | Y.append(yk) 436 | if len(S) > m : 437 | S.popleft() 438 | Y.popleft() 439 | Sarray = np.asarray(S).T 440 | Yarray = np.asarray(Y).T 441 | STS = np.transpose(Sarray).dot(Sarray) 442 | L = np.transpose(Sarray).dot(Yarray) 443 | D = np.diag(-np.diag(L)) 444 | L = np.tril(L, -1) 445 | 446 | thet = yTy/sTy 447 | W = np.hstack([Yarray, thet*Sarray]) 448 | M = np.linalg.inv(np.hstack([np.vstack([D, L]), np.vstack([L.T, thet*STS])])) 449 | 450 | return [W, M, thet] 451 | 452 | def L_BFGS_B(x0, f, df, l, u, m=10,\ 453 | epsg = 1e-5, epsf = 1e7, max_iter = 50,\ 454 | alpha_linesearch = 1e-4, beta_linesearch = 0.9,\ 455 | max_steplength = 1e8,\ 456 | xtol_minpack = 1e-5, max_iter_linesearch = 30, eps_SY = 2.2e-16): 457 | """ 458 | Solves bound constrained optimization problems by using the compact formula 459 | of the limited memory BFGS updates. 460 | 461 | :param x0: initial guess 462 | :type sk: np.array 463 | 464 | :param f: cost function to optimize f(x) 465 | :type f: function returning float 466 | 467 | :param df: gradient of cost function to optimize f'(x) 468 | :type df: function returning np.array 469 | 470 | :param l: the lower bound of x 471 | :type l: np.array 472 | 473 | :param u: the upper bound of x 474 | :type u: np.array 475 | 476 | :param m: Maximum size of lists for L-BFGS Hessian approximation 477 | :type m: integer 478 | 479 | :param epsg: Tolerance on projected gradient: programs converges when 480 | P(x-g, l, u)epsg and i