1、function errmsg,Z,X,t,c,fail =BNB18(fun,x0,xstat,xl,xu,A,B,Aeq,Beq,nonlcon,setts,options1,options2,maxSQPit,varargin);% ?D?6 y1? £ Di ?6 ?a ?§?,.?心 u ' u ? 一？£ ?u MATLAB5.3?De 1 o ?£ ?De Optimizat ion toolbox 2.0?§ 3?% Minimize F(x)%subject to: xlb <= x <=xub%A*x &

2、lt;= B%Aeq*x=Beq%C(x)<=0%Ceq(x)=0%x(i)?e ?aa ?D?± ?a ?£?e y£?o 1 i ? 一？心% e 1 o ? e ?%errmsg,Z,X=BNB18('fun',x0,xstat,xl,xu,A,B,Aeq,Beq,'nonlcon',setts)%fun£o M?t? £ ?± i e ?x?D? ?± e o e yf=fun(x)%x0:a D?o a ?£?+ i e ?± ?a ?3?心%xstat &

3、#163; o a D?o ? ?£ ?xstat(i)=0 ± i e ?x(i)?a & ?D?± ?a ?£?1 ± i e ?e y£?2± i e ?1 i ? 一 ？ %xl£o a D?6 a ?£ ?± i e ?± ?a ?%xu:a D?o a ?£?+ i e ?± ?a ?e ?%A:? o , ± i e ?D?2? fi e e ? e ? e y%B:a D?o ? ?, ± i e ?D?2? e e ?

4、e ? e ?%Aeq:? o , ± i e ?D? i e e ? e ? e y%Beg:a D?o ? ?, ± i e ?D?2? e e ? 6 ? 6 6 ?心%nonlcon:M?t?£?± i e ? - ?D?e ?o e yC,Ceq=nonlin(x),?DC(x)?a2? e e ? e % Ceq(x)?ae e ? e ?%setts: ? " e e ?%errmsq:，心？' i ? o i a e ?%Z: ? ?+ 6 o 6 yx ?D?a%X:心？x ?6 ?a%d y i a% max x1*

5、x2*x3% -x1+2*x2+2*x3>=0% x1+2*x2+2*x3<=72% 10<=x2<=20% x1-x2=10% ?6 D' Mo e ydiscfun.m%function f=discfun(x)%f=-x(1)*x(2)*x(3);%?6 ?a% clear;x0=25,15,10'xstat=1 1 1'% xl=20 10 -10'xu=30 20 20'% A=1 -2 -2;1 2 2;B=0 72'Aeq=1 -1 0;Beq=10;% err,Z,X=BNB18('discfun&

6、#39;,x0,xstat,xl,xu,A,B,Aeq,Beq);% XMAX=X',ZMAX=-Z% BNB18 Finds the constrained minimum of a function of several possibly integer variables.% Usage: errmsg,Z,X,t,c,fail =%BNB18(fun,x0,xstatus,xlb,xub,A,B,Aeq,Beq,nonlcon,settings,options1,options2,maxSQPiter,P1,P2,.)% BNB solves problems of the f

7、orm:% Minimize F(x) subject to: xlb <= x0 <=xub%A*x <= B Aeq*x=Beq%C(x)<=0 Ceq(x)=0%x(i) is continuous for xstatus(i)=0%x(i) integer for xstatus(i)=1%x(i) fixed for xstatus(i)=2% BNB uses:% Optimization Toolbox Version 2.0 (R11) 09-Oct-1998% From this toolbox fmincon.m is called. For mor

8、e info type help fmincon.% fun is the function to be minimized and should return a scalar.F(x)=feval(fun,x).% x0 is the starting point for x. x0 should be a column vector.% xstatus is a column vector describing the status of every variable x(i).% xlb and xub are column vectors with lower and upper b

9、ounds for x.% A and Aeq are matrices for the linear constrains.% B and Beq are column vectors for the linear constrains.% nonlcon is the function for the nonlinear constrains.% C(x);Ceq(x)=feval(nonlcon,x). Both C(x) and Ceq(x) should be column vectors.% errmsg is a string containing an error messag

10、e if BNB found an error in the input.% Z is the scalar result of the minimization, X the values of the accompanying variables.% t is the time elapsed while the algorithm BNB has run, c is the numberof BNB cycles and% fail is the number of unsolved leaf sub-problems.% settings is a row vector with se

11、ttings for BNB:% settings(1) (standard 0) if 1: use phase 1 by relaxation. This sometimes makes the algorithm% faster, because phase 1 means the algorithm first checks if there is a feasible solution% for a sub-problem before trying to find a best solution. If there is no feasible solution BNB% will

12、 not try to find a best solution.% settings(2) (standard 0) if 1: if the sub-problem did not converge do not branch. If a sub-% problem did not converge this means BNB did not find a solution for it.Normally BNB will% branch the problem so it can try again to find a solution.% A sub-problem that is

13、a leaf of the branch-and-bound-three can not be branched. If such% a problem does not converge it will be considered unfeasible and the parameter fail will be% raised by one.% settings(3) (standard 0) if 1: if 1 a sub-problem that did not converge but did return a feasible% point will be considered

14、convergent. This might be useful if fmincon is having a hard time with% a certain problem but you do want some results.% options1 and options2 are options structures for phase 1 and phase 2.% For details about the options structure type help optimset.% maxSQPiter is a global variable used by fmincon

15、 (if modified as described in bnb18.m).% maxSQPiter is 1000 by default.% P1,P2,. are parameters to be passed to fun and nonlcon.% F(x)=feval(fun,x,P1,P2,.). C(x);Ceq(x)=feval(nonlcon,x,P1,P2,.).% Type edit BNB18 for more info.% E.C. Kuipers% e-mail E.C.Kuiperscpedu.rug.nl% FI-Lab% Applied Physics% R

16、ijksuniversiteit Groningen % To get rid of bugs and to stop fmincon from hanging make the following chances: % In optim/private/nlconst.m($Revision:1.20 $ $Date:1998/08/2413:46:15$):% Get EXITFLAG independent of verbosity.% After the lines:disp(' less than 2*options.TolFun butconstraints are not

17、 satisfied.')%end%EXITFLAG = -1;%end%end%status=1;% add the line: if (strncmp(howqp, 'i',1) & mg > 0), EXITFLAG = -1; end; % % In optim/private/qpsub.m ($Revision: 1.21 $ $Date: 1998/09/01 21:37:56 $):% Stop qpsub from hanging.% After the line: % Andy Grace 7-9-90. Mary Ann Branch

18、 9-30-96.% add the line: global maxSQPiter;% and changed the line: maxSQPiters = Inf;elseworks% to the line: if exist('maxSQPiter','var'),maxSQPiters = maxSQPiter;maxSQPiters=inf; end;% I guess there was a reason to put maxSQPiters at infinity, but this fine for me.global maxSQPiter;

19、% STEP 0 CHECKING INPUT Z=; X=; t=0; c=0; fail=0;if nargin<2, errmsg=if isempty(fun), errmsg=if isempty(x0), errmsg='BNB needs at least 2 input arguments.'return ; end ;'No fun found.''No x0 found.' return; return ;end ;elseif size(x0,2)>1, errmsg= xstatus=zeros(size(x0

20、);if nargin>2 & isempty(xstat)if all(size(xstat)<=size(x0) xstatus(1:size(xstat)=xstat;else errmsg= 'xstatus end ;'x0 must be a column vector.'must be a column vector thesame sizereturn ; end ;as x0.' ; returnif any(xstatus=round(xstatus) | xstatus<0 | 2<xstatus)errms

21、g= end ;'xstatus must consist of the integers 0,1 en 2.'returnend ;xlb=zeros(size(x0);xlb(find(xstatus=0)=-inf;if nargin>3 & isempty(xl)if all(size(xl)<=size(x0) xlb(1:size(xl,1)=xl;else errmsg= 'xlb must be a column vector the same size as x0.' returnend ;end ;if any(x0<

22、;xlb)errmsg= 'x0 must be in the range xlb <= x0.' return ;elseif any(xstatus=1 & (isfinite(xlb) | xlb=round(xlb)errmsg= 'xlb(i) must be an integer if x(i) is an integer variabele.'return ;end ;xlb(find(xstatus=2)=x0(find(xstatus=2);xub=ones(size(x0);xub(find(xstatus=0)=inf;if

23、nargin>4 & isempty(xu)if all(size(xu)<=size(x0)xub(1:size(xu,1)=xu;else errmsg= 'xub must be a column vector the same size as x0.' returnend ;end ;if any(x0>xub)errmsg= 'x0 must be in the range x0 <=xub.' return ;elseif any(xstatus=1 & (isfinite(xub) | xub=round(x

24、ub)errmsg= 'xub(i) must be an integer if x(i) is an integer variabale.' return end ;xub(find(xstatus=2)=x0(find(xstatus=2);if nargin>5errmsg= 'Matrix A not correct.1), errmsg= 'Column vector B not'A and B should only be nonemptyif isempty(A) & size(A,2)=size(x0,1), return

25、; end ;else A=; end ; if nargin>6if isempty(B) & any(size(B)=size(A,1) correct.' ; return ; end ;else B=; end ;if isempty(A) & isempty(B), errmsg= together.' ; return ; end ;if isempty(B) & isempty(A), B=zeros(size(A,1),1);end ;if nargin>7 & isempty(Aeq)if size(Aeq,2)=s

26、ize(x0,1), errmsg='Matrix Aeq not correct.' returnend ;else Aeq=; end ;if nargin>8if isempty(Beq) & any(size(Beq)=size(Aeq,1) 1), errmsg='Columnvector Beq not correct.' return ; end ;else Beq=; end ;only be nonemptyif isempty(Aeq) & isempty(Beq), errmsg= 'Aeq and Beq s

27、hould together' ; return ; end ;if isempty(Beq) & isempty(Aeq), Beq=zeros(size(Aeq,1),1);end ;if nargin<10, nonlcon='' ; end ;settings = 0 0 0;if nargin>10 & isempty(setts)if all(size(setts)<=size(settings)settings(setts=0)=setts(setts=0);else errmsg= 'settings shoul

28、d be a row vector of length 3.' ; return ; end ; end ;if nargin<12, options1=;end ;options1=optimset(optimset('fmincon'),options1);if nargin<13, options2=;end ;options2=optimset(optimset('fmincon'),options2);if nargin<14, maxSQPiter=1000;elseif isnumeric(maxSQPit) &

29、all(size(maxSQPit)=1 & maxSQPit>0 &round(maxSQPit)=maxSQPitmaxSQPiter=maxSQPit;else errmsg= 'maxSQPiter must be an integer >0' return ; end ;eval( 'z=' ,fun, '(x0,varargin:);', 'errmsg=''fun caused error.''return;' );if isempty(nonlcon)ev

30、al( 'C, Ceq=' ,nonlcon, '(x0,varargin:);', 'errmsg=''nonlconcaused error.'' return;');if size(C,2)>1| size(Ceq,2)>1, errmsg= 'C en Ceq must be column vectors.' ;return ; end ;end ;% STEP 1 INITIALISATIONcurrentwarningstate=warning;warning off ;ti

31、c;lx = size(x0,1);z_incumbent=inf;x_incumbent=inf*ones(size(x0);I =ceil(sum(log2(xub(find(xstatus=1)-xlb(find(xstatus=1)+1)+size(find(xstatus=1),1)+1);stackx0=zeros(lx,I);stackx0(:,1)=x0;stackxlb=zeros(lx,I);stackxlb(:,1)=xlb;stackxub=zeros(lx,I);stackxub(:,1)=xub;stacksize=1;xchoice=zeros(size(x0);

32、if isempty(Aeq)j=0;for i=1:size(Aeq,1)if Beq(i)=1 & all(Aeq(i,:)=0 | Aeq(i,:)=1)J=find(Aeq(i,:)=1);if all(xstatus(J)=0 & xchoice(J)=0 & xlb(J)=0 & xub(J)=1)if all(xstatus(J)=2) | all(x0(J(find(xstatus(J)=2)=0)j=j+1;xchoice(J)=j;if sum(x0(J)=0, errmsg='x0 not correct.' return

33、; end ;end ;end ;end ;end ;end ;errx=optimget(options2,'TolX' );errcon=optimget(options2,'TolCon' );fail=0;c=0;% STEP 2 TERMINIATIONwhile stacksize>0c=c+1;% STEP 3 LOADING OF CSPx0=stackx0(:,stacksize);xlb=stackxlb(:,stacksize);xub=stackxub(:,stacksize);x0(find(x0<xlb)=xlb(find

34、(x0<xlb);x0(find(x0>xub)=xub(find(x0>xub);stacksize=stacksize-1;% STEP 4 RELAXATION% PHASE 1con=BNBCON(x0,A,B,Aeq,Beq,xlb,xub,nonlcon,varargin:);if abs(con)>errcon & settings(1)=0x1 dummyfeasflag=fmincon( '0' ,x0,A,B,Aeq,Beq,xlb,xub,nonlcon,options1,varargin :);if settings(3)

35、 & feasflag=0con=BNBCON(x1,A,B,Aeq,Beq,xlb,xub,nonlcon,varargin:);if con<errcon, feasflag=1;end ;end ;else x1=x0; feasflag=1;end ;% PHASE 2if feasflag>0x zconvflag=fmincon(fun,x1,A,B,Aeq,Beq,xlb,xub,nonlcon,options2,varargin :);if settings(3) & convflag=0con=BNBCON(x,A,B,Aeq,Beq,xlb,xu

36、b,nonlcon,varargin:);if con<errcon, convflag=1;end ;end ;else convflag=feasflag;end ;% STEP 5 FATHOMINGK = find(xstatus=1 & xlb=xub);separation=1;if convflag<0 | (convflag=0 & settings(2)% FC 1separation=0;elseif z>=z_incumbent & convflag>0% FC 2separation=0;elseif all(abs(ro

37、und(x(K)-x(K)<errx) & convflag>0% FC 3z_incumbent = z;x_incumbent = x;separation = 0;end ;% STEP 6 SELECTIONif separation = 1 & isempty(K)dzsep=-1;for i=1:size(K,1)dxsepc = abs(round(x(K(i)-x(K(i);if dxsepc>=errx | convflag=0xsepc = x; xsepc(K(i)=round(x(K(i);dzsepc = abs(feval(fun,

38、xsepc,varargin:)-z);if dzsepc>dzsepdzsep=dzsepc;ixsep=K(i);end ;end ;end ;% STEP 7 SEPARATIONif xchoice(ixsep)=0% XCHOICE=0branch=1;domain=xlb(ixsep) xub(ixsep);while branch=1xboundary=(domain(1)+domain(2)/2;if x(ixsep)<xboundarydomainA=domain(1) floor(xboundary);domainB=floor(xboundary+1) dom

39、ain(2); elsedomainA=floor(xboundary+1) domain(2);domainB=domain(1) floor(xboundary);end ;stacksize=stacksize+1;stackx0(:,stacksize)=x;stackxlb(:,stacksize)=xlb;stackxlb(ixsep,stacksize)=domainB(1);stackxub(:,stacksize)=xub;stackxub(ixsep,stacksize)=domainB(2);if domainA(1)=domainA(2)stacksize=stacksize+1;stackx0(:,stacksize)=x;stackxlb(:,stacksize)=xlb;stackxlb(ixsep,stacksize)=doma