1/* The MIT License (MIT) 2 * 3 * Copyright (c) 2022-2024 Rick Workman 4 * 5 * Permission is hereby granted, free of charge, to any person obtaining a copy 6 * of this software and associated documentation files (the "Software"), to deal 7 * in the Software without restriction, including without limitation the rights 8 * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 9 * copies of the Software, and to permit persons to whom the Software is 10 * furnished to do so, subject to the following conditions: 11 * 12 * The above copyright notice and this permission notice shall be included in all 13 * copies or substantial portions of the Software. 14 * 15 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 16 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 17 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 18 * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 19 * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 20 * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE 21 * SOFTWARE. 22 */ 23:- module(clpBNR_toolkit, % SWI module declaration 24 [ 25 iterate_until/3, % general purpose iterator 26 mid_split_one/1, % contractor to split largest interval at midpoint 27 mid_split/1, % contractor to split an interval at midpoint 28 taylor_contractor/2, % build cf_contractor based on Taylor expansion 29 taylor_merged_contractor/2, % build merged Taylor cf_contractor from list of equations 30 cf_contractor/2, % execute cf_contractor 31 cf_solve/1, cf_solve/2, % a solve predicate for centre form contractors 32 33 integrate/3, integrate/4, % simple numerical integration 34 boundary_values/2, boundary_values/3, boundary_values/4, % solve boundary value problems 35 36 lin_minimum/3, % find minimum of linear problem using library(simplex) 37 lin_maximum/3, % find maximum of linear problem using library(simplex) 38 lin_minimize/3, % lin_minimum/3 plus bind vars to solution minimizers 39 lin_maximize/3, % lin_maximum/3 plus bind vars to solution maximizers 40 41 local_minima/1, % apply KT constraints for objective function expression (OFE) 42 local_maxima/1, % semantically equivalent to local_minima/1 43 local_minima/2, % apply KT constraints for minima with constraints 44 local_maxima/2 % apply KT constraints for maxima with constraints 45 ]).
56:- use_module(library(apply),[maplist/3]). 57:- use_module(library(apply_macros)). % compiler support for `maplist`, helps a bit 58:- use_module(library(clpBNR)). 59:- use_module(library(simplex)). 60 61% sandboxing for SWISH 62:- multifile(sandbox:safe_primitive/1). 63 64% messages for noisy failure 65fail_msg_(FString,Args) :- 66 debug(clpBNR,FString,Args), fail. 67 68:- set_prolog_flag(optimise,true). % for arithmetic, this module only
small/2 and Goal midsplit/1 :
?- X::real(-1,1),iterate_until(10,small(X,0),mid_split(X)),format("X = ~w\n",X),fail;true.
X = _6288{real(-1,-1r2)}
X = _6288{real(-1r2,0)}
X = _6288{real(0,1r2)}
X = _6288{real(1r2,1)}
true.
The specific intended use case is to provide an iterator for meta-contractors such as the centre-form contractor such as midsplit/1 (example above) or as constructed by taylor_contractor/2 as in:
?- X::real,taylor_contractor({X**4-4*X**3+4*X**2-4*X+3==0},T),
iterate_until(50,small(X),(T,mid_split_one([X]))),format("X = ~w\n",X),fail;true.
X = _150{real(0.999999999926943,1.00000000007306)}
X = _150{real(2.999999999484828,3.0000000005152105)}
true.
(Aside: For some problems, solving with Taylor contractors can be a faster and more precise alternative to clpBNR:solve/1.)
*/
92% 93% General purpose iterator: execute Goal a maximum of N times or until Test succeeds 94% 95iterate_until(N,Test,Goal) :- N>0, !, 96 , 97 N1 is N-1, 98 ( 99 -> true 100 ; iterate_until(N1,Test,Goal) 101 ). 102iterate_until(_N,_,_). % non-positive N --> exit 103 104sandbox:safe_meta(clpBNR_toolkit:iterate_until(_N,Test,Goal), [Test, Goal]).
mid_split for details of interval splitting for this predicate.
113mid_split_one(Xs) :- 114 select_split(Xs,X), % select largest interval with largest width 115 mid_split(X). % split it
mid_split(X) :-
M is midpoint(X),
({X=<M} ; {M=<X}).
Note that mid_split succeeds if X is a number, but doesn't do anything.
Use clpBNR:small as a pre-test to avoid splitting intervals which are already small enough.
131mid_split(X) :- 132 (number(X) % optimise number case 133 -> true 134 ; (small(X) 135 -> true 136 ; midpoint(X,M), % fails if not an interval 137 ({X=<M} ; {M=<X}) % possible choicepoint, Note: can split on solution leaving CP 138 ) 139 ). 140% 141% select interval with largest width 142% 143select_split([X],X) :- !. % select last remaining element 144select_split([X1,X2|Xs],X) :- % compare widths and discard one interval 145 delta(X1,D1), 146 delta(X2,D2), 147 (D1 >= D2 148 -> select_split([X1|Xs],X) 149 ; select_split([X2|Xs],X) 150 ).
taylor_contractor. In normal usage, a direct call to cf_contractor does appear; instead use cf_contractor or in a Goal for iterate_until/3.
159% 160% centred form contractor 161% 162% Bind the values of As to the midpoints of Xs. To support repetitive application 163% of the contractor (required by the iterator), the contractor should not permanently 164% bind anything so findall/3 will be used to achieve this "forward checking" 165% (as suggested in [CLIP]). After the call to findall, the bounds of the resulting list 166% of narrowed domains (XDs) are then applied to Xs. 167% 168% This contractor can be used with any "centred form", e.g., Newton or Krawczyk, since it 169% only depends on intervals and their midpoints, hence its name `cf_contractor`. The 170% details which distinguish the variety of centred form are built into the variables' 171% constraints. 172% 173cf_contractor(Xs,As) :- 174 findall(Ds,(maplist(bind_to_midpoint,Xs,As),maplist(cf_domain,Xs,Ds)),[XDs]), 175 maplist(set_domain,Xs,XDs). 176 177bind_to_midpoint(X,A) :- A is float(midpoint(X)). 178 179cf_domain(X,D) :- 180 number(X) -> D = X ; domain(X,D). % in case X narrowed to a point 181 182set_domain(X,D) :- 183 number(D) -> X = D ; X::D.
cf_solve/2 using default precision.
*/clpBNR_default_precision); otherwise fails.
This is done by using iterate_until/3 limited to a count determined by the flag clpBNR_iteration_limit. Examples:
?- X::real, taylor_contractor({X**4-4*X**3+4*X**2-4*X+3==0},T), cf_solve(T).
T = cf_contractor([X], [_A]),
X:: 1.000000000...,
_A::real(-1.0Inf, 1.0Inf) ;
T = cf_contractor([X], [_A]),
X:: 3.00000000...,
_A::real(-1.0Inf, 1.0Inf) ;
false.
?- taylor_contractor({2*X1+5*X1**3+1==X2*(1+X2), 2*X2+5*X2**3+1==X1*(1+X1)},T), cf_solve(T).
T = cf_contractor([X2, X1], [_A, _B]),
X1:: -0.42730462...,
X2:: -0.42730462...,
_B::real(-1.0Inf, 1.0Inf),
_A::real(-1.0Inf, 1.0Inf) ;
false.
217cf_solve(T) :- 218 current_prolog_flag(clpBNR_default_precision,P), 219 cf_solve(T,P). 220cf_solve(cf_contractor(Xs,As),P) :- 221 current_prolog_flag(clpBNR_iteration_limit,L), 222 Count is L div 10, % heuristic - primitive iteration limit/10 223 cf_iterate_(Count,Xs,As,P). 224 225cf_iterate_(Count,Xs,As,P) :- 226 Count > 0, 227 \+ small(Xs,P), % at least one var not narrow enough 228 !, 229 cf_contractor(Xs,As), % execute contractor 230 select_split(Xs,X), % select widest 231 (small(X,P) % still wide enough to split? 232 -> true % no, we're done 233 ; cf_split(X,Pt), % yes, split it 234 ({X=<Pt} ; {Pt=<X}), 235 Count1 is Count-1, 236 cf_iterate_(Count1,Xs,As,P) % and iterate 237 ). 238cf_iterate_(_,_,_,_). % done (Count=<0 or all small Xs) 239 240cf_split(X,Pt) :- 241 range(X,[L,H]), 242 cf_split_point(L,H,X,M), % modest attempt to find non-solution 243 !, 244 (M = 1.5NaN, L<0,0<H -> Pt = 0 ; Pt = M). % (-inf,inf) case: if NaN and spans 0, use 0 245 246cf_split_point(L,H,X,M) :- M is L/2 + H/2, \+ X = M. % midpoint, not a solution 247cf_split_point(L,H,X,M) :- M is 0.625*L + 0.375*H, \+ X = M. % 0.375*width, not a solution 248cf_split_point(L,H,_,M) :- M is 0.375*L + 0.625*H. % 0.625*width, may be a solution
== or =:=) constraints Constraints; otherwise fails. Example:
?- taylor_contractor({X**4-4*X**3+4*X**2-4*X+3==0},T).
T = cf_contractor([X], [_A]),
X::real(-1.509169756145379, 4.18727500493995),
_A::real(-1.0Inf, 1.0Inf).
Use the contractor with cf_solve to search for solutions, as in:
?- X::real,taylor_contractor({X**4-4*X**3+4*X**2-4*X+3==0},T), cf_solve(T).
T = cf_contractor([X], [_A]),
X:: 1.000000000...,
_A::real(-1.0Inf, 1.0Inf) ;
T = cf_contractor([X], [_A]),
X:: 3.00000000...,
_A::real(-1.0Inf, 1.0Inf) ;
false.
Multiple equality constraints are supported, as in this example of the Broyden banded problem (N=2):
?- taylor_contractor({2*X1+5*X1**3+1==X2*(1+X2), 2*X2+5*X2**3+1==X1*(1+X1)},T), cf_solve(T).
T = cf_contractor([X2, X1], [_A, _B]),
X1:: -0.42730462...,
X2:: -0.42730462...,
_B::real(-1.0Inf, 1.0Inf),
_A::real(-1.0Inf, 1.0Inf) ;
false.
Centre form contractors can converge faster than the general purpose builtin fixed point iteration provided by solve/1.
285% 286% build a cf_contractor for a multivariate expression based on Taylor expansion 287% 288taylor_contractor({E1=:=E2},CF) :- 289 taylor_contractor({E1==E2},CF). 290taylor_contractor({E1==E2},cf_contractor(Xs,As)) :- 291 Exp=E1-E2, 292 term_variables(Exp,Xs), % original arguments, bound to TXs on call 293 make_EQ_(Exp,TEQ), % original constraint with arguments 294 % build constraint list starting with Z's and ending with TEQ and DEQ () 295 T::real(0,1), 296 make_As_and_Zs_(Xs,T,As,Zs,Cs,[TEQ,DEQ]), % T per Z 297 % now build Taylor constraint, DEQ 298 copy_term_nat(Exp,AExp), % copy of original constraint with As 299 term_variables(AExp,As), 300 sum_diffs(Xs, As, Zs, Zs, Exp, AExp, DEQ), % add on D(Z)'s' 301 % make any vars in original equation and contractor arguments finite real intervals 302 !, 303 Xs::real, % all vars are intervals 304 {Cs}. % apply constraints 305taylor_contractor({Es},CF) :- 306 taylor_merged_contractor({Es},CF), % list or sequence 307 !. 308taylor_contractor(Eq,_) :- 309 fail_msg_('Invalid constraint for Taylor contractor: ~w',[Eq]). 310 311make_As_and_Zs_([],_,[],[],Tail,Tail). 312make_As_and_Zs_([X|Xs],T,[A|As],[Z|Zs],[Z==A+T*(X-A)|CZs],Tail) :- 313 make_As_and_Zs_(Xs,T,As,Zs,CZs,Tail). 314 315sum_diffs([], [], [], _AllZs, _Exp, ExpIn, EQ) :- make_EQ_(ExpIn,EQ). 316sum_diffs([X|Xs], [A|As], [Z|Zs], AllZs, Exp, AExp, DEQ) :- 317 copy_term_nat(Exp,NExp), % copy expression and replace Xs by Zs 318 term_variables(NExp,AllZs), 319 partial_derivative(NExp,Z,DZ), % differentiate wrt. Z and add to generated expression 320 sum_diffs(Xs, As, Zs, AllZs, Exp, AExp+DZ*(X-A), DEQ). 321 322% map expression Exp to an equation equivalent to Exp==0 with numeric RHS 323make_EQ_(Exp,LHS==RHS) :- % turn expression into equation equivalent to Exp==0. 324 make_EQ_(Exp,LHS,RHS). 325 326make_EQ_(E,E,0) :- var(E), !. 327make_EQ_(X+Y,X,SY) :- number(Y), !, SY is -Y. 328make_EQ_(X-Y,X,Y) :- number(Y), !. 329make_EQ_(X+Y,Y,SX) :- number(X), !, SX is -X. 330make_EQ_(X-Y,SY,SX) :- number(X), !, SX is -X, negate_sum_(Y,SY). 331make_EQ_(X+Y,LHS+Y,RHS) :- !, make_EQ_(X,LHS,RHS). 332make_EQ_(X-Y,LHS-Y,RHS) :- !, make_EQ_(X,LHS,RHS). 333make_EQ_(E,E,0). % default (non +/- subexpression) 334 335negate_sum_(Y,-Y) :- var(Y), !. 336negate_sum_(X+Y,NX-Y) :- !, negate_sum_(X,NX). 337negate_sum_(X-Y,NX+Y) :- !, negate_sum_(X,NX). 338negate_sum_(E,-E).
== or =:=) constraint in Constraints; otherwise fails.
347% 348% build a cf_contractor by merging a list of cf_contractor's 349% 350taylor_merged_contractor({Es},T) :- 351 cf_list(Es,Ts), 352 cf_merge(Ts,T). 353 354cf_list([],[]) :- !. 355cf_list([C|Cs],[CF|CFs]) :- !, 356 cf_list(C, CF), 357 cf_list(Cs,CFs). 358cf_list((C,Cs),[CF|CFs]) :- !, 359 cf_list(C, CF), 360 cf_list(Cs,CFs). 361cf_list(C,CF) :- 362 taylor_contractor({C},CF). 363 364cf_merge(CFs,CF) :- cf_merge(CFs,cf_contractor([],[]),CF). 365 366cf_merge([],CF,CF). 367cf_merge([CF|CFs],CFIn,CFOut) :- 368 cf_merge(CF,CFIn,CFNxt), 369 cf_merge(CFs,CFNxt,CFOut). 370cf_merge(cf_contractor(Xs,As),cf_contractor(XsIn,AsIn),cf_contractor(XsOut,AsOut)) :- 371 cf_add(Xs,As,XsIn,AsIn,XsOut,AsOut). 372 373cf_add([],[],Xs,As,Xs,As). 374cf_add([X|Xs],[A|As],XsIn,AsIn,XsOut,AsOut) :- 375 var_existing(XsIn,AsIn,X,A), !, 376 cf_add(Xs,As,XsIn,AsIn,XsOut,AsOut). 377cf_add([X|Xs],[A|As],XsIn,AsIn,XsOut,AsOut) :- 378 cf_add(Xs,As,[X|XsIn],[A|AsIn],XsOut,AsOut). 379 380var_existing([Xex|Xs],[Aex|As], X,A) :- Xex==X -> Aex=A ; var_existing(Xs,As,X,A).
integrate/4 with default precision.
*/The number of integration steps (= 2**P) is determined by the precision parameter P (default is value of environment flag clpBNR_default_precision). Example of use with increasing precision values:
?- X::real(0.0,1.0), F=X**2, between(2,10,P),integrate(F,X,RV,P), range(RV,R), format('~w:~w\n',[P,R]), fail.
2:[0.328125,0.359375]
3:[0.33203125,0.33984375]
4:[0.3330078125,0.3349609375]
5:[0.333251953125,0.333740234375]
6:[0.33331298828125,0.33343505859375]
7:[0.3333282470703125,0.3333587646484375]
8:[0.3333320617675781,0.3333396911621094]
9:[0.33333301544189453,0.33333492279052734]
10:[0.33333325386047363,0.33333373069763184]
false.
*/
408% integrate(F,X,R) where X is an interval over which to integrate and F = f(X) 409% Note that integrate(F,X,R) is equivalent to boundary_values(X,[dV(_,F,0,R]) 410integrate(F,X,R) :- 411 current_prolog_flag(clpBNR_default_precision,P), 412 integrate(F,X,R,P,_). 413integrate(F,X,R,P) :- 414 integrate(F,X,R,P,_). 415integrate(F,X,R,P,Steps) :- % internal arity 5 for development 416 compound(F), % F must be an expression in X 417 interval(X), % X must be an interval 418 integer(P), P>0, % P must be positive integer 419 !, % args OK, commit 420 boundary_values(X,[dV(_,F,0,R)],P,Steps). % use integration in `boundary_values` 421integrate(F,X,R,P,_Steps) :- 422 fail_msg_('Invalid argument(s): ~w',[integrate(F,X,R,P)]).
boundary_values/4 with default precision and discarding steps list.
*/boundary_values/4 discarding steps list.
*/dV(Y, Fxy, Yi, Yf). The initial and final values of the independent variable for the purposes of the boundary value problem are specified by the lower and upper values of the domain of X. The arguments of for dvar/4
are:
The optional third argument P defines a precision value, a positive integer (default = environment flag clpBNR_default_precision), which controls the the numerical integration; larger P means smaller step size.
The arity 4 version has an additional (final) argument which is unified with a list of the step values generated by the integration; each value is a tuple of the form (X,Ys).
This predicate fails if any of the arguments are invalid (generates an error message if clpBNR debug topic is enabled) or if a solution to the boundary value problem cannot be found. Examples for X in the range 0..1 and derivative of Y = -2*X*Y:
?- X::real(0,1), boundary_values(X,[dV(Y, -2*X*Y,1,Yf)]). X::real(0, 1), Yf:: 0.368... . ?- X::real(0,1),boundary_values(X,[dV(Y, -2*X*Y,1,Yf)],9). X::real(0, 1), Yf:: 0.36788... . ?- X::real(0,1), boundary_values(X,[dV(Y, -2*X*Y,Yi,1/e)]). X::real(0, 1), Yi:: 1.00... . ?- debug(clpBNR). true. ?- X=42, boundary_values(X,[dV(Y, -2*X*Y,Yi,1/e)]). % Invalid argument(s): boundary_values(42,[dV(_10836,-2*42*_10836,_10850,1/e)],6) false.
As with any application requiring numerical integration, care must be taken to avoid instability problems (more discussion in A Guide to CLP(BNR). */
472boundary_values(X,YDefs) :- 473 current_prolog_flag(clpBNR_default_precision,P), 474 boundary_values_(X,YDefs,P,_). 475 476boundary_values(X,YDefs,P) :- 477 boundary_values_(X,YDefs,P,_). 478 479boundary_values(X,YDefs,P,Steps) :- 480 boundary_values_(X,YDefs,P,Steps/_). 481 482boundary_values_(X,YDefs,P,Out/[(Xf,Yfs)]) :- 483 integer(P), P>0, % P must be positive integer 484 domain(X,Xdom), Xdom =.. [_Type,Xi,Xf], % X must be an interval 485 eval_dvars(YDefs,Ys,Yis,Yfs,Ydoms), % too many args for maplist 486 maplist(fXY_lambda(X,Ys),YDefs,Fxys), % list of partial derivative lambda args 487 (maplist(total_derivative_(Fxys),Fxys,DFxys) % list of connective derivative lambda args 488 -> true 489 ; DFxys = none % F non-differentiable(?), use euler step 490 ), 491 !, % args all good, commit 492 integrate_(P,Fxys,DFxys,(Xi,Yis),(Xf,Yfs),Ydoms,Out/[(Xf,Yfs)]). 493boundary_values_(X,YDefs,P,_) :- 494 fail_msg_('Invalid argument(s): ~w',[boundary_values(X,YDefs,P)]). 495 496eval_dvars([],[],[],[],[]). 497eval_dvars([dV(Y, _PD, Lexp, Uexp)|YDefs],[Y|Ys],[Yi|Yis],[Yf|Yfs],[Ydom|Ydoms]) :- 498 (domain(Y,Ydom) -> true ; Ydom = real), % Ydom defaults to real 499 (var(Lexp) -> Yi=Lexp, Yi::Ydom ; Yi is Lexp), 500 (var(Uexp) -> Yf=Uexp, Yf::Ydom ; Yf is Uexp), 501 eval_dvars(YDefs,Ys,Yis,Yfs,Ydoms). 502 503% construct Lambda args for Fxy 504fXY_lambda(X,Ys,dV(_Y,Fxy,_,_),FxyArgs) :- 505 lambda_for_(Fxy,X,Ys,FxyArgs). 506 507% construct Lambda args for derivative function of Fxy from Lambda of Fxy 508 509total_derivative_(Fxys,_Free/Ps,DxyArgs) :- !, % ignore free variables 510 total_derivative_(Fxys,Ps,DxyArgs). 511total_derivative_(Fxys,[X,Ys,Fxy],DxyArgs) :- 512 partial_derivative(Fxy,X,DFDX), % clpBNR built-in 513 sumYpartials(Fxys,Ys,Fxy,0,DYsum), 514 simplify_sum_(DFDX, DYsum, DExp), !, 515 lambda_for_(DExp,X,Ys,DxyArgs). 516 517sumYpartials([],[],_Fxy,Acc,Acc). 518sumYpartials([_Free/FxyI|FxyIs],YIs,Fxy,Acc,Sum) :- !, 519 sumYpartials([FxyI|FxyIs],YIs,Fxy,Acc,Sum). 520sumYpartials([[_X,_Ys,FxyI]|FxyIs],[YI|YIs],Fxy,Acc,Sum) :- 521 partial_derivative(Fxy,YI,DFDYI), 522 (number(DFDYI), DFDYI =:= 0 -> NxtAcc = Acc ; simplify_sum_(Acc,FxyI*DFDYI,NxtAcc)), 523 !, 524 sumYpartials(FxyIs,YIs,Fxy,NxtAcc,Sum). 525 526simplify_sum_(X,Y,Y) :- number(X),X=:=0. 527simplify_sum_(X,Y,X) :- number(Y),Y=:=0. 528simplify_sum_(X,Y,X+Y). 529 530% construct args for Lambda expression 531lambda_for_(Fxy,X,Ys,Args) :- 532 Lambda_parms = [X,Ys,Fxy], 533 term_variables(Fxy,FVs), 534 exclude(var_member_([X|Ys]),FVs,EVs), % EVs = free variables 535 (comma_op_(EVs,EV) -> Args = {EV}/Lambda_parms ; Args = Lambda_parms). 536 537var_member_([L|Ls],E) :- L==E -> true ; var_member_(Ls,E). 538 539comma_op_([X],X). % assumes use in if-then 540comma_op_([X|Xs],(X,Y)) :- comma_op_(Xs,Y). 541 542% integration loop 543integrate_(0, Fxys, Dxys, Initial, Final, Ydomains, [Initial|Ps]/Ps) :- !, 544 % select integration step 545 ( Dxys == none 546 -> step_euler(Fxys, Initial, Final, Ydomains) 547 ; step_trap(Fxys, Dxys, Initial, Final, Ydomains) 548 ). 549integrate_(P, Fxys, Dxys, Initial, Final, Ydomains, L/E) :- 550 % create interpolation point and integrate two halves 551 interpolate_(Initial, Final, Ydomains, Middle), 552 Pn is P - 1, 553 integrate_(Pn, Fxys, Dxys, Initial, Middle, Ydomains, L/M), 554 integrate_(Pn, Fxys, Dxys, Middle, Final, Ydomains, M/E). 555 556interpolate_((X0,_Y0s), (X1,_Y1s), Ydomains, (XM,YMs)) :- 557 XM is (X0 + X1)/2, % XM is midpoint of (X0,X1) 558 maplist(::,YMs,Ydomains). % corresponding YMs 559 560step_euler(Fxys, (X0,Y0), (X1,Y1), Ydoms) :- 561 X01:: real(X0,X1), % range of X in step 562 maplist(lambda_constrain_(X01,Y01),Fxys,Fs), % approx f' over X0 563 Dx is X1 - X0, % assumed (X1>X0) 564 DX :: real(0,Dx), % range for estimate 565 euler_constraints(Y0,Y1,Y01,Ydoms,Dx,DX,Fs,In/In,Cs/[]), % flatten with diff list 566 {Cs}. 567 568step_trap(Fxys, Dxys, (X0,Y0), (X1,Y1), Ydoms) :- 569 X01:: real(X0,X1), % range of X in step 570 maplist(lambda_constrain_(X0,Y0),Fxys,F0s), % F0s = slopes at X0 571 maplist(lambda_constrain_(X1,Y1),Fxys,F1s), % F1s = slopes at X1 572 maplist(lambda_constrain_(X01,Y01),Dxys,Ds), % approx f' over X0 573 Dx is X1 - X0, % assumed (X1>X0) 574 DX :: real(0,Dx), % range for estimate 575 trap_constraints(Y0,Y1,Y01,Ydoms,Dx,DX,F0s,F1s,Ds,In/In,Cs/[]), % flatten with diff list 576 {Cs}. %%, absolve(Y1,2). 577 578lambda_constrain_(X,Ys,Args,F) :- % reorder args for yall: >> 579 yall: >>(Args,true,X,Ys,F). % avoid meta-call (basically just makes copy) 580% known safe since lambda Goal=true 581sandbox:safe_primitive(clpBNR_toolkit:lambda_constrain_(_X,_Ys,_Args,_F)). 582 583/* see Carleton notes: 584https://www.softwarepreservation.org/projects/prolog/bnr/doc/Older-Introduction_to_CLP%28BNR%29-1995.pdf/view 585*/ 586euler_constraints([],[],[],[],_Dx,_DX,[],In,In). 587euler_constraints([Y0|Y0s],[Y1|Y1s],[Y01|Y01s],[Ydom|Ydoms],Dx,DX,[F|Fs], 588 In/[FM <= F, % FM = slope inclusion 589 Y01 - Y0 is DX*FM, 590 Y1 - Y0 is Dx*FM 591 |Cs], 592 Out) :- 593 Y01:: Ydom, 594 euler_constraints(Y0s,Y1s,Y01s,Ydoms,Dx,DX,Fs,In/Cs,Out). 595 596trap_constraints([],[],[],[],_Dx,_DX,[],[],[],In,In). 597trap_constraints([Y0|Y0s],[Y1|Y1s],[Y01|Y01s],[Ydom|Ydoms],Dx,DX,[F0|F0s],[F1|F1s],[D|Ds], 598 % use `is` to circumvent `simplify` 599 In/[FM <= (F0+F1)/2, % FM = average slope using step endpoints (one-way) % Note that the following must not be symbolically simplified to eliminate D 8*DR is D - D, % 4*deltaR == (D-D)/2 (for error term) Y01 - Y0 is DX*(FM + DR*DX), Y1 - Y0 is Dx*(FM + DR*Dx) |Cs], Out) :- Y01:: Ydom, DR::real, trap_constraints(Y0s,Y1s,Y01s,Ydoms,Dx,DX,F0s,F1s,Ds,In/Cs,Out).
X*C (or C*X) are permitted since the actual computation is done using library(simplex). Narrowing of minimizers (variables in ObjF) is limited to that constrained by the Min result to accomodate multiple sets of minimizers. (See lin_minimize/3 to use minimizers used to derive Min.) A solution generator, e.g., clpBNR:solve/1 can be used to search for alternative sets of minimizers. "Universal Mines" example from the User Guide:
?- [M_Idays,M_IIdays,M_IIIdays]::integer(0,7),
lin_minimum(20*M_Idays+22*M_IIdays+18*M_IIIdays,
{4*M_Idays+6*M_IIdays+M_IIIdays>=54,4*M_Idays+4*M_IIdays+6*M_IIIdays>=65}, Min).
Min = 284,
M_Idays::integer(2, 7),
M_IIdays::integer(4, 7),
M_IIIdays::integer(2, 7).
?- [M_Idays,M_IIdays,M_IIIdays]::integer(0,7),
lin_minimum(20*M_Idays+22*M_IIdays+18*M_IIIdays,
{4*M_Idays+6*M_IIdays+M_IIIdays>=54,4*M_Idays+4*M_IIdays+6*M_IIIdays>=65}, Min),
solve([M_Idays,M_IIdays,M_IIIdays]).
M_Idays = 2,
M_IIdays = 7,
M_IIIdays = 5,
Min = 284 ;
false.
For linear systems, lin_minimum/3, lin_maximum/3 can be significantly faster than using the more general purpose clpBNR:global_minimum/3, clpBNR:global_maximum/3
lin_minimum/3 for finding global maxima.
644lin_minimum(ObjF,{Constraints},MinValue) :- 645 lin_minimum_(ObjF,{Constraints},MinValue,false). 646 647lin_maximum(ObjF,{Constraints},MinValue) :- 648 lin_maximum_(ObjF,{Constraints},MinValue,false).
lin_minimum/3 except variables in ObjF will be narrowed to the values used in calculating the final value of Min. Any other sets of minimizers corresponding to Min are removed from the solution space. "Universal Mines" example from the User Guide:
?- [M_Idays,M_IIdays,M_IIIdays]::integer(0,7),
lin_minimize(20*M_Idays+22*M_IIdays+18*M_IIIdays,
{4*M_Idays+6*M_IIdays+M_IIIdays>=54,4*M_Idays+4*M_IIdays+6*M_IIIdays>=65}, Min).
M_Idays = 2,
M_IIdays = 7,
M_IIIdays = 5,
Min = 284.
lin_maximum/3 except variables in ObjF will be narrowed to the values used in calculating the final value of Max. Any other sets of minimizers corresponding to Min are removed from the solution space.
671lin_minimize(ObjF,{Constraints},MinValue) :- 672 lin_minimum_(ObjF,{Constraints},MinValue,true). 673 674lin_maximize(ObjF,{Constraints},MinValue) :- 675 lin_maximum_(ObjF,{Constraints},MinValue,true). 676 677 678lin_minimum_(ObjF,{Constraints},MinValue,BindVars) :- 679 init_simplex_(ObjF,Constraints,Objective,S0,Vs), 680 (minimize(Objective,S0,S) 681 -> objective(S,MinValue), {ObjF == MinValue}, 682 (BindVars == true 683 -> bind_vars_(Vs,S) 684 ; remove_names_(Vs), 685 {Constraints} % apply constraints 686 ) 687 ; fail_msg_('Failed to minimize: ~w',[ObjF]) 688 ). 689 690lin_maximum_(ObjF,{Constraints},MaxValue,BindVars) :- 691 init_simplex_(ObjF,Constraints,Objective,S0,Vs), 692 (maximize(Objective,S0,S) 693 -> objective(S,MaxValue), {ObjF == MaxValue}, 694 (BindVars == true 695 -> bind_vars_(Vs,S) 696 ; remove_names_(Vs), 697 {Constraints} % apply constraints 698 ) 699 ; fail_msg_('Failed to maximize: ~w',[ObjF]) 700 ). 701 702init_simplex_(ObjF,Constraints,Objective,S,Vs) :- 703 gen_state(S0), 704 term_variables((ObjF,Constraints),Vs), 705 (Vs = [] 706 -> fail_msg_('No variables in expression to optimize: ~w',[ObjF]) 707 ; sim_constraints_(Constraints,S0,S1), 708 _::real(_,Max), % max value to constrain for simplex 709 constrain_ints_(Vs,Max,S1,S), 710 (map_simplex_(ObjF,T/T,Objective/[]) 711 -> true 712 ; fail_msg_('Illegal linear objective: ~w',[ObjF]) 713 ) 714 ). 715 716% use an attribute to associate a var with a unique simplex variable name 717simplex_var_(V,SV) :- var(V), 718 (get_attr(V,clpBNR_toolkit,SV) 719 -> true 720 ; statistics(inferences,VID), SV = var(VID), put_attr(V,clpBNR_toolkit,SV) 721 ). 722 723% Name attribute removed before exit. 724remove_names_([]). 725remove_names_([V|Vs]) :- 726 del_attr(V,clpBNR_toolkit), 727 remove_names_(Vs). 728 729attr_unify_hook(var(_),_). % unification always does nothing and succeeds 730 731constrain_ints_([],_,S,S). 732constrain_ints_([V|Vs],Max,Sin,Sout) :- 733 % Note: library(simplex) currently constrains all variables to be non-negative 734 simplex_var_(V,SV), % corresponding simplex variable name 735 % if not already an interval, make it one with finite non-negative value 736 (domain(V,D) -> true ; V::real(0,_), domain(V,D)), 737 (D == boolean -> Dom = integer(0,1); Dom = D), 738 Dom =.. [Type,L,H], 739 (Type == integer -> constraint(integral(SV),Sin,S1) ; S1 = Sin), 740 (L < 0 741 -> % apply non-negativity condition 742 ({V >= 0} -> L1 = 0 ; fail_msg_('Negative vars not supported by \'simplex\': ~w',[V])) 743 ; L1 = L 744 ), 745 % explicitly constrain any vars not (0,Max-eps) 746 (L1 > 0 -> constraint([SV] >= L1,S1,S2) ; S2 = S1), % L1 not negative 747 (H < Max -> constraint([SV] =< H,S2,SNxt) ; SNxt = S2), 748 constrain_ints_(Vs,Max,SNxt,Sout). 749 750bind_vars_([],_). 751bind_vars_([V|Vs],S) :- 752 % Note: skip anything nonvar (already bound due to active constraints) 753 (simplex_var_(V,SV) -> variable_value(S,SV,V) ; true), 754 bind_vars_(Vs,S). 755 756% clpBNR constraints have already been applied so worst errors have been detected 757sim_constraints_([],S,S) :- !. 758sim_constraints_([C|Cs],Sin,Sout) :- !, 759 sim_constraints_(C, Sin,Snxt), 760 sim_constraints_(Cs,Snxt,Sout). 761sim_constraints_((C,Cs),Sin,Sout) :- !, 762 sim_constraints_(C, Sin,Snxt), 763 sim_constraints_(Cs,Snxt,Sout). 764sim_constraints_(C,Sin,Sout) :- 765 sim_constraint_(C,SC), 766 constraint(SC,Sin,Sout). % from simplex 767 768sim_constraint_(C,SC) :- 769 C=..[Op,LHS,RHS], % decompose 770 constraint_op(Op,COp), % acceptable to simplex 771 number(RHS), RHS >= 0, % requirement of simplex 772 map_simplex_(LHS,T/T,Sim/[]), % map to simplex list of terms 773 !, 774 SC=..[COp,Sim,RHS]. % recompose 775sim_constraint_(C,_) :- 776 fail_msg_('Illegal linear constraint: ~w',[C]). 777 778map_simplex_(T,CT/[S|Tail],CT/Tail) :- 779 map_simplex_term_(T,S), 780 !. 781map_simplex_(A+B,Cin,Cout) :- !, 782 map_simplex_(A,Cin,Cnxt), 783 map_simplex_(B,Cnxt,Cout). 784map_simplex_(A-B,Cin,Cout) :- !, 785 map_simplex_(A,Cin,Cnxt), 786 map_simplex_(-B,Cnxt,Cout). 787 788map_simplex_term_(V,1*SV) :- simplex_var_(V,SV), !. 789map_simplex_term_(-T,NN*V) :- !, 790 map_simplex_term_(T,N*V), 791 NN is -N. 792map_simplex_term_(N*V,N*SV) :- number(N), simplex_var_(V,SV), !. 793map_simplex_term_(V*N,N*SV) :- number(N), simplex_var_(V,SV). 794 795constraint_op(==,=). 796constraint_op(=<,=<). 797constraint_op(>=,>=).
local_minima should be executed prior to a call to clpBNR:global_minimum using the same objective function, e.g.,
?- X::real(0,10), OF=X**3-6*X**2+9*X+6, local_minima(OF), global_minimum(OF,Z). OF = X**3-6*X**2+9*X+6, X:: 3.00000000000000..., Z:: 6.000000000000... .
Using any local optima predicate can significantly improve performance compared to searching for global optima (clpBNR:global_*) without local constraints.
local_maxima should be executed prior to a call to clpBNR:global_maximum using the same objective function, e.g.,
?- X::real(0,10), OF=X**3-6*X**2+9*X+6, local_maxima(OF), global_maximum(OF,Z). OF = X**3-6*X**2+9*X+6, X:: 1.000000000000000..., Z:: 10.0000000000000... .
825% 826% local_minima/1, % apply KT constraints for objective function expression (OFE) 827% local_maxima/1, % semantically equivalent to local_minima/1 828% 829local_minima(ObjExp) :- 830 local_optima_(min,ObjExp,[]). 831 832local_maxima(ObjExp) :- 833 local_optima_(max,ObjExp,[]).
local_minima should be executed prior to a call to clpBNR:global_minimum using the same objective function, e.g.,
?- [X1,X2]::real, OF=X1**4*exp(-0.01*(X1*X2)**2),
local_minima(OF,{2*X1**2+X2**2==10}), global_minimum(OF,Z), solve([X1,X2]).
OF = X1**4*exp(-0.01*(X1*X2)**2),
X1::real(-1.703183936003284e-108, 1.703183936003284e-108),
X2:: -3.16227766016838...,
Z:: 0.0000000000000000... ;
OF = X1**4*exp(-0.01*(X1*X2)**2),
X1::real(-1.703183936003284e-108, 1.703183936003284e-108),
X2:: 3.16227766016838...,
Z:: 0.0000000000000000... .
local_maxima should be executed prior to a call to clpBNR:global_maximum using the same objective function, e.g.,
?- [X1,X2]::real,OF=X1**4*exp(-0.01*(X1*X2)**2),
local_maxima(OF,{2*X1**2+X2**2==10}), global_maximum(OF,Z),solve([X1,X2]).
OF = X1**4*exp(-0.01*(X1*X2)**2),
X1:: -2.23606797749979...,
X2:: 0.0000000000000000...,
Z:: 25.0000000000000... ;
OF = X1**4*exp(-0.01*(X1*X2)**2),
X1:: 2.23606797749979...,
X2:: 0.0000000000000000...,
Z:: 25.0000000000000... .
872% 873% local_minima/2, % apply KT constraints for minima with constraints 874% local_maxima/2 % apply KT constraints for maxima with constraints 875% 876local_minima(ObjExp,{Constraints}) :- 877 local_optima_(min,ObjExp,Constraints). 878 879local_maxima(ObjExp,{Constraints}) :- 880 local_optima_(max,ObjExp,Constraints). 881 882 883local_optima_(MinMax,ObjF,Constraints) :- 884 local_optima_(MinMax,ObjF,Constraints,Cs), % generate constraints 885 {Cs}. % then apply 886 887local_optima_(MinMax,ObjF,Constraints,[Constraints,GCs,LCs]) :- 888 lagrangian_(Constraints,MinMax,ObjF,LObjF,LCs), 889 term_variables((Constraints,ObjF),Vs), 890 gradient_constraints_(Vs,GCs,LObjF). 891 892gradient_constraints_([],[],_Exp). 893gradient_constraints_([X|Xs],[C|Cs],Exp) :- 894 partial_derivative(Exp,X,D), 895 (number(D) -> C=[] ; C=(D==0)), % no constraint if PD is a constant 896 gradient_constraints_(Xs,Cs,Exp). 897 898% for each constraint add to Lagrangian expression with auxiliary KKT constraints 899lagrangian_(C,MinMax,Exp,LExp, LC) :- nonvar(C), 900 kt_constraint_(C,CExp, LC), % generate langrange term with multiplier 901 lexp(MinMax,Exp,CExp,LExp), 902 !. 903lagrangian_([],_,L,L,[]). 904lagrangian_([C|Cs],MinMax,Exp,LExp,[LC|LCs]) :- 905 lagrangian_(C, MinMax,Exp,NExp,LC), 906 lagrangian_(Cs,MinMax,NExp,LExp,LCs). 907lagrangian_((C,Cs),MinMax,Exp,LExp,[LC|LCs]) :- 908 lagrangian_(C,MinMax,Exp,NExp,LC), 909 lagrangian_(Cs,MinMax,NExp,LExp,LCs). 910 911lexp(min,Exp,CExp,Exp+CExp). 912lexp(max,Exp,CExp,Exp-CExp). 913 914kt_constraint_(LHS == RHS, M*(LHS-RHS), []) :- 915 M::real. % finite multiplier only 916kt_constraint_(LHS =< RHS, MGx, MGx==0) :- 917 MGx = M*(LHS-RHS), M::real(0,_). % positive multiplier and KKT non-negativity condition 918kt_constraint_(LHS >= RHS, Exp, LC) :- 919 kt_constraint_(RHS =< LHS, Exp, LC). % >= convert to =<
clpBNR_toolkit: Toolkit of various utilities used for solving problems with clpBNR
CLP(BNR) (
library(clpBNR)) is a CLP over the domain of real numbers extended with ±∞. This module contains a number of useful utilities for specific problem domains like the optimization of linear systems, enforcing local optima conditions, and constructing centre form contractors to improve performance (e.g., Taylor extensions of constraints). For more detailed discussion, see A Guide to CLP(BNR) (HTML version included with this pack in directorydocs/).Documentation for exported predicates follows. The "custom" types include:
clpBNRattribute