多目标非线性规划程序Matlab完整版
多目标非线性规划程序
M a t l a b
Document serial number【NL89WT-NY98YT-NC8CB-NNUUT-NUT108】
f u n c t i o n[e r r m s g,Z,X,t,c,f a i l]=
BNB18(fun,x0,xstat,xl,xu,A,B,Aeq,Beq,nonlcon,setts,options1,options2,maxSQPit,varargin );
%·Dêy1￡Díóa·§¨μü′ú·¨￡úDê1ó￡DèOptimization toolbox §3
% Minimize F(x)
%subject to: xlb <= x <=xub
% A*x <= B
% Aeq*x=Beq
% C(x)<=0
% Ceq(x)=0
%
% x(i)éaáD±á￡êy￡ò1ì¨μ
% ê1óê
%[errmsg,Z,X]=BNB18('fun',x0,xstat,xl,xu,A,B,Aeq,Beq,'nonlcon',setts)
%fun￡o Mt￡±íê×Dˉ±êoˉêyf=fun(x)
%x0: áDòá￡±íê±á3μ
%xstat￡o áDòá￡xstat(i)=0±íêx(i)aáD±á￡1±íêêy￡2±íê1ì¨μ
%xl￡o áDòá￡±íê±á
%xu: áDòá￡±íê±áé
%A: ó, ±íêD2μèêêμêy
%B: áDòá, ±íêD2μèêêé
%Aeq: ó, ±íêDμèêêμêy
%Beg: áDòá, ±íêD2μèêêóòμ
%nonlcon: Mt￡±íê·Dêoˉêy[C,Ceq]=nonlin(x),DC(x)a2μèêê,
% Ceq(x)aμèêê
%setts: ·¨éè
%errmsq: ·μ′íóìáê
%Z: ·μ±êoˉêy×Dμ
%X: ·μ×óa
%
%àyìa
% max x1*x2*x3
% -x1+2*x2+2*x3>=0
% x1+2*x2+2*x3<=72
% 10<=x2<=20
% x1-x2=10
% èD′ Moˉêy
% function f=discfun(x)
% f=-x(1)*x(2)*x(3);
%óa
% 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',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 form:
% 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 (R11) 09-Oct-1998
% From this toolbox is called. For more 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 bounds 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 message 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 number of BNB cycles and
% fail is the number of unsolved leaf sub-problems.
%
% settings is a row vector with settings 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 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 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 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 (if modified as described in .
% 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.
% . Kuipers
% FI-Lab
% Applied Physics
% Rijksuniversiteit Groningen
% To get rid of bugs and to stop fmincon from hanging make the following chances:
%
% In optim/private/ ($Revision: $ $Date: 1998/08/24 13:46:15 $):
% Get EXITFLAG independent of verbosity.
% After the lines: disp(' less than 2* but constraints are not satisfied.')
% end
% EXITFLAG = -1;
% end
% end
% status=1;
% add the line: if (strncmp(howqp, 'i',1) & mg > 0), EXITFLAG = -1; end;
%
% In optim/private/ ($Revision: $ $Date: 1998/09/01 21:37:56 $):
% Stop qpsub from hanging.
% After the line: % Andy Grace 7-9-90. Mary Ann Branch 9-30-96.
% add the line: global maxSQPiter;
% and changed the line: maxSQPiters = Inf;
% to the line: if exist('maxSQPiter','var'), maxSQPiters = maxSQPiter; else maxSQPiters=inf; end;
% I guess there was a reason to put maxSQPiters at infinity, but this works fine for me.
global maxSQPiter;
% STEP 0 CHECKING INPUT
Z=[]; X=[]; t=0; c=0; fail=0;
if nargin<2, errmsg='BNB needs at least 2 input arguments.'; return; end;
if isempty(fun), errmsg='No fun found.'; return; end;
if isempty(x0), errmsg='No x0 found.'; return;
elseif size(x0,2)>1, errmsg='x0 must be a column vector.'; return; end;
xstatus=zeros(size(x0));
if nargin>2 & ~isempty(xstat)
if all(size(xstat)<=size(x0))
xstatus(1:size(xstat))=xstat;
else errmsg='xstatus must be a column vector the same size as x0.'; return;
end;
if any(xstatus~=round(xstatus) | xstatus<0 | 2 errmsg='xstatus must consist of the integers 0,1 en 2.'; return; end; end; 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.'; return; end; end; if any(x0 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 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.'; return; end; end; if any(x0>xub) errmsg='x0 must be in the range x0 <=xub.'; return; elseif any(xstatus==1 & (~isfinite(xub) | xub~=round(xub))) 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>5 if ~isempty(A) & size(A,2)~=size(x0,1), errmsg='Matrix A not correct.'; return; end; else A=[]; end; if nargin>6 if ~isempty(B) & any(size(B)~=[size(A,1) 1]), errmsg='Column vector B not correct.'; return; end; else B=[]; end; if isempty(A) & ~isempty(B), errmsg='A and B should only be nonempty together.'; return; end; if isempty(B) & ~isempty(A), B=zeros(size(A,1),1); end; if nargin>7 & ~isempty(Aeq) if size(Aeq,2)~=size(x0,1), errmsg='Matrix Aeq not correct.'; return; end; else Aeq=[]; end; if nargin>8 if ~isempty(Beq) & any(size(Beq)~=[size(Aeq,1) 1]), errmsg='Column vector Beq not correct.'; return; end; else Beq=[]; end; if isempty(Aeq) & ~isempty(Beq), errmsg='Aeq and Beq should only be nonempty 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 should 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) & all(size(maxSQPit))==1 & maxSQPit>0 & round(maxSQPit)==maxSQPit maxSQPiter=maxSQPit; else errmsg='maxSQPiter must be an integer >0'; return; end; eval(['z=',fun,'(x0,varargin{:});'],'errmsg=''fun caused error.''; return;'); if ~isempty(nonlcon) eval(['[C, Ceq]=',nonlcon,'(x0,varargin{:});'],'errmsg=''nonlcon caused error.''; return;'); if size(C,2)>1 | size(Ceq,2)>1, errmsg='C en Ceq must be column vectors.'; return; end; end; % STEP 1 INITIALISATION currentwarningstate=warning; warning off; tic; 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)); 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; end; end; end; end; end; end; errx=optimget(options2,'TolX'); errcon=optimget(options2,'TolCon'); fail=0; c=0; % STEP 2 TERMINIATION while stacksize>0 c=c+1; % STEP 3 LOADING OF CSP x0=stackx0(:,stacksize); xlb=stackxlb(:,stacksize); xub=stackxub(:,stacksize); x0(find(x0 x0(find(x0>xub))=xub(find(x0>xub)); stacksize=stacksize-1; % STEP 4 RELAXATION % PHASE 1 con=BNBCON(x0,A,B,Aeq,Beq,xlb,xub,nonlcon,varargin{:}); if abs(con)>errcon & settings(1)~=0 [x1 dummy feasflag]=fmincon('0',x0,A,B,Aeq,Beq,xlb,xub,nonlcon,options1,varargin{:}); if settings(3) & feasflag==0 con=BNBCON(x1,A,B,Aeq,Beq,xlb,xub,nonlcon,varargin{:}); if con end; else x1=x0; feasflag=1; end; % PHASE 2 if feasflag>0 [x z convflag]=fmincon(fun,x1,A,B,Aeq,Beq,xlb,xub,nonlcon,options2,varargin{:}); if settings(3) & convflag==0 con=BNBCON(x,A,B,Aeq,Beq,xlb,xub,nonlcon,varargin{:}); if con end; else convflag=feasflag; end; % STEP 5 FATHOMING K = find(xstatus==1 & xlb~=xub); separation=1; if convflag<0 | (convflag==0 & settings(2)) % FC 1 separation=0; elseif z>=z_incumbent & convflag>0 % FC 2 separation=0; elseif all(abs(round(x(K))-x(K)) % FC 3 z_incumbent = z; x_incumbent = x; separation = 0; end; % STEP 6 SELECTION if 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==0 xsepc = x; xsepc(K(i))=round(x(K(i))); dzsepc = abs(feval(fun,xsepc,varargin{:})-z); if dzsepc>dzsep dzsep=dzsepc; ixsep=K(i); end; end; end; % STEP 7 SEPARATION if xchoice(ixsep)==0 % XCHOICE==0 branch=1; domain=[xlb(ixsep) xub(ixsep)]; while branch==1 xboundary=(domain(1)+domain(2))/2; if x(ixsep) domainA=[domain(1) floor(xboundary)]; domainB=[floor(xboundary+1) domain(2)]; else domainA=[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)=domainA(1); stackxub(:,stacksize)=xub; stackxub(ixsep,stacksize)=domainA(2); branch=0; else domain=domainA; branch=1; end; end; else % XCHOICE~=0 L=find(xchoice==xchoice(ixsep)); M=intersect(K,L); [dummy,N]=sort(x(M)); part1=M(N(1:floor(size(N)/2))); part2=M(N(floor(size(N)/2)+1:size(N))); stacksize=stacksize+1; stackx0(:,stacksize)=x; O = (1-sum(stackx0(part1,stacksize)))/size(part1,1); stackx0(part1,stacksize)=stackx0(part1,stacksize)+O; stackxlb(:,stacksize)=xlb; stackxub(:,stacksize)=xub; stackxub(part2,stacksize)=0; stacksize=stacksize+1; stackx0(:,stacksize)=x; O = (1-sum(stackx0(part2,stacksize)))/size(part2,1); stackx0(part2,stacksize)=stackx0(part2,stacksize)+O; stackxlb(:,stacksize)=xlb; stackxub(:,stacksize)=xub; stackxub(part1,stacksize)=0; if size(part2,1)==1, stackxlb(part2,stacksize)=1; end; end; elseif separation==1 & isempty(K) fail=fail+1; end; end; % STEP 8 OUTPUT t=toc; Z = z_incumbent; X = x_incumbent; errmsg=''; eval(['warning ',currentwarningstate]); function CON=BNBCON(x,A,B,Aeq,Beq,xlb,xub,nonlcon,varargin); if isempty(A), CON1=[]; else CON1 = max(A*x-B,0); end; if isempty(Aeq), CON2=[]; else CON2 = abs(Aeq*x-Beq); end; CON3 = max(xlb-x,0); CON4 = max(x-xub,0); if isempty(nonlcon) CON5=[]; CON6=[]; else [C Ceq]=feval(nonlcon,x,varargin{:}); CON5 = max(C,0); CON6 = abs(Ceq); end; CON = max([CON1; CON2; CON3; CON4; CON5; CON6]);