Documentation Trust Region Subroutine Algorithm
Documentation for
Trust Region Subroutine Algorithm
Charles Fortin and Henry Wolkowicz
Dept. Comb. & Opt., Univ. of Waterloo
This file is available with URL:
http://orion.math.uwaterloo.ca/~hwolkowi/henry/software/trustreg.d/docs.html
Variable Definitions, Tolerances, Initialization,
Call:
[xopt,fopt,lopt,lambdamin,muup,mulow,iter,flag] = newtrust(a,s,dgaptol)
solves: min q(x) = x'Ax  2a'x
such that s^2x'x >=0 (Lagr. mult. lopt <=0)
which is the trust region subproblem. This can be used within a
trust region algorithm for min f(x). It is specifically designed for
large sparse problems. The algorithm is based on the one developed in the paper
A Semidefinite Framework for Trust Region Subproblems with Applications to
Large Scale Minimization
(Franz Rendl and Henry Wolkowicz, Math. Progr. 1997., Math. Review
files needed: Afun.m,Dfun.m,conj_grad.m, and this file newtrust.m
BEFORE USING NEWTRUST, THE FOLLOWING INSTRUCTION MUST BE ENTERED
>> global A D counter
VARIABLES
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 INPUT:
 A is an n by n symmetric matrix. It is assumed to be in
sparse format. It is a global variable in order
to find the minimum eigenvalue using a Lanczos routine.
 a is a column vector  it is changed to full format below.
 s>0 is a scalar. It is the radius of the trust region.
 dgaptol is the desired duality gap tolerance.
(The user should decrease it as one gets closer
to the optimum of min f(x))
 par is a vector that contains initial estimates if
they are available, par=[lambda lambdat t], where lambda is an eigenvalue
of A, t is the parameter in D(t) and lambdat is corresponding
eigenvalue. Also, xparvec on input is eigenvector corresponding
to lambda while tparvec it eigenvector of D(t) corresponding to
lambdat. ?????not done yet????
 OUTPUT:
 xopt approximate optimum of the quadratic (nvector).
 fopt approximate optimal value of the quadratic.
... value is < gap(1,1)+dgaptol if assigned a value;
... takes value 99999 if not assigned a value.
 lopt ... dual opt multiplier vector,
 lambdamin ... minimum eigenvalue of A
 iter ... number of iterations (eigenvalue calls)
 flag ... a boolean variable to know if solution is
in the interior of the ball.
(flag ==2) and if the hard case (case 2)
(or near hard case (case2)) occurs
(flag ==1) and otherwise (flag == 0).

OTHER VARIABLES OF INTEREST:
 iter ... number of iterations  success occurred if < iterbnd
 gap ... gap, optimal value is in interval [gap(1,1),gap(1,2)]
on output, vector gap is:
gap=[mulow muup]
lower and upper bounds on mu^* (the optimum
of the quadratic objective )
ptbad(1,3) ptgood(1,3)
lower and upper bounds on lambda^* (multiplier)
low high lower and upper bounds on t^*
tarn1f t0f]
 tarn1f ... = cputimetarn10;
save initial eigenvalue cputime for statistics
 t0f ... = cputime  t0; total computation time
 topt ... right tvalue to generate opt. solution;
t=99999 means that lopt=0 used or
0 iterations with no optimal t,
i.e. an unconstrained optimum was found.
t is parameter in eigenvalue problem.
 ptgood, ptbad variables: ... used for saving interpolation data
 bestxf ... current best approximation to the xoptimum
 bestfxf ... current best approximation to the optimum of f(x)
corresponding to bestxf only (not muup)
 global A D counter
TOLERANCES
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Tolerances  these can and should be changed by the user.
The tolerance for the duality gap is for significant figures,
i.e. it is a relative duality gap.
The significant figures of agreement are computed for
the duality gap as follows:
signif = max( log_10 primaldual/(primal+1) , 0 )
The tolerances for ***LANCZOS** should be significantly higher
than those used here, e.g. 4 here and 7 for Lanczos
Stopping criteria: (i) feasibility, x > (1+normtol)s
(ii) feasibility dbestgap= (bestfxfmulow)/(bestfxf+1)
> 2*dgaptol
bestfxf is the current best opt. value corresponding to
the current best feasible point bestxf, while muup is the current
best upper bound of the optimal value.
(iii) duality gap, gap(2)gap(1)/(gap(2)+1)
> dgaptol
(iv) iteration count, iter <= iterbnd
(v) interval of uncertainty for t is > zerotol
Loop while: ( ((i) and (ii)) or (iii)) and ((iv) and (v))
i.e. while ( (t1&t2)  t3) & (t4 & t5),
So stop: ( (~t1~t2) & ~t3)  (~t4  ~t5)
Inverse Interpolation Starts
if side>0, last iterate was on the GOOD SIDE
if sum(ptgood(:,4)) >=2, two titerates with data good side
find interpolant using two points  then check for only
one point type interpolant using point from bad side
if (lambda<0)& (min(ptgood(1:2,2)) > (10^(side1)*zerotol) ),
try to avoid too many on good side??????
correct sign of multiplier
lambda(t) is concave and nondecreasing in t
???what about >0?? interpol using lambda data to
get to <=0 quickly?????
INVERSE INTERPOLATION  interpolate on good side
Let t(psi) be the inverse function of
psi(t), then we linearly interpolate
the value t(0), which is t^*, using the two points
from the good side.
tint=ptgood(1:2,1);
yint=ptgood(1:2,2);
f1=1/yint(1) + 1/ystar;
f2=1/yint(2) + 1/ystar;
Slope on the good side should always guarantee f1f2 not =0.
slope=(tint(1)tint(2))/(f1f2);
tt=tint(1)  f1* slope; linear inverse interpolation
if newlow <=tt&tt<=newhigh, roundoff error problem check
We now decide whether to chose the new point
t=tt;
tind=[tind 3]; indicator for t value  for debugging
end
elseif side>4, We need a point on the bad side if too many ???
if side>4&(min(ptgood(1:2,2)) < (10^(side1)*zerotol) ),
too many iterates on good side and it is near bad case
t=.95*newlow+.05*newhigh;
????use weights from k' if indexb>0?????
tind=[tind 4]; indicator for t value  for debugging
end
elseif sum(ptgood(:,4)) >=1 & indexb>0 >= 1,
if ptgood(1,2)>zerotol, not near hard case
interpolate with one point from each side if not hard case
tintg=ptgood(1,1);
tintb=ptbad(1,1);
psig=1/ptgood(1,2)+sqrt(ss1);
psib=1/ptbad(1,2)+sqrt(ss1);
tt=(tintg*psibtintb*psig)/(psibpsig);
if newlow <= tt & tt <= newhigh,
t=tt;
tind=[tind 31];
end
else
use spline on k(t)  to do still
tind=[tind 32];
end
end end of if sum(ptgood) >=2 else >=1
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