Nondifferentiable Optimization Algorithms
7 Methods for nonconvex problems
Nonconvex minimization problems are solved in NOA by natural extensions of the methods described in Sections 3, 4 and 6, see Kiwiel (1985a, 1985d, 1986b, 1986c, 1988a).
For simplicity, let us consider the problem of minimizing a locally Lipschitzian function f on SL
.
In the nonconvex case the subgradient gf (y) may be used for modelling f around z only when y is close to z (we no longer have f _>f
(.
; y ) ). The subgradient locality measurewith a parameter 7, > 0 indicates how much gf (y) differs from being a subgradient of f a t z . At the k-th iteration the algorithm uses the following modification of (6) for finding dk via (7)
f k ( z ) = f ( z k ) + m a x { - u f ( z k ; $ ) + ( g f ( # ) , 2 - z k ) : j~
~ f } .
In the convex case with 7, = 0 this approximation is equivalent to (6)
(since f ( z k ) 2 f ( z k ; y ~ ) ) . For 7, > 0 the local subgradients with small weights af ( z k ; y ~ ) tend t o influence dk more strongly than the nonlocal ones.
The above definition of a/ is rather arbitrary (cf. Mifflin, 1982), and it is not clear how the value of 7, should reflect the degree of nonconvexity of f (in theory any 7, > 0 will do).
Of course, for convex f 7, = 0 is best. Larger values of 7, are essential for concluding that z k
is optimal because
jk
indicates that f has no feasible descent directions a t z k . On the other hand, a large 7, may cause that after a serious step all the past subgradients will become inactive a t the search direction finding. Then the algorithm will be forced to accumulate local subgradients by performing many null steps.It is, therefore, reassuring to observe that 7, = 1 seems t o work quite well in practice (cf.
Kiwiel, 1988a). However, it may be necessary to scale the variables so that they are of order 1 a t the solution (to justify the Euclidean norm in (17)). Since automatic scaling could be dangerous, it is not implemented in NOA, but we intend to pursue this subject in the future.
Another feature of the nonconvex case is the need for line searches. Two cases are possible when a line search explores how well
jk
agrees with f between z k and z k+
dk. Either it is possible to make a serious step by finding a stepsize t i E ( 0 , 1 ] such that the next iterate zk+' = z k+
t i d k has a significantly lower objective value than z k,
or a null step zk+' = z k ( t i = 0 ) which evaluates f and gf at the new trial point yk+' = z k + t k d k,
with t k E ( 0 , 11, should improve the next model fk+' that will yield a better dk+'.More specifically, a serious step t i = t k > 0 is taken if
whereas a null step occurs with 0 = t i < t k
5 f
andwhere mL
,
mR,
mu andi
are positive parameters. A simple procedure for finding t i and t k is given in Kiwiel ( 1 9 8 6 ~ ) for the case of mL+
mu < mR < 1 andf 5
1. Since the aim of a null step is to detect discontinuities of gf on the segment [ z k,
zk+' ],
this procedure requires that f and gf be consistent in the sense thatl i m s ~ ~ ( ~ ~ ( z + t ' d ) , d )
>
l i m i n f [ f ( z + t i d ) - f ( z ) ] / t ii-+m s + m
for all z , d E Rn
,
{ ti ) c R+,
ti1
0.
In practice we use mL = 0.1, mR = 0.5, mu = 0.1,
f
= 0.1 and simple quadratic interpolation for diminishing trial stepsizes (see Remark 3.3.5 in Kiwiel, 1985d). Yet our crude procedure seems t o be quite efficient; it requires on average less than two function evaluations (cf. Kiwiel, 1988a). On the other hand, our experience with more sophisticated procedures that insist on directional minimizations (cf. Mifflin, 1984) is quite negative. The resulting increase in the number of f-evaluations is not usually offset by a reduction in the number of iterations. This is not efficient in applications where the cost of one f-evaluation may dominate the effort in auxiliary operations (mainly a t quadratic programming) per iteration.We should add that in practice we employ the locality measures a:,j = max
{
l/(zk) - i ( z k ; Y')I,
78(~;)'}
that over-estimate as ( z k ; y j ) by using the upper estimate 8; =
-
ZJ1 + x!i/
(zit' - zi)of Izk - yjl, which can be updated without storing y ~ .
The extensions to the nonconvex case of the methods of Sections 4 and 6 follow the lines sketched above. We only add that all the problem functions should satisfy the semidiffer- entiability condition (18). In fact the convergence analysis of Kiwiel (1988a) requires the
equality constraints t o be continuously differentiable, but we have managed t o solve many problems with nondifferentiable equality constraints.
We may add t h a t each method of NOA has another version that uses subgradient deletion rules instead of subgradient locality measures for localizing the past subgradient information (see, for instance, Kiwiel, 1985a and Kiwiel, 1 9 8 6 ~ ) . It is not clear which version is preferable, since their merits are problem-dependent. We intend t o clarify this situation in the near future.
8 Conclusions
We have presented an overview of several NDO algorithms t h a t are implemented in the system NOA. The emphasis has been laid on practical difficulties, but they can only be resolved by further theoretical work. We hope, therefore, t h a t this paper will contribute t o the development of NDO methods.
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