Compute a perturbation needed for a given symmetric matrix to be pos. def.
I.e., compute a lower estimate of the smallest (negative) eigenvalue (if any).
Input:
M ... symmetric matrix to find 'pert>=0' so that M+pert*I>0 (pos.def.)
lb ... (>=0) lower bound on pert, i.e., find pert>=lb>=0,
e.g., current known feasibility of other matrix constraints
ub ... (>0, Inf if not known), upper bound on pert which is known that
M+ub*I>0, typically the penalty parameter
pstart ... if lb==0, pstart is the first perturbation to try if M not posdef
pstop ... tolerance of the requested perturbation, the exact perturbation
will be in interval [pert,pert+ptol]
Output:
pert ... estimated perturbation within the given tolerance unless
M+lb*I>=0 is already pos def
TODO
how to make pstart&ptol dynamic to save some time...0001 % Compute a perturbation needed for a given symmetric matrix to be pos. def. 0002 % I.e., compute a lower estimate of the smallest (negative) eigenvalue (if any). 0003 % Input: 0004 % M ... symmetric matrix to find 'pert>=0' so that M+pert*I>0 (pos.def.) 0005 % lb ... (>=0) lower bound on pert, i.e., find pert>=lb>=0, 0006 % e.g., current known feasibility of other matrix constraints 0007 % ub ... (>0, Inf if not known), upper bound on pert which is known that 0008 % M+ub*I>0, typically the penalty parameter 0009 % pstart ... if lb==0, pstart is the first perturbation to try if M not posdef 0010 % pstop ... tolerance of the requested perturbation, the exact perturbation 0011 % will be in interval [pert,pert+ptol] 0012 % Output: 0013 % pert ... estimated perturbation within the given tolerance unless 0014 % M+lb*I>=0 is already pos def 0015 % 0016 % TODO 0017 % how to make pstart&ptol dynamic to save some time... 0018 function [pert] = feasm(M, lb, ub, pstart, pstop) 0019 0020 nfactor=0; 0021 0022 % Get ordering if M really sparse & shuffle M to avoid recomputing 0023 [n m] = size(M); 0024 Missparse = n>10 && issparse(M) && nnz(M)<0.15*n*n; 0025 % nnz() used directly and behind issparse() because otherwise it 0026 % counts nonzero elements even in the dense matrix 0027 0028 if (Missparse) 0029 perm=amd(M); 0030 M=M(perm,perm); 0031 I=speye(n,n); 0032 else 0033 % it is usually faster to compute with dense matrices in dense format 0034 M=full(M); 0035 I=eye(n,n); 0036 end 0037 0038 % first quick try 0039 pert=max(0.0, lb); 0040 [R,p] = chol(M+pert*I); 0041 %pert 0042 nfactor=nfactor+1; 0043 if (p==0) 0044 % first match -> go 0045 return; 0046 end 0047 0048 % guess a reasonable step how to increase the perturbation 0049 if (ub<Inf) 0050 % perturbation with ub should be pos. def; with 'pert' wasnt --> pert<ub 0051 %pstep=min(20,max((ub-pert)/4, 2)); 0052 pstep=min(50,max(1.01*realpow(ub/pert,1/4),2)); 0053 % in 4 steps jump above the known perturbation 0054 else 0055 pstep=10; 0056 end 0057 %pstep 0058 pertlowstart = pert; 0059 pert=max(pert,pstart/pstep); 0060 0061 while (p~=0) 0062 pertlow = pert; 0063 pert=pert*pstep; 0064 [R,p] = chol(M+pert*I); 0065 %disp(sprintf('up pert=%e (%i)',pert,p)); 0066 nfactor=nfactor+1; 0067 end 0068 pertup=pert; 0069 0070 if (nfactor==2) 0071 % this is probably not necessary but it the upper loop finishes after one 0072 % step, perlow might be actually higher than the one which failed in the 0073 % "first quick try" 0074 pertlow=pertlowstart; 0075 end 0076 %nfactor 0077 0078 pstopn = max(pstop,norm(pertlow)*pstop); 0079 0080 while (pertup-pertlow>pstopn) 0081 %pert=pertlow + (pertup-pertlow)/3; 0082 pert=(pertlow+pertup)/2; 0083 %pert=realsqrt(pertlow*pertup); 0084 [R,p] = chol(M+pert*I); 0085 %disp(sprintf('down pert=%e (%i)',pert,p)); 0086 nfactor=nfactor+1; 0087 if (p==0) 0088 pertup=pert; 0089 else 0090 pertlow=pert; 0091 end 0092 end 0093 pert = pertup; % return upper bound on the perturbation (=overestimate) 0094 % (=lower bound on the (negative) eigenvalue) 0095 0096 %nfactor 0097 end 0098