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