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 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 ; mid_split(X), % yes, split it 234 Count1 is Count-1, 235 cf_iterate_(Count1,Xs,As,P) % and iterate 236 ). 237cf_iterate_(_,_,_,_). % done (Count=<0 or all small Xs)
== 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.
274% 275% build a cf_contractor for a multivariate expression based on Taylor expansion 276% 277taylor_contractor({E1=:=E2},CF) :- 278 taylor_contractor({E1==E2},CF). 279taylor_contractor({E1==E2},cf_contractor(Xs,As)) :- 280 Exp=E1-E2, 281 term_variables(Exp,Xs), % original arguments, bound to TXs on call 282 make_EQ_(Exp,TEQ), % original constraint with arguments 283 % build constraint list starting with Z's and ending with TEQ and DEQ () 284 T::real(0,1), 285 make_As_and_Zs_(Xs,T,As,Zs,Cs,[TEQ,DEQ]), % T per Z 286 % now build Taylor constraint, DEQ 287 copy_term_nat(Exp,AExp), % copy of original constraint with As 288 term_variables(AExp,As), 289 sum_diffs(Xs, As, Zs, Zs, Exp, AExp, DEQ), % add on D(Z)'s' 290 % make any vars in original equation and contractor arguments finite real intervals 291 !, 292 Xs::real, % all vars are intervals 293 {Cs}. % apply constraints 294taylor_contractor({Es},CF) :- 295 taylor_merged_contractor({Es},CF), % list or sequence 296 !. 297taylor_contractor(Eq,_) :- 298 fail_msg_('Invalid constraint for Taylor contractor: ~w',[Eq]). 299 300make_As_and_Zs_([],_,[],[],Tail,Tail). 301make_As_and_Zs_([X|Xs],T,[A|As],[Z|Zs],[Z==A+T*(X-A)|CZs],Tail) :- 302 make_As_and_Zs_(Xs,T,As,Zs,CZs,Tail). 303 304sum_diffs([], [], [], _AllZs, _Exp, ExpIn, EQ) :- make_EQ_(ExpIn,EQ). 305sum_diffs([X|Xs], [A|As], [Z|Zs], AllZs, Exp, AExp, DEQ) :- 306 copy_term_nat(Exp,NExp), % copy expression and replace Xs by Zs 307 term_variables(NExp,AllZs), 308 partial_derivative(NExp,Z,DZ), % differentiate wrt. Z and add to generated expression 309 sum_diffs(Xs, As, Zs, AllZs, Exp, AExp+DZ*(X-A), DEQ). 310 311% map expression Exp to an equation equivalent to Exp==0 with numeric RHS 312make_EQ_(Exp,LHS==RHS) :- % turn expression into equation equivalent to Exp==0. 313 make_EQ_(Exp,LHS,RHS). 314 315make_EQ_(E,E,0) :- var(E), !. 316make_EQ_(X+Y,X,SY) :- number(Y), !, SY is -Y. 317make_EQ_(X-Y,X,Y) :- number(Y), !. 318make_EQ_(X+Y,Y,SX) :- number(X), !, SX is -X. 319make_EQ_(X-Y,SY,SX) :- number(X), !, SX is -X, negate_sum_(Y,SY). 320make_EQ_(X+Y,LHS+Y,RHS) :- !, make_EQ_(X,LHS,RHS). 321make_EQ_(X-Y,LHS-Y,RHS) :- !, make_EQ_(X,LHS,RHS). 322make_EQ_(E,E,0). % default (non +/- subexpression) 323 324negate_sum_(Y,-Y) :- var(Y), !. 325negate_sum_(X+Y,NX-Y) :- !, negate_sum_(X,NX). 326negate_sum_(X-Y,NX+Y) :- !, negate_sum_(X,NX). 327negate_sum_(E,-E).
== or =:=) constraint in Constraints; otherwise fails.
336% 337% build a cf_contractor by merging a list of cf_contractor's 338% 339taylor_merged_contractor({Es},T) :- 340 cf_list(Es,Ts), 341 cf_merge(Ts,T). 342 343cf_list([],[]) :- !. 344cf_list([C|Cs],[CF|CFs]) :- !, 345 cf_list(C, CF), 346 cf_list(Cs,CFs). 347cf_list((C,Cs),[CF|CFs]) :- !, 348 cf_list(C, CF), 349 cf_list(Cs,CFs). 350cf_list(C,CF) :- 351 taylor_contractor({C},CF). 352 353cf_merge(CFs,CF) :- cf_merge(CFs,cf_contractor([],[]),CF). 354 355cf_merge([],CF,CF). 356cf_merge([CF|CFs],CFIn,CFOut) :- 357 cf_merge(CF,CFIn,CFNxt), 358 cf_merge(CFs,CFNxt,CFOut). 359cf_merge(cf_contractor(Xs,As),cf_contractor(XsIn,AsIn),cf_contractor(XsOut,AsOut)) :- 360 cf_add(Xs,As,XsIn,AsIn,XsOut,AsOut). 361 362cf_add([],[],Xs,As,Xs,As). 363cf_add([X|Xs],[A|As],XsIn,AsIn,XsOut,AsOut) :- 364 var_existing(XsIn,AsIn,X,A), !, 365 cf_add(Xs,As,XsIn,AsIn,XsOut,AsOut). 366cf_add([X|Xs],[A|As],XsIn,AsIn,XsOut,AsOut) :- 367 cf_add(Xs,As,[X|XsIn],[A|AsIn],XsOut,AsOut). 368 369var_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.
*/
397% integrate(F,X,R) where X is an interval over which to integrate and F = f(X) 398% Note that integrate(F,X,R) is equivalent to boundary_values(X,[dV(_,F,0,R]) 399integrate(F,X,R) :- 400 current_prolog_flag(clpBNR_default_precision,P), 401 integrate(F,X,R,P,_). 402integrate(F,X,R,P) :- 403 integrate(F,X,R,P,_). 404integrate(F,X,R,P,Steps) :- % internal arity 5 for development 405 compound(F), % F must be an expression in X 406 interval(X), % X must be an interval 407 integer(P), P>0, % P must be positive integer 408 !, % args OK, commit 409 boundary_values(X,[dV(_,F,0,R)],P,Steps). % use integration in `boundary_values` 410integrate(F,X,R,P,_Steps) :- 411 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). */
461boundary_values(X,YDefs) :- 462 current_prolog_flag(clpBNR_default_precision,P), 463 boundary_values_(X,YDefs,P,_). 464 465boundary_values(X,YDefs,P) :- 466 boundary_values_(X,YDefs,P,_). 467 468boundary_values(X,YDefs,P,Steps) :- 469 boundary_values_(X,YDefs,P,Steps/_). 470 471boundary_values_(X,YDefs,P,Out/[(Xf,Yfs)]) :- 472 integer(P), P>0, % P must be positive integer 473 domain(X,Xdom), Xdom =.. [_Type,Xi,Xf], % X must be an interval 474 eval_dvars(YDefs,Ys,Yis,Yfs,Ydoms), % too many args for maplist 475 maplist(fXY_lambda(X,Ys),YDefs,Fxys), % list of partial derivative lambda args 476 (maplist(total_derivative_(Fxys),Fxys,DFxys) % list of connective derivative lambda args 477 -> true 478 ; DFxys = none % F non-differentiable(?), use euler step 479 ), 480 !, % args all good, commit 481 integrate_(P,Fxys,DFxys,(Xi,Yis),(Xf,Yfs),Ydoms,Out/[(Xf,Yfs)]). 482boundary_values_(X,YDefs,P,_) :- 483 fail_msg_('Invalid argument(s): ~w',[boundary_values(X,YDefs,P)]). 484 485eval_dvars([],[],[],[],[]). 486eval_dvars([dV(Y, _PD, Lexp, Uexp)|YDefs],[Y|Ys],[Yi|Yis],[Yf|Yfs],[Ydom|Ydoms]) :- 487 (domain(Y,Ydom) -> true ; Ydom = real), % Ydom defaults to real 488 (var(Lexp) -> Yi=Lexp, Yi::Ydom ; Yi is Lexp), 489 (var(Uexp) -> Yf=Uexp, Yf::Ydom ; Yf is Uexp), 490 eval_dvars(YDefs,Ys,Yis,Yfs,Ydoms). 491 492% construct Lambda args for Fxy 493fXY_lambda(X,Ys,dV(_Y,Fxy,_,_),FxyArgs) :- 494 lambda_for_(Fxy,X,Ys,FxyArgs). 495 496% construct Lambda args for derivative function of Fxy from Lambda of Fxy 497 498total_derivative_(Fxys,_Free/Ps,DxyArgs) :- !, % ignore free variables 499 total_derivative_(Fxys,Ps,DxyArgs). 500total_derivative_(Fxys,[X,Ys,Fxy],DxyArgs) :- 501 partial_derivative(Fxy,X,DFDX), % clpBNR built-in 502 sumYpartials(Fxys,Ys,Fxy,0,DYsum), 503 simplify_sum_(DFDX, DYsum, DExp), !, 504 lambda_for_(DExp,X,Ys,DxyArgs). 505 506sumYpartials([],[],_Fxy,Acc,Acc). 507sumYpartials([_Free/FxyI|FxyIs],YIs,Fxy,Acc,Sum) :- !, 508 sumYpartials([FxyI|FxyIs],YIs,Fxy,Acc,Sum). 509sumYpartials([[_X,_Ys,FxyI]|FxyIs],[YI|YIs],Fxy,Acc,Sum) :- 510 partial_derivative(Fxy,YI,DFDYI), 511 (number(DFDYI), DFDYI =:= 0 -> NxtAcc = Acc ; simplify_sum_(Acc,FxyI*DFDYI,NxtAcc)), 512 !, 513 sumYpartials(FxyIs,YIs,Fxy,NxtAcc,Sum). 514 515simplify_sum_(X,Y,Y) :- number(X),X=:=0. 516simplify_sum_(X,Y,X) :- number(Y),Y=:=0. 517simplify_sum_(X,Y,X+Y). 518 519% construct args for Lambda expression 520lambda_for_(Fxy,X,Ys,Args) :- 521 Lambda_parms = [X,Ys,Fxy], 522 term_variables(Fxy,FVs), 523 exclude(var_member_([X|Ys]),FVs,EVs), % EVs = free variables 524 (comma_op_(EVs,EV) -> Args = {EV}/Lambda_parms ; Args = Lambda_parms). 525 526var_member_([L|Ls],E) :- L==E -> true ; var_member_(Ls,E). 527 528comma_op_([X],X). % assumes use in if-then 529comma_op_([X|Xs],(X,Y)) :- comma_op_(Xs,Y). 530 531% integration loop 532integrate_(0, Fxys, Dxys, Initial, Final, Ydomains, [Initial|Ps]/Ps) :- !, 533 % select integration step 534 ( Dxys == none 535 -> step_euler(Fxys, Initial, Final, Ydomains) 536 ; step_trap(Fxys, Dxys, Initial, Final, Ydomains) 537 ). 538integrate_(P, Fxys, Dxys, Initial, Final, Ydomains, L/E) :- 539 % create interpolation point and integrate two halves 540 interpolate_(Initial, Final, Ydomains, Middle), 541 Pn is P - 1, 542 integrate_(Pn, Fxys, Dxys, Initial, Middle, Ydomains, L/M), 543 integrate_(Pn, Fxys, Dxys, Middle, Final, Ydomains, M/E). 544 545interpolate_((X0,_Y0s), (X1,_Y1s), Ydomains, (XM,YMs)) :- 546 XM is (X0 + X1)/2, % XM is midpoint of (X0,X1) 547 maplist(::,YMs,Ydomains). % corresponding YMs 548 549step_euler(Fxys, (X0,Y0), (X1,Y1), Ydoms) :- 550 X01:: real(X0,X1), % range of X in step 551 maplist(lambda_constrain_(X01,Y01),Fxys,Fs), % approx f' over X0 552 Dx is X1 - X0, % assumed (X1>X0) 553 DX :: real(0,Dx), % range for estimate 554 euler_constraints(Y0,Y1,Y01,Ydoms,Dx,DX,Fs,In/In,Cs/[]), % flatten with diff list 555 {Cs}. 556 557step_trap(Fxys, Dxys, (X0,Y0), (X1,Y1), Ydoms) :- 558 X01:: real(X0,X1), % range of X in step 559 maplist(lambda_constrain_(X0,Y0),Fxys,F0s), % F0s = slopes at X0 560 maplist(lambda_constrain_(X1,Y1),Fxys,F1s), % F1s = slopes at X1 561 maplist(lambda_constrain_(X01,Y01),Dxys,Ds), % approx f' over X0 562 Dx is X1 - X0, % assumed (X1>X0) 563 DX :: real(0,Dx), % range for estimate 564 trap_constraints(Y0,Y1,Y01,Ydoms,Dx,DX,F0s,F1s,Ds,In/In,Cs/[]), % flatten with diff list 565 {Cs}. %%, absolve(Y1,2). 566 567lambda_constrain_(X,Ys,Args,F) :- % reorder args for yall: >> 568 yall: >>(Args,true,X,Ys,F). % avoid meta-call (basically just makes copy) 569% known safe since lambda Goal=true 570sandbox:safe_primitive(clpBNR_toolkit:lambda_constrain_(_X,_Ys,_Args,_F)). 571 572/* see Carleton notes: 573https://www.softwarepreservation.org/projects/prolog/bnr/doc/Older-Introduction_to_CLP%28BNR%29-1995.pdf/view 574*/ 575euler_constraints([],[],[],[],_Dx,_DX,[],In,In). 576euler_constraints([Y0|Y0s],[Y1|Y1s],[Y01|Y01s],[Ydom|Ydoms],Dx,DX,[F|Fs], 577 In/[FM <= F, % FM = slope inclusion 578 Y01 - Y0 is DX*FM, 579 Y1 - Y0 is Dx*FM 580 |Cs], 581 Out) :- 582 Y01:: Ydom, 583 euler_constraints(Y0s,Y1s,Y01s,Ydoms,Dx,DX,Fs,In/Cs,Out). 584 585trap_constraints([],[],[],[],_Dx,_DX,[],[],[],In,In). 586trap_constraints([Y0|Y0s],[Y1|Y1s],[Y01|Y01s],[Ydom|Ydoms],Dx,DX,[F0|F0s],[F1|F1s],[D|Ds], 587 % use `is` to circumvent `simplify` 588 In/[FM <= (F0+F1)/2, % FM = average slope using step endpoints (one-way) 589 % Note that the following must not be symbolically simplified to eliminate D 590 8*DR is D - D, % 4*deltaR == (D-D)/2 (for error term) 591 Y01 - Y0 is DX*(FM + DR*DX), 592 Y1 - Y0 is Dx*(FM + DR*Dx) 593 |Cs], 594 Out) :- 595 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.
633lin_minimum(ObjF,{Constraints},MinValue) :- 634 lin_minimum_(ObjF,{Constraints},MinValue,false). 635 636lin_maximum(ObjF,{Constraints},MinValue) :- 637 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.
660lin_minimize(ObjF,{Constraints},MinValue) :- 661 lin_minimum_(ObjF,{Constraints},MinValue,true). 662 663lin_maximize(ObjF,{Constraints},MinValue) :- 664 lin_maximum_(ObjF,{Constraints},MinValue,true). 665 666 667lin_minimum_(ObjF,{Constraints},MinValue,BindVars) :- 668 init_simplex_(ObjF,Constraints,Objective,S0,Vs), 669 (minimize(Objective,S0,S) 670 -> objective(S,MinValue), {ObjF == MinValue}, 671 (BindVars == true 672 -> bind_vars_(Vs,S) 673 ; remove_names_(Vs), 674 {Constraints} % apply constraints 675 ) 676 ; fail_msg_('Failed to minimize: ~w',[ObjF]) 677 ). 678 679lin_maximum_(ObjF,{Constraints},MaxValue,BindVars) :- 680 init_simplex_(ObjF,Constraints,Objective,S0,Vs), 681 (maximize(Objective,S0,S) 682 -> objective(S,MaxValue), {ObjF == MaxValue}, 683 (BindVars == true 684 -> bind_vars_(Vs,S) 685 ; remove_names_(Vs), 686 {Constraints} % apply constraints 687 ) 688 ; fail_msg_('Failed to maximize: ~w',[ObjF]) 689 ). 690 691init_simplex_(ObjF,Constraints,Objective,S,Vs) :- 692 gen_state(S0), 693 term_variables((ObjF,Constraints),Vs), 694 (Vs = [] 695 -> fail_msg_('No variables in expression to optimize: ~w',[ObjF]) 696 ; sim_constraints_(Constraints,S0,S1), 697 _::real(_,Max), % max value to constrain for simplex 698 constrain_ints_(Vs,Max,S1,S), 699 (map_simplex_(ObjF,T/T,Objective/[]) 700 -> true 701 ; fail_msg_('Illegal linear objective: ~w',[ObjF]) 702 ) 703 ). 704 705% use an attribute to associate a var with a unique simplex variable name 706simplex_var_(V,SV) :- var(V), 707 (get_attr(V,clpBNR_toolkit,SV) 708 -> true 709 ; statistics(inferences,VID), SV = var(VID), put_attr(V,clpBNR_toolkit,SV) 710 ). 711 712% Name attribute removed before exit. 713remove_names_([]). 714remove_names_([V|Vs]) :- 715 del_attr(V,clpBNR_toolkit), 716 remove_names_(Vs). 717 718attr_unify_hook(var(_),_). % unification always does nothing and succeeds 719 720constrain_ints_([],_,S,S). 721constrain_ints_([V|Vs],Max,Sin,Sout) :- 722 % Note: library(simplex) currently constrains all variables to be non-negative 723 simplex_var_(V,SV), % corresponding simplex variable name 724 % if not already an interval, make it one with finite non-negative value 725 (domain(V,D) -> true ; V::real(0,_), domain(V,D)), 726 (D == boolean -> Dom = integer(0,1); Dom = D), 727 Dom =.. [Type,L,H], 728 (Type == integer -> constraint(integral(SV),Sin,S1) ; S1 = Sin), 729 (L < 0 730 -> % apply non-negativity condition 731 ({V >= 0} -> L1 = 0 ; fail_msg_('Negative vars not supported by \'simplex\': ~w',[V])) 732 ; L1 = L 733 ), 734 % explicitly constrain any vars not (0,Max-eps) 735 (L1 > 0 -> constraint([SV] >= L1,S1,S2) ; S2 = S1), % L1 not negative 736 (H < Max -> constraint([SV] =< H,S2,SNxt) ; SNxt = S2), 737 constrain_ints_(Vs,Max,SNxt,Sout). 738 739bind_vars_([],_). 740bind_vars_([V|Vs],S) :- 741 % Note: skip anything nonvar (already bound due to active constraints) 742 (simplex_var_(V,SV) -> variable_value(S,SV,V) ; true), 743 bind_vars_(Vs,S). 744 745% clpBNR constraints have already been applied so worst errors have been detected 746sim_constraints_([],S,S) :- !. 747sim_constraints_([C|Cs],Sin,Sout) :- !, 748 sim_constraints_(C, Sin,Snxt), 749 sim_constraints_(Cs,Snxt,Sout). 750sim_constraints_((C,Cs),Sin,Sout) :- !, 751 sim_constraints_(C, Sin,Snxt), 752 sim_constraints_(Cs,Snxt,Sout). 753sim_constraints_(C,Sin,Sout) :- 754 sim_constraint_(C,SC), 755 constraint(SC,Sin,Sout). % from simplex 756 757sim_constraint_(C,SC) :- 758 C=..[Op,LHS,RHS], % decompose 759 constraint_op(Op,COp), % acceptable to simplex 760 number(RHS), RHS >= 0, % requirement of simplex 761 map_simplex_(LHS,T/T,Sim/[]), % map to simplex list of terms 762 !, 763 SC=..[COp,Sim,RHS]. % recompose 764sim_constraint_(C,_) :- 765 fail_msg_('Illegal linear constraint: ~w',[C]). 766 767map_simplex_(T,CT/[S|Tail],CT/Tail) :- 768 map_simplex_term_(T,S), 769 !. 770map_simplex_(A+B,Cin,Cout) :- !, 771 map_simplex_(A,Cin,Cnxt), 772 map_simplex_(B,Cnxt,Cout). 773map_simplex_(A-B,Cin,Cout) :- !, 774 map_simplex_(A,Cin,Cnxt), 775 map_simplex_(-B,Cnxt,Cout). 776 777map_simplex_term_(V,1*SV) :- simplex_var_(V,SV), !. 778map_simplex_term_(-T,NN*V) :- !, 779 map_simplex_term_(T,N*V), 780 NN is -N. 781map_simplex_term_(N*V,N*SV) :- number(N), simplex_var_(V,SV), !. 782map_simplex_term_(V*N,N*SV) :- number(N), simplex_var_(V,SV). 783 784constraint_op(==,=). 785constraint_op(=<,=<). 786constraint_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... .
814% 815% local_minima/1, % apply KT constraints for objective function expression (OFE) 816% local_maxima/1, % semantically equivalent to local_minima/1 817% 818local_minima(ObjExp) :- 819 local_optima_(min,ObjExp,[]). 820 821local_maxima(ObjExp) :- 822 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... .
861% 862% local_minima/2, % apply KT constraints for minima with constraints 863% local_maxima/2 % apply KT constraints for maxima with constraints 864% 865local_minima(ObjExp,{Constraints}) :- 866 local_optima_(min,ObjExp,Constraints). 867 868local_maxima(ObjExp,{Constraints}) :- 869 local_optima_(max,ObjExp,Constraints). 870 871 872local_optima_(MinMax,ObjF,Constraints) :- 873 local_optima_(MinMax,ObjF,Constraints,Cs), % generate constraints 874 {Cs}. % then apply 875 876local_optima_(MinMax,ObjF,Constraints,[Constraints,GCs,LCs]) :- 877 lagrangian_(Constraints,MinMax,ObjF,LObjF,LCs), 878 term_variables((Constraints,ObjF),Vs), 879 gradient_constraints_(Vs,GCs,LObjF). 880 881gradient_constraints_([],[],_Exp). 882gradient_constraints_([X|Xs],[C|Cs],Exp) :- 883 partial_derivative(Exp,X,D), 884 (number(D) -> C=[] ; C=(D==0)), % no constraint if PD is a constant 885 gradient_constraints_(Xs,Cs,Exp). 886 887% for each constraint add to Lagrangian expression with auxiliary KKT constraints 888lagrangian_(C,MinMax,Exp,LExp, LC) :- nonvar(C), 889 kt_constraint_(C,CExp, LC), % generate langrange term with multiplier 890 lexp(MinMax,Exp,CExp,LExp), 891 !. 892lagrangian_([],_,L,L,[]). 893lagrangian_([C|Cs],MinMax,Exp,LExp,[LC|LCs]) :- 894 lagrangian_(C, MinMax,Exp,NExp,LC), 895 lagrangian_(Cs,MinMax,NExp,LExp,LCs). 896lagrangian_((C,Cs),MinMax,Exp,LExp,[LC|LCs]) :- 897 lagrangian_(C,MinMax,Exp,NExp,LC), 898 lagrangian_(Cs,MinMax,NExp,LExp,LCs). 899 900lexp(min,Exp,CExp,Exp+CExp). 901lexp(max,Exp,CExp,Exp-CExp). 902 903kt_constraint_(LHS == RHS, M*(LHS-RHS), []) :- 904 M::real. % finite multiplier only 905kt_constraint_(LHS =< RHS, MGx, MGx==0) :- 906 MGx = M*(LHS-RHS), M::real(0,_). % positive multiplier and KKT non-negativity condition 907kt_constraint_(LHS >= RHS, Exp, LC) :- 908 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