optimize.lisp

   1 ;;; -*- mode: lisp -*-
   2 ;;; Copyright (c) 2005--2007, by A.J. Rossini <blindglobe@gmail.com>
   3 ;;; See COPYRIGHT file for any additional restrictions (BSD license).
   4 ;;; Since 1991, ANSI was finally finished.  Edited for ANSI Common Lisp.
   5
   6 (in-package :cl-user)
   7
   8 (defpackage :lisp-stat-optimize
   9  (:use :common-lisp
  10        :lisp-stat-object-system
  11        :lisp-stat-types
  12        :lisp-stat-basics
  13        :lisp-stat-matrix
  14        :lisp-stat-linalg)
  15  (:shadowing-import-from :lisp-stat-object-system
  16                          slot-value call-method call-next-method)
  17  (:export
  18      ;; derivatives
  19      numgrad numhess
  20
  21      ;; optimization
  22      newtonmax nelmeadmax))
  23
  24
  25
  26 ;;; FIXME:AJR: There is a need to figure out the proper symbols to
  27 ;;; export.  more importantly should there be any specialty package
  28 z;;; that are exported for maximization?
  29
  30 (in-package :lisp-stat-optimize)
  31
  32 (defvar *maximize-callback-function* nil
  33   "Used in generic optimization to determine function name -- symbol or string?")
  34
  35 (defvar *maximize-callback-arg* nil
  36   "args to function to maximize")
  37
  38 ;;;;
  39 ;;;; minfo basics (internal??)
  40 ;;;;
  41
  42 (defun init-minfo-ipar-values (n ipars &key
  43                                (TRUE 1)
  44                                (FALSE 0)
  45                                (k 0)
  46                                (m 0)
  47                                (itnlimit -1)
  48                                (backtrack TRUE)
  49                                (verbose 0)
  50                                (vals_suppl FALSE)
  51                                (exptilt TRUE)
  52                                (count 0)
  53                                (termcode 0))
  54   "Initialize ipars (iteration parameters) by destructive modification."
  55   (setf (aref ipars 0) n)
  56   (setf (aref ipars 1) m)
  57   (setf (aref ipars 2) k)
  58   (setf (aref ipars 3) itnlimit)
  59   (setf (aref ipars 4) backtrack)
  60   (setf (aref ipars 5) verbose)
  61   (setf (aref ipars 6) vals_suppl)
  62   (setf (aref ipars 7) exptilt)
  63   (setf (aref ipars 8) count)
  64   (setf (aref ipars 9) termcode))
  65
  66 (defun init-minfo-dpar-values (h dpars &key
  67                                (typf 1.0)
  68                                (gradtol -1.0)
  69                                (steptol -1.0)
  70                                (maxstep -1.0)
  71                                (dflt 0.0)
  72                                (tilt 0.0)
  73                                (newtilt 0.0)
  74                                (hessadd 0.0))
  75   "Initialize dpars (derivative parameters) by destructive modification."
  76   (setf (aref dpars 0) typf)
  77   (setf (aref dpars 1) h)
  78   (setf (aref dpars 2) gradtol)
  79   (setf (aref dpars 3) steptol)
  80   (setf (aref dpars 4) maxstep)
  81   (setf (aref dpars 5) dflt)
  82   (setf (aref dpars 6) tilt)
  83   (setf (aref dpars 7) newtilt)
  84   (setf (aref dpars 8) hessadd))
  85
  86 (defun init-minfo-internals (n h internals)
  87   (let ((ipars (aref internals 8))
  88         (dpars (aref internals 9)))
  89     (init-minfo-ipar-values n ipars)
  90     (init-minfo-dpar-values h dpars)))
  91
  92 (defun new-minfo-internals (f x &key scale ((:derivstep h) -1.0))
  93   (check-sequence x)
  94   (check-real x)
  95   (check-one-real h)
  96   (let ((n (length x)))
  97     (when scale
  98           (check-sequence scale)
  99           (check-real scale)
 100           (if (/= n (length scale)) (error "scale and x not the same length")))
 101     (let ((internals (make-array 12)))
 102       (setf (aref internals 0) f)
 103       (setf (aref internals 3) (if (consp x) (copy-list x) (coerce x 'list)))
 104       (setf (aref internals 4)
 105             (if scale (copy-seq scale) (make-array n :initial-element 1.0)))
 106       (setf (aref internals 5) (make-list (+ 1 n (* n n))))
 107       (setf (aref internals 8) (make-array 10))
 108       (setf (aref internals 9) (make-array 9))
 109       (init-minfo-internals n h internals)
 110       internals)))
 111
 112 (defun minfo-maximize (internals &optional verbose)
 113   "This function does what?"
 114   (let* ((f (aref internals 0))
 115          (x (aref internals 3))
 116          (fvals (aref internals 5))
 117          (n (length x))
 118          (v (if verbose (if (integerp verbose) verbose 1) -1)))
 119     (setf (aref internals 3) (copy-list x))
 120     (setf (aref internals 5) (copy-list fvals))
 121     (let ((*maximize-callback-function* f)
 122           (*maximize-callback-arg* (make-list n)))
 123       (let* ((x (aref internals 3))
 124              (scale (aref internals 4))
 125              (fvals (aref internals 5))
 126              (ip (aref internals 8))
 127              (dp (aref internals 9))
 128              (px (la-data-to-vector x mode-re))
 129              (pscale (la-data-to-vector scale mode-re))
 130              (pfvals (la-vector (length fvals) mode-re))
 131              (pip (la-data-to-vector ip mode-in))
 132              (pdp (la-data-to-vector dp mode-re)))
 133         (unwind-protect
 134             (progn
 135               (base-minfo-maximize px pfvals pscale pip pdp v)) ;; access to C
 136           (la-vector-to-data px n mode-re x)
 137           (la-vector-to-data pfvals (+ 1 n (* n n)) mode-re fvals)
 138           (la-vector-to-data pip (length ip) mode-in ip)
 139           (la-vector-to-data pdp (length dp) mode-re dp))
 140         (get-buf)))))
 141
 142
 143
 144 ;;;;
 145 ;;;; Mode Info Prototype
 146 ;;;;
 147
 148 (defproto minfo-proto '(internals))
 149
 150 #+xlisp (send minfo-proto :add-method :isnew #'|minfo-isnew|)
 151 #+xlisp (send minfo-proto :add-method :maximize #'|minfo-maximize|)
 152 #+xlisp (send minfo-proto :add-method :loglaplace #'|minfo-loglap|)
 153 #-xlisp
 154 (defmeth minfo-proto :isnew (&rest args)
 155   (setf (slot-value 'internals) (apply #'new-minfo-internals args)))
 156 #-xlisp
 157 (defmeth minfo-proto :maximize (&rest args)
 158   (apply #'minfo-maximize (slot-value 'internals) args))
 159
 160 (defmeth minfo-proto :x () (aref (slot-value 'internals) 3))
 161 (defmeth minfo-proto :scale () (aref (slot-value 'internals) 4))
 162 (defmeth minfo-proto :derivstep () (aref (aref (slot-value 'internals) 9) 1))
 163 (defmeth minfo-proto :tilt () (aref (aref (slot-value 'internals) 9) 6))
 164
 165 (defmeth minfo-proto :f (&optional (val nil set))
 166   (when set
 167         (send self :set-no-vals-supplied)
 168         (setf (aref (slot-value 'internals) 0) val))
 169   (aref (slot-value 'internals) 0))
 170
 171 (defmeth minfo-proto :set-no-vals-supplied ()
 172   (setf (aref (aref (slot-value 'internals) 8) 6) 0))
 173
 174 (defmeth minfo-proto :exptilt (&optional (val nil set))
 175   (if set
 176       (let ((old (send self :exptilt)))
 177         (setf (aref (aref (slot-value 'internals) 8) 7) (if val 1 0))
 178         (if (and (not (or (and old val) (and (not old) (not val))))
 179                  (/= (send self :tilt) 0.0))
 180             (send self :set-no-vals-supplied))))
 181   (= 1 (aref (aref (slot-value 'internals) 8) 7)))
 182
 183 (defmeth minfo-proto :newtilt (&optional (val nil set))
 184   (when set
 185         (setf (aref (aref (slot-value 'internals) 9) 7) (float val))
 186         (if (/= (send self :tilt) 0.0) (send self :set-no-vals-supplied)))
 187   (aref (aref (slot-value 'internals) 9) 7))
 188
 189 (defmeth minfo-proto :gfuns (&optional (val nil set))
 190   (when set
 191         (if (or (not (consp val))
 192                 (not (every #'functionp val)))
 193             (error "not all functions"))
 194         (setf (aref (slot-value 'internals) 1) val)
 195         (setf (aref (aref (slot-value 'internals) 8) 1) (length val))
 196         (setf (aref (slot-value 'internals) 10) (repeat 1.0 (length val)))
 197         (if (/= (send self :tilt) 0.0) (send self :set-no-vals-supplied)))
 198   (aref (slot-value 'internals) 1))
 199
 200 (defmeth minfo-proto :cfuns (&optional (val nil set))
 201   (when set
 202         (if (or (not (consp val))
 203                 (not (every #'functionp val)))
 204             (error "not all functions"))
 205         (setf (aref (slot-value 'internals) 2) val)
 206         (setf (aref (aref (slot-value 'internals) 8) 2) (length val))
 207         (setf (aref (slot-value 'internals) 7) (repeat 0.0 (length val)))
 208         (setf (aref (slot-value 'internals) 11) (repeat 0.0 (length val)))
 209         (send self :set-no-vals-supplied))
 210   (aref (slot-value 'internals) 2))
 211
 212 (defmeth minfo-proto :ctarget (&optional (val nil set))
 213   (when set
 214         (if (/= (length val) (length (send self :ctarget)))
 215             (error "bad target length"))
 216         (setf (aref (slot-value 'internals) 7) val))
 217   (aref (slot-value 'internals) 7))
 218
 219 (defmeth minfo-proto :fvals ()
 220   (let* ((fv (aref (slot-value 'internals) 5))
 221          (n (length (send self :x)))
 222          (val (select fv 0))
 223          (grad (select fv (iseq 1 n)))
 224          (hess (matrix (list n n) (select fv (iseq (+ 1 n) (+ n (* n n)))))))
 225     (list val grad hess)))
 226
 227 (defmeth minfo-proto :copy ()
 228   (let ((obj (make-object minfo-proto))
 229         (internals (copy-seq (slot-value 'internals))))
 230     (dotimes (i (length internals))
 231              (let ((x (aref internals i)))
 232                (if (sequencep x)
 233                    (setf (aref internals i) (copy-seq x)))))
 234     (send obj :add-slot 'internals internals)
 235     obj))
 236
 237 (defmeth minfo-proto :derivscale ()
 238   (let* ((step (^ machine-epsilon (/ 1 6)))
 239          (hess (numhess (send self :f) (send self :x) (send self :scale) step))
 240          (scale (pmax (abs (send self :x)) (sqrt (abs (/ (diagonal hess)))))))
 241     (setf hess (numhess (send self :f) (send self :x) scale step))
 242     (setf scale (pmax (abs (send self :x)) (sqrt (abs (/ (diagonal hess))))))
 243     (setf (aref (slot-value 'internals) 4) scale)
 244     (setf (aref (aref (slot-value 'internals) 9) 1) step)))
 245
 246 (defmeth minfo-proto :verbose (&optional (val nil set))
 247   (when set
 248         (setf (aref (aref (slot-value 'internals) 8) 5)
 249               (cond ((integerp val) val)
 250                     ((null val) 0)
 251                     (t 1))))
 252   (aref (aref (slot-value 'internals) 8) 5))
 253
 254 (defmeth minfo-proto :backtrack (&optional (val nil set))
 255   (if set (setf (aref (aref (slot-value 'internals) 8) 4) (if val 1 0)))
 256   (aref (aref (slot-value 'internals) 8) 4))
 257
 258 (defmeth minfo-proto :maxiter (&optional (val nil set))
 259     (if set (setf (aref (aref (slot-value 'internals) 8) 3)
 260                   (if (integerp val) val -1)))
 261     (aref (aref (slot-value 'internals) 8) 3))
 262
 263 (defmeth minfo-proto :tiltscale (&optional (val nil set))
 264   (when set
 265         (if (/= (length val) (length (send self :gfuns)))
 266             (error "wrong size tilt scale sequence"))
 267         (setf (aref (slot-value 'internals) 10) val))
 268   (aref (slot-value 'internals) 10))
 269
 270 ;;;;
 271 ;;;;
 272 ;;;; Newton's Method with Backtracking
 273 ;;;;
 274 ;;;;
 275
 276 (defun newtonmax (f start &key
 277                     scale
 278                     (derivstep -1.0)
 279                     (count-limit -1)
 280                     (verbose 1)
 281                     return-derivs)
 282 "Args:(f start &key scale derivstep (verbose 1) return-derivs)
 283 Maximizes F starting from START using Newton's method with backtracking.
 284 If RETURN-DERIVS is NIL returns location of maximum; otherwise returns
 285 list of location, unction value, gradient and hessian at maximum.
 286 SCALE should be a list of the typical magnitudes of the parameters.
 287 DERIVSTEP is used in numerical derivatives and VERBOSE controls printing
 288 of iteration information. COUNT-LIMIT limits the number of iterations"
 289   (let ((verbose (if verbose (if (integerp verbose) verbose 1) 0))
 290         (minfo (send minfo-proto :new f start
 291                      :scale scale :derivstep derivstep)))
 292     (send minfo :maxiter count-limit)
 293     (send minfo :derivscale)
 294     (send minfo :maximize verbose)
 295     (if return-derivs
 296         (cons (send minfo :x) (- (send minfo :fvals)))
 297         (send minfo :x))))
 298
 299 ;;;;
 300 ;;;;
 301 ;;;; Nelder-Mead Simplex Method
 302 ;;;;
 303 ;;;;
 304
 305 (defun nelmeadmax (f start &key
 306                      (size 1)
 307                      (epsilon (sqrt machine-epsilon))
 308                      (count-limit 500)
 309                      (verbose t)
 310                      (alpha 1.0)
 311                      (beta 0.5)
 312                      (gamma 2.0)
 313                      (delta 0.5))
 314 "Args: (f start &key (size 1) (epsilon (sqrt machine-epsilon))
 315           (count-limit 500) (verbose t) alpha beta gamma delta)
 316 Maximizes F using the Nelder-Mead simplex method. START can be a
 317 starting simplex - a list of N+1 points, with N=dimension of problem,
 318 or a single point. If start is a single point you should give the
 319 size of the initial simplex as SIZE, a sequence of length N. Default is
 320 all 1's. EPSILON is the convergence tolerance. ALPHA-DELTA can be used to
 321 control the behavior of simplex algorithm."
 322     (let ((s (send simplex-proto :new f start size)))
 323       (do ((best (send s :best-point) (send s :best-point))
 324            (count 0 (+ count 1))
 325            next)
 326           ((or (< (send s :relative-range) epsilon) (>= count count-limit))
 327            (if (and verbose (>= count count-limit))
 328                (format t "Iteration limit exceeded.~%"))
 329            (send s :point-location (send s :best-point)))
 330           (setf next (send s :extrapolate-from-worst (- alpha)))
 331           (if (send s :is-worse best next)
 332               (setf next (send s :extrapolate-from-worst gamma))
 333               (when (send s :is-worse next (send s :second-worst-point))
 334                     (setf next (send s :extrapolate-from-worst beta))
 335                     (if (send s :is-worse next (send s :worst-point))
 336                         (send s :shrink-to-best delta))))
 337           (if verbose
 338               (format t "Value = ~10g~%"
 339                       (send s :point-value (send s :best-point)))))))
 340
 341
 342 ;;;
 343 ;;; Simplex Prototype
 344 ;;;
 345
 346 (defproto simplex-proto '(f simplex))
 347
 348 ;;;
 349 ;;; Simplex Points
 350 ;;;
 351
 352 (defmeth simplex-proto :make-point (x)
 353   (let ((f (send self :f)))
 354     (if f
 355         (let ((val (funcall f x)))
 356           (cons (if (consp val) (car val) val) x))
 357         (cons nil x))))
 358
 359 (defmeth simplex-proto :point-value (x) (car x))
 360
 361 (defmeth simplex-proto :point-location (x) (cdr x))
 362
 363 (defmeth simplex-proto :is-worse (x y)
 364   (< (send self :point-value x) (send self :point-value y)))
 365
 366 ;;;
 367 ;;; Making New Simplices
 368 ;;;
 369
 370 (defmeth simplex-proto :isnew (f start &optional size)
 371   (send self :simplex start size)
 372   (send self :f f))
 373
 374 ;;;
 375 ;;; Slot Accessors and Mutators
 376 ;;;
 377
 378 (defmeth simplex-proto :simplex (&optional new size)
 379   (if new
 380       (let ((simplex
 381              (if (and (consp new) (sequencep (car new)))
 382                  (if (/= (length new) (+ 1 (length (car new))))
 383                      (error "bad simplex data")
 384                      (copy-list new))
 385                  (let* ((n (length new))
 386                         (size (if size size (repeat 1 n)))
 387                      ;   (pts (- (* 2 (uniform-rand (repeat n (+ n 1)))) 1)))
 388                         (diag (* 2 size (- (random (repeat 2 n)) .5)))
 389                         (pts (cons (repeat 0 n)
 390                                    (mapcar #'(lambda (x) (coerce x 'list))
 391                                            (column-list (diagonal diag))))))
 392                    (mapcar #'(lambda (x) (reduce #'+ (list (* size x) new))) pts)))))
 393         (setf (slot-value 'simplex)
 394               (mapcar #'(lambda (x) (send self :make-point x)) simplex))
 395         (send self :sort-simplex)))
 396   (slot-value 'simplex))
 397
 398 (defmeth simplex-proto :f (&optional f)
 399   (when f
 400         (setf (slot-value 'f) f)
 401         (let ((simplex
 402                (mapcar #'(lambda (x) (send self :point-location x))
 403                        (send self :simplex))))
 404           (send self :simplex simplex)))
 405   (slot-value 'f))
 406
 407 (defmeth simplex-proto :sort-simplex ()
 408   (if (send self :f)
 409       (setf (slot-value 'simplex)
 410             (sort (slot-value 'simplex)
 411                   #'(lambda (x y) (send self :is-worse x y))))))
 412
 413 ;;;
 414 ;;; Other Methods Using List Representation of SImplex
 415 ;;;
 416
 417 (defmeth simplex-proto :best-point () (car (last (send self :simplex))))
 418 (defmeth simplex-proto :worst-point () (first (send self :simplex)))
 419 (defmeth simplex-proto :second-worst-point () (second (send self :simplex)))
 420 (defmeth simplex-proto :replace-point (new old)
 421   (let* ((simplex (send self :simplex))
 422          (n (position old simplex)))
 423     (when n
 424           (setf (nth n simplex) new)
 425           (send self :sort-simplex))))
 426 (defmeth simplex-proto :mean-opposite-face (x)
 427   (let ((face (mapcar #'(lambda (x) (send self :point-location x))
 428                       (remove x (send self :simplex)))))
 429     (/ (reduce #'+ face) (length face))))
 430
 431 ;;;
 432 ;;; Iteration Step Methods
 433 ;;;
 434
 435 (defmeth simplex-proto :extrapolate-from-worst (fac)
 436   (let* ((worst (send self :worst-point))
 437          (wloc (send self :point-location worst))
 438          (delta (- (send self :mean-opposite-face worst) wloc))
 439          (new (send self :make-point (+ wloc (* (- 1 fac) delta)))))
 440     (if (send self :is-worse worst new) (send self :replace-point new worst))
 441     new))
 442
 443 (defmeth simplex-proto :shrink-to-best (fac)
 444   (let* ((best (send self :best-point))
 445          (bloc (send self :point-location best)))
 446     (dolist (x (copy-list (send self :simplex)))
 447             (if (not (eq x best))
 448                 (send self :replace-point
 449                       (send self :make-point
 450                             (+ bloc
 451                                (* fac
 452                                   (- (send self :point-location x) bloc))))
 453                       x)))))
 454
 455 (defmeth simplex-proto :relative-range ()
 456   (let ((best (send self :point-value (send self :best-point)))
 457         (worst (send self :point-value (send self :worst-point))))
 458     (* 2 (/ (abs (- best worst)) (+ 1 (abs best) (abs worst))))))
 459
 460
 461
 462
 463 ;;;;
 464 ;;;; Maximization and Numerical Derivatives
 465 ;;;;
 466
 467
 468 (defun data2double (n data ptr)
 469   (declare (fixnum n))
 470   (let* ((seq (compound-data-seq data))
 471          (elem (make-next-element seq)))
 472     (if (/= (length seq) n) (error "bad data size"))
 473     (dotimes (i n)
 474       (declare (fixnum i))
 475       (la-put-double ptr i (get-next-element elem i)))))
 476
 477 (defun maximize-callback (n px pfval pgrad phess pderivs)
 478   (la-vector-to-data px n mode-re *maximize-callback-arg*)
 479   (let* ((val (funcall *maximize-callback-function* *maximize-callback-arg*))
 480          (derivs (if (consp val) (- (length val) 1) 0)))
 481     (la-put-integer pderivs 0 derivs)
 482     (la-put-double pfval 0 (if (consp val) (first val) val))
 483     (if (<= 1 derivs) (data2double n (second val) pgrad))
 484     (if (<= 2 derivs) (data2double (* n n) (third val) phess))))
 485
 486 (defun numgrad (f x &optional scale (h -1.0))
 487 "Args: (f x &optional scale derivstep)
 488 Computes the numerical gradient of F at X."
 489   (check-sequence x)
 490   (check-real x)
 491   (when scale
 492         (check-sequence scale)
 493         (check-real scale))
 494   (check-one-real h)
 495   (let* ((n (length x))
 496          (result (make-list n)))
 497     (if (and scale (/= n (length scale)))
 498         (error "scale not the same length as x"))
 499     (let ((*maximize-callback-function* f)
 500           (*maximize-callback-arg* (make-list n)))
 501       (let ((px (la-data-to-vector x mode-re))
 502             (pgrad (la-vector n mode-re))
 503             (pscale (la-data-to-vector
 504                      (if scale scale (make-list n :initial-element 1.0))
 505                      mode-re)))
 506         (unwind-protect
 507             (progn
 508               (numgrad-front n px pgrad h pscale)
 509               (la-vector-to-data pgrad n mode-re result))
 510           (la-free-vector px)
 511           (la-free-vector pgrad)
 512           (la-free-vector pscale))))
 513       result))
 514
 515 (defun numhess (f x &optional scale (h -1.0) all)
 516 "Args: (f x &optional scale derivstep)
 517 Computes the numerical Hessian matrix of F at X."
 518   (check-sequence x)
 519   (check-real x)
 520   (when scale
 521         (check-sequence scale)
 522         (check-real scale))
 523   (check-one-real h)
 524   (let* ((n (length x))
 525          (result (if all
 526                      (list nil (make-list n) (make-array (list n n)))
 527                      (make-array (list n n)))))
 528     (if (and scale (/= n (length scale)))
 529         (error "scale not the same length as x"))
 530     (let ((*maximize-callback-function* f)
 531           (*maximize-callback-arg* (make-list n)))
 532       (let ((hess-data (compound-data-seq (if all (third result) result)))
 533             (px (la-data-to-vector x mode-re))
 534             (pf (la-vector 1 mode-re))
 535             (pgrad (la-vector n mode-re))
 536             (phess (la-vector (* n n) mode-re))
 537             (pscale (la-data-to-vector
 538                      (if scale scale (make-list n :initial-element 1.0))
 539                      mode-re)))
 540         (unwind-protect
 541             (progn
 542               (numhess-front n px pf pgrad phess h pscale)
 543               (when all
 544                     (setf (first result) (la-get-double pf 0))
 545                     (la-vector-to-data pgrad n mode-re (second result)))
 546               (la-vector-to-data phess (* n n) mode-re hess-data))
 547           (la-free-vector pf)
 548           (la-free-vector px)
 549           (la-free-vector pgrad)
 550           (la-free-vector phess)
 551           (la-free-vector pscale))))
 552     result))
 553