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 ;;;
  40 ;;; CFFI support using library for optimization work.
  41 ;;;
  42
  43 ;; There is a problem with this particular approach, in terms of
  44 ;; circular dependencies.  We can not have this out-of-object call
  45 ;; into optimize, at least not from here.
  46 (cffi:defcallback ccl-maximize-callback :void ((n :int)
  47                                                (px :pointer)
  48                                                (pfval :pointer)
  49                                                (pgrad :pointer)
  50                                                (phess :pointer)
  51                                                (pderivs :pointer))
  52    (lisp-stat-optimize::maximize-callback n px pfval pgrad phess pderivs))
  53
  54 (cffi:defcfun ("register_maximize_callback" register-maximize-callback)
  55     :void (x :pointer))
  56 (register-maximize-callback (cffi:callback ccl-maximize-callback))
  57
  58 (cffi:defcfun ("ccl_numgrad_front" ccl-numgrad-front)
  59     :int (x :int) (y :pointer) (z :pointer) (u :double) (v :pointer))
  60 (defun numgrad-front (x y z u v)
  61   (ccl-numgrad-front x y z (float u 1d0) v))
  62
  63 (cffi:defcfun ("ccl_numhess_front" ccl-numhess-front)
  64     :int (x :int) (y :pointer) (z :pointer) (u :pointer) (v :pointer) (w :double) (a :pointer))
  65 (defun numhess-front (x y z u v w a)
  66   (ccl-numhess-front x y z u v (float w 1d0) a))
  67
  68 (cffi:defcfun ("ccl_minfo_maximize" ccl-minfo-maximize)
  69     :int (x :pointer) (y :pointer) (z :pointer) (u :pointer) (v :pointer) (w :int))
  70 (defun base-minfo-maximize (x y z u v w)
  71   (ccl-minfo-maximize x y z u v w))
  72
  73
  74
  75 ;;;;
  76 ;;;; minfo basics (internal??)
  77 ;;;;
  78
  79 (defun init-minfo-ipar-values (n ipars &key
  80                                (TRUE 1)
  81                                (FALSE 0)
  82                                (k 0)
  83                                (m 0)
  84                                (itnlimit -1)
  85                                (backtrack TRUE)
  86                                (verbose 0)
  87                                (vals_suppl FALSE)
  88                                (exptilt TRUE)
  89                                (count 0)
  90                                (termcode 0))
  91   "Initialize ipars (iteration parameters) by destructive modification."
  92   (setf (aref ipars 0) n)
  93   (setf (aref ipars 1) m)
  94   (setf (aref ipars 2) k)
  95   (setf (aref ipars 3) itnlimit)
  96   (setf (aref ipars 4) backtrack)
  97   (setf (aref ipars 5) verbose)
  98   (setf (aref ipars 6) vals_suppl)
  99   (setf (aref ipars 7) exptilt)
 100   (setf (aref ipars 8) count)
 101   (setf (aref ipars 9) termcode))
 102
 103 (defun init-minfo-dpar-values (h dpars &key
 104                                (typf 1.0)
 105                                (gradtol -1.0)
 106                                (steptol -1.0)
 107                                (maxstep -1.0)
 108                                (dflt 0.0)
 109                                (tilt 0.0)
 110                                (newtilt 0.0)
 111                                (hessadd 0.0))
 112   "Initialize dpars (derivative parameters) by destructive modification."
 113   (setf (aref dpars 0) typf)
 114   (setf (aref dpars 1) h)
 115   (setf (aref dpars 2) gradtol)
 116   (setf (aref dpars 3) steptol)
 117   (setf (aref dpars 4) maxstep)
 118   (setf (aref dpars 5) dflt)
 119   (setf (aref dpars 6) tilt)
 120   (setf (aref dpars 7) newtilt)
 121   (setf (aref dpars 8) hessadd))
 122
 123 (defun init-minfo-internals (n h internals)
 124   (let ((ipars (aref internals 8))
 125         (dpars (aref internals 9)))
 126     (init-minfo-ipar-values n ipars)
 127     (init-minfo-dpar-values h dpars)))
 128
 129 (defun new-minfo-internals (f x &key scale ((:derivstep h) -1.0))
 130   (check-sequence x)
 131   (check-real x)
 132   (check-one-real h)
 133   (let ((n (length x)))
 134     (when scale
 135           (check-sequence scale)
 136           (check-real scale)
 137           (if (/= n (length scale)) (error "scale and x not the same length")))
 138     (let ((internals (make-array 12)))
 139       (setf (aref internals 0) f)
 140       (setf (aref internals 3) (if (consp x) (copy-list x) (coerce x 'list)))
 141       (setf (aref internals 4)
 142             (if scale (copy-seq scale) (make-array n :initial-element 1.0)))
 143       (setf (aref internals 5) (make-list (+ 1 n (* n n))))
 144       (setf (aref internals 8) (make-array 10))
 145       (setf (aref internals 9) (make-array 9))
 146       (init-minfo-internals n h internals)
 147       internals)))
 148
 149 (defun minfo-maximize (internals &optional verbose)
 150   "This function does what?"
 151   (let* ((f (aref internals 0))
 152          (x (aref internals 3))
 153          (fvals (aref internals 5))
 154          (n (length x))
 155          (v (if verbose (if (integerp verbose) verbose 1) -1)))
 156     (setf (aref internals 3) (copy-list x))
 157     (setf (aref internals 5) (copy-list fvals))
 158     (let ((*maximize-callback-function* f)
 159           (*maximize-callback-arg* (make-list n)))
 160       (let* ((x (aref internals 3))
 161              (scale (aref internals 4))
 162              (fvals (aref internals 5))
 163              (ip (aref internals 8))
 164              (dp (aref internals 9))
 165              (px (la-data-to-vector x mode-re))
 166              (pscale (la-data-to-vector scale mode-re))
 167              (pfvals (la-vector (length fvals) mode-re))
 168              (pip (la-data-to-vector ip mode-in))
 169              (pdp (la-data-to-vector dp mode-re)))
 170         (unwind-protect
 171             (progn
 172               (base-minfo-maximize px pfvals pscale pip pdp v)) ;; access to C
 173           (la-vector-to-data px n mode-re x)
 174           (la-vector-to-data pfvals (+ 1 n (* n n)) mode-re fvals)
 175           (la-vector-to-data pip (length ip) mode-in ip)
 176           (la-vector-to-data pdp (length dp) mode-re dp))
 177         (get-buf)))))
 178
 179
 180
 181 ;;;;
 182 ;;;; Mode Info Prototype
 183 ;;;;
 184
 185 (defproto minfo-proto '(internals))
 186
 187 #+xlisp (send minfo-proto :add-method :isnew #'|minfo-isnew|)
 188 #+xlisp (send minfo-proto :add-method :maximize #'|minfo-maximize|)
 189 #+xlisp (send minfo-proto :add-method :loglaplace #'|minfo-loglap|)
 190 #-xlisp
 191 (defmeth minfo-proto :isnew (&rest args)
 192   (setf (slot-value 'internals) (apply #'new-minfo-internals args)))
 193 #-xlisp
 194 (defmeth minfo-proto :maximize (&rest args)
 195   (apply #'minfo-maximize (slot-value 'internals) args))
 196
 197 (defmeth minfo-proto :x () (aref (slot-value 'internals) 3))
 198 (defmeth minfo-proto :scale () (aref (slot-value 'internals) 4))
 199 (defmeth minfo-proto :derivstep () (aref (aref (slot-value 'internals) 9) 1))
 200 (defmeth minfo-proto :tilt () (aref (aref (slot-value 'internals) 9) 6))
 201
 202 (defmeth minfo-proto :f (&optional (val nil set))
 203   (when set
 204         (send self :set-no-vals-supplied)
 205         (setf (aref (slot-value 'internals) 0) val))
 206   (aref (slot-value 'internals) 0))
 207
 208 (defmeth minfo-proto :set-no-vals-supplied ()
 209   (setf (aref (aref (slot-value 'internals) 8) 6) 0))
 210
 211 (defmeth minfo-proto :exptilt (&optional (val nil set))
 212   (if set
 213       (let ((old (send self :exptilt)))
 214         (setf (aref (aref (slot-value 'internals) 8) 7) (if val 1 0))
 215         (if (and (not (or (and old val) (and (not old) (not val))))
 216                  (/= (send self :tilt) 0.0))
 217             (send self :set-no-vals-supplied))))
 218   (= 1 (aref (aref (slot-value 'internals) 8) 7)))
 219
 220 (defmeth minfo-proto :newtilt (&optional (val nil set))
 221   (when set
 222         (setf (aref (aref (slot-value 'internals) 9) 7) (float val))
 223         (if (/= (send self :tilt) 0.0) (send self :set-no-vals-supplied)))
 224   (aref (aref (slot-value 'internals) 9) 7))
 225
 226 (defmeth minfo-proto :gfuns (&optional (val nil set))
 227   (when set
 228         (if (or (not (consp val))
 229                 (not (every #'functionp val)))
 230             (error "not all functions"))
 231         (setf (aref (slot-value 'internals) 1) val)
 232         (setf (aref (aref (slot-value 'internals) 8) 1) (length val))
 233         (setf (aref (slot-value 'internals) 10) (repeat 1.0 (length val)))
 234         (if (/= (send self :tilt) 0.0) (send self :set-no-vals-supplied)))
 235   (aref (slot-value 'internals) 1))
 236
 237 (defmeth minfo-proto :cfuns (&optional (val nil set))
 238   (when set
 239         (if (or (not (consp val))
 240                 (not (every #'functionp val)))
 241             (error "not all functions"))
 242         (setf (aref (slot-value 'internals) 2) val)
 243         (setf (aref (aref (slot-value 'internals) 8) 2) (length val))
 244         (setf (aref (slot-value 'internals) 7) (repeat 0.0 (length val)))
 245         (setf (aref (slot-value 'internals) 11) (repeat 0.0 (length val)))
 246         (send self :set-no-vals-supplied))
 247   (aref (slot-value 'internals) 2))
 248
 249 (defmeth minfo-proto :ctarget (&optional (val nil set))
 250   (when set
 251         (if (/= (length val) (length (send self :ctarget)))
 252             (error "bad target length"))
 253         (setf (aref (slot-value 'internals) 7) val))
 254   (aref (slot-value 'internals) 7))
 255
 256 (defmeth minfo-proto :fvals ()
 257   (let* ((fv (aref (slot-value 'internals) 5))
 258          (n (length (send self :x)))
 259          (val (select fv 0))
 260          (grad (select fv (iseq 1 n)))
 261          (hess (matrix (list n n) (select fv (iseq (+ 1 n) (+ n (* n n)))))))
 262     (list val grad hess)))
 263
 264 (defmeth minfo-proto :copy ()
 265   (let ((obj (make-object minfo-proto))
 266         (internals (copy-seq (slot-value 'internals))))
 267     (dotimes (i (length internals))
 268              (let ((x (aref internals i)))
 269                (if (sequencep x)
 270                    (setf (aref internals i) (copy-seq x)))))
 271     (send obj :add-slot 'internals internals)
 272     obj))
 273
 274 (defmeth minfo-proto :derivscale ()
 275   (let* ((step (^ machine-epsilon (/ 1 6)))
 276          (hess (numhess (send self :f) (send self :x) (send self :scale) step))
 277          (scale (pmax (abs (send self :x)) (sqrt (abs (/ (diagonal hess)))))))
 278     (setf hess (numhess (send self :f) (send self :x) scale step))
 279     (setf scale (pmax (abs (send self :x)) (sqrt (abs (/ (diagonal hess))))))
 280     (setf (aref (slot-value 'internals) 4) scale)
 281     (setf (aref (aref (slot-value 'internals) 9) 1) step)))
 282
 283 (defmeth minfo-proto :verbose (&optional (val nil set))
 284   (when set
 285         (setf (aref (aref (slot-value 'internals) 8) 5)
 286               (cond ((integerp val) val)
 287                     ((null val) 0)
 288                     (t 1))))
 289   (aref (aref (slot-value 'internals) 8) 5))
 290
 291 (defmeth minfo-proto :backtrack (&optional (val nil set))
 292   (if set (setf (aref (aref (slot-value 'internals) 8) 4) (if val 1 0)))
 293   (aref (aref (slot-value 'internals) 8) 4))
 294
 295 (defmeth minfo-proto :maxiter (&optional (val nil set))
 296     (if set (setf (aref (aref (slot-value 'internals) 8) 3)
 297                   (if (integerp val) val -1)))
 298     (aref (aref (slot-value 'internals) 8) 3))
 299
 300 (defmeth minfo-proto :tiltscale (&optional (val nil set))
 301   (when set
 302         (if (/= (length val) (length (send self :gfuns)))
 303             (error "wrong size tilt scale sequence"))
 304         (setf (aref (slot-value 'internals) 10) val))
 305   (aref (slot-value 'internals) 10))
 306
 307 ;;;;
 308 ;;;;
 309 ;;;; Newton's Method with Backtracking
 310 ;;;;
 311 ;;;;
 312
 313 (defun newtonmax (f start &key
 314                     scale
 315                     (derivstep -1.0)
 316                     (count-limit -1)
 317                     (verbose 1)
 318                     return-derivs)
 319 "Args:(f start &key scale derivstep (verbose 1) return-derivs)
 320 Maximizes F starting from START using Newton's method with backtracking.
 321 If RETURN-DERIVS is NIL returns location of maximum; otherwise returns
 322 list of location, unction value, gradient and hessian at maximum.
 323 SCALE should be a list of the typical magnitudes of the parameters.
 324 DERIVSTEP is used in numerical derivatives and VERBOSE controls printing
 325 of iteration information. COUNT-LIMIT limits the number of iterations"
 326   (let ((verbose (if verbose (if (integerp verbose) verbose 1) 0))
 327         (minfo (send minfo-proto :new f start
 328                      :scale scale :derivstep derivstep)))
 329     (send minfo :maxiter count-limit)
 330     (send minfo :derivscale)
 331     (send minfo :maximize verbose)
 332     (if return-derivs
 333         (cons (send minfo :x) (- (send minfo :fvals)))
 334         (send minfo :x))))
 335
 336 ;;;;
 337 ;;;;
 338 ;;;; Nelder-Mead Simplex Method
 339 ;;;;
 340 ;;;;
 341
 342 (defun nelmeadmax (f start &key
 343                      (size 1)
 344                      (epsilon (sqrt machine-epsilon))
 345                      (count-limit 500)
 346                      (verbose t)
 347                      (alpha 1.0)
 348                      (beta 0.5)
 349                      (gamma 2.0)
 350                      (delta 0.5))
 351 "Args: (f start &key (size 1) (epsilon (sqrt machine-epsilon))
 352           (count-limit 500) (verbose t) alpha beta gamma delta)
 353 Maximizes F using the Nelder-Mead simplex method. START can be a
 354 starting simplex - a list of N+1 points, with N=dimension of problem,
 355 or a single point. If start is a single point you should give the
 356 size of the initial simplex as SIZE, a sequence of length N. Default is
 357 all 1's. EPSILON is the convergence tolerance. ALPHA-DELTA can be used to
 358 control the behavior of simplex algorithm."
 359     (let ((s (send simplex-proto :new f start size)))
 360       (do ((best (send s :best-point) (send s :best-point))
 361            (count 0 (+ count 1))
 362            next)
 363           ((or (< (send s :relative-range) epsilon) (>= count count-limit))
 364            (if (and verbose (>= count count-limit))
 365                (format t "Iteration limit exceeded.~%"))
 366            (send s :point-location (send s :best-point)))
 367           (setf next (send s :extrapolate-from-worst (- alpha)))
 368           (if (send s :is-worse best next)
 369               (setf next (send s :extrapolate-from-worst gamma))
 370               (when (send s :is-worse next (send s :second-worst-point))
 371                     (setf next (send s :extrapolate-from-worst beta))
 372                     (if (send s :is-worse next (send s :worst-point))
 373                         (send s :shrink-to-best delta))))
 374           (if verbose
 375               (format t "Value = ~10g~%"
 376                       (send s :point-value (send s :best-point)))))))
 377
 378
 379 ;;;
 380 ;;; Simplex Prototype
 381 ;;;
 382
 383 (defproto simplex-proto '(f simplex))
 384
 385 ;;;
 386 ;;; Simplex Points
 387 ;;;
 388
 389 (defmeth simplex-proto :make-point (x)
 390   (let ((f (send self :f)))
 391     (if f
 392         (let ((val (funcall f x)))
 393           (cons (if (consp val) (car val) val) x))
 394         (cons nil x))))
 395
 396 (defmeth simplex-proto :point-value (x) (car x))
 397
 398 (defmeth simplex-proto :point-location (x) (cdr x))
 399
 400 (defmeth simplex-proto :is-worse (x y)
 401   (< (send self :point-value x) (send self :point-value y)))
 402
 403 ;;;
 404 ;;; Making New Simplices
 405 ;;;
 406
 407 (defmeth simplex-proto :isnew (f start &optional size)
 408   (send self :simplex start size)
 409   (send self :f f))
 410
 411 ;;;
 412 ;;; Slot Accessors and Mutators
 413 ;;;
 414
 415 (defmeth simplex-proto :simplex (&optional new size)
 416   (if new
 417       (let ((simplex
 418              (if (and (consp new) (sequencep (car new)))
 419                  (if (/= (length new) (+ 1 (length (car new))))
 420                      (error "bad simplex data")
 421                      (copy-list new))
 422                  (let* ((n (length new))
 423                         (size (if size size (repeat 1 n)))
 424                      ;   (pts (- (* 2 (uniform-rand (repeat n (+ n 1)))) 1)))
 425                         (diag (* 2 size (- (random (repeat 2 n)) .5)))
 426                         (pts (cons (repeat 0 n)
 427                                    (mapcar #'(lambda (x) (coerce x 'list))
 428                                            (column-list (diagonal diag))))))
 429                    (mapcar #'(lambda (x) (reduce #'+ (list (* size x) new))) pts)))))
 430         (setf (slot-value 'simplex)
 431               (mapcar #'(lambda (x) (send self :make-point x)) simplex))
 432         (send self :sort-simplex)))
 433   (slot-value 'simplex))
 434
 435 (defmeth simplex-proto :f (&optional f)
 436   (when f
 437         (setf (slot-value 'f) f)
 438         (let ((simplex
 439                (mapcar #'(lambda (x) (send self :point-location x))
 440                        (send self :simplex))))
 441           (send self :simplex simplex)))
 442   (slot-value 'f))
 443
 444 (defmeth simplex-proto :sort-simplex ()
 445   (if (send self :f)
 446       (setf (slot-value 'simplex)
 447             (sort (slot-value 'simplex)
 448                   #'(lambda (x y) (send self :is-worse x y))))))
 449
 450 ;;;
 451 ;;; Other Methods Using List Representation of SImplex
 452 ;;;
 453
 454 (defmeth simplex-proto :best-point () (car (last (send self :simplex))))
 455 (defmeth simplex-proto :worst-point () (first (send self :simplex)))
 456 (defmeth simplex-proto :second-worst-point () (second (send self :simplex)))
 457 (defmeth simplex-proto :replace-point (new old)
 458   (let* ((simplex (send self :simplex))
 459          (n (position old simplex)))
 460     (when n
 461           (setf (nth n simplex) new)
 462           (send self :sort-simplex))))
 463 (defmeth simplex-proto :mean-opposite-face (x)
 464   (let ((face (mapcar #'(lambda (x) (send self :point-location x))
 465                       (remove x (send self :simplex)))))
 466     (/ (reduce #'+ face) (length face))))
 467
 468 ;;;
 469 ;;; Iteration Step Methods
 470 ;;;
 471
 472 (defmeth simplex-proto :extrapolate-from-worst (fac)
 473   (let* ((worst (send self :worst-point))
 474          (wloc (send self :point-location worst))
 475          (delta (- (send self :mean-opposite-face worst) wloc))
 476          (new (send self :make-point (+ wloc (* (- 1 fac) delta)))))
 477     (if (send self :is-worse worst new) (send self :replace-point new worst))
 478     new))
 479
 480 (defmeth simplex-proto :shrink-to-best (fac)
 481   (let* ((best (send self :best-point))
 482          (bloc (send self :point-location best)))
 483     (dolist (x (copy-list (send self :simplex)))
 484             (if (not (eq x best))
 485                 (send self :replace-point
 486                       (send self :make-point
 487                             (+ bloc
 488                                (* fac
 489                                   (- (send self :point-location x) bloc))))
 490                       x)))))
 491
 492 (defmeth simplex-proto :relative-range ()
 493   (let ((best (send self :point-value (send self :best-point)))
 494         (worst (send self :point-value (send self :worst-point))))
 495     (* 2 (/ (abs (- best worst)) (+ 1 (abs best) (abs worst))))))
 496
 497
 498
 499
 500 ;;;;
 501 ;;;; Maximization and Numerical Derivatives
 502 ;;;;
 503
 504
 505 (defun data2double (n data ptr)
 506   (declare (fixnum n))
 507   (let* ((seq (compound-data-seq data))
 508          (elem (make-next-element seq)))
 509     (if (/= (length seq) n) (error "bad data size"))
 510     (dotimes (i n)
 511       (declare (fixnum i))
 512       (la-put-double ptr i (get-next-element elem i)))))
 513
 514 (defun maximize-callback (n px pfval pgrad phess pderivs)
 515   (la-vector-to-data px n mode-re *maximize-callback-arg*)
 516   (let* ((val (funcall *maximize-callback-function* *maximize-callback-arg*))
 517          (derivs (if (consp val) (- (length val) 1) 0)))
 518     (la-put-integer pderivs 0 derivs)
 519     (la-put-double pfval 0 (if (consp val) (first val) val))
 520     (if (<= 1 derivs) (data2double n (second val) pgrad))
 521     (if (<= 2 derivs) (data2double (* n n) (third val) phess))))
 522
 523 (defun numgrad (f x &optional scale (h -1.0))
 524 "Args: (f x &optional scale derivstep)
 525 Computes the numerical gradient of F at X."
 526   (check-sequence x)
 527   (check-real x)
 528   (when scale
 529         (check-sequence scale)
 530         (check-real scale))
 531   (check-one-real h)
 532   (let* ((n (length x))
 533          (result (make-list n)))
 534     (if (and scale (/= n (length scale)))
 535         (error "scale not the same length as x"))
 536     (let ((*maximize-callback-function* f)
 537           (*maximize-callback-arg* (make-list n)))
 538       (let ((px (la-data-to-vector x mode-re))
 539             (pgrad (la-vector n mode-re))
 540             (pscale (la-data-to-vector
 541                      (if scale scale (make-list n :initial-element 1.0))
 542                      mode-re)))
 543         (unwind-protect
 544             (progn
 545               (numgrad-front n px pgrad h pscale)
 546               (la-vector-to-data pgrad n mode-re result))
 547           (la-free-vector px)
 548           (la-free-vector pgrad)
 549           (la-free-vector pscale))))
 550       result))
 551
 552 (defun numhess (f x &optional scale (h -1.0) all)
 553 "Args: (f x &optional scale derivstep)
 554 Computes the numerical Hessian matrix of F at X."
 555   (check-sequence x)
 556   (check-real x)
 557   (when scale
 558         (check-sequence scale)
 559         (check-real scale))
 560   (check-one-real h)
 561   (let* ((n (length x))
 562          (result (if all
 563                      (list nil (make-list n) (make-array (list n n)))
 564                      (make-array (list n n)))))
 565     (if (and scale (/= n (length scale)))
 566         (error "scale not the same length as x"))
 567     (let ((*maximize-callback-function* f)
 568           (*maximize-callback-arg* (make-list n)))
 569       (let ((hess-data (compound-data-seq (if all (third result) result)))
 570             (px (la-data-to-vector x mode-re))
 571             (pf (la-vector 1 mode-re))
 572             (pgrad (la-vector n mode-re))
 573             (phess (la-vector (* n n) mode-re))
 574             (pscale (la-data-to-vector
 575                      (if scale scale (make-list n :initial-element 1.0))
 576                      mode-re)))
 577         (unwind-protect
 578             (progn
 579               (numhess-front n px pf pgrad phess h pscale)
 580               (when all
 581                     (setf (first result) (la-get-double pf 0))
 582                     (la-vector-to-data pgrad n mode-re (second result)))
 583               (la-vector-to-data phess (* n n) mode-re hess-data))
 584           (la-free-vector pf)
 585           (la-free-vector px)
 586           (la-free-vector pgrad)
 587           (la-free-vector phess)
 588           (la-free-vector pscale))))
 589     result))
 590