minimize
f
( z ) , subject F ( z )5
0 (33)with a convex
f
and a convex F satisfying the Slater condition ( F ( z S ) < O for some z S ) may be solved in NOA2 by the constraint linearization method (Kiwiel, 1987), which is frequently more efficient than the algorithms of the preceding section.K . Kiwiel, A . Stachurski
-
55 - Theoretical guide t o NOA2 the algorithm sets p8=pk--' ifp"<pk--';
otherwisek -k
is the k-th polyhedral approximation to the improvement function
H ( z ; z ~ )
=maz{
f ( z ) -f (
z k ) , F ( z ) }for all
z.^ k k
Thus, if F(zk)<O, we wish t o find a feasible
( F
( z +dk)<O ) direction of descent( r ~ ~ ~ t d ~ j <
f ( z k )),
whereas for F(zk)>O,d k
should be a descent direction for F a t z k( F ( 2
+ d )<o),
since then we would like t o decrease the constraint violation.K . Kiwiel, A . Stachurski - 56 - Theoretical guide to N O A 2 The algorithm runs in two phases. At phase I successive points x are infeasible, and k the line search rules of Section 2.1 are applied t o F. Finding a feasible xk starts phase 11, in which the line search rules are augmented t o ensure feasibility of successive iterates. Of course, phase I will be omitted if the initial point x1 is feasible.
The algorithm requires the Slater constraint qualification (F(xS)<O for some xS);
otherwise, it may terminate a t a point x t h a t is an approximate minimizer of k
F.
The algorithm is, in general, more reliable than the exact penalty methods of Sec- tions 2.3 and 2.4, because it does not need t o choose penalty coefficients. Unfortunately, its convergence may be slower, since it cannot approach the boundary of the feasible set a t a fast rate.
Additional linear constraints are handled as in Section 2.2.
2.6. M e t h o d s f o r n o n c o n v e x p r o b l e m s
Minimization problems with nonconvex objectives and constraints are solved in NOA2 by natural extensions (Kiwiel, 1985a, 1985b, 1986a, 1986c) of the methods for con- vex minimization described in the preceding sections. Except for the constraint lineariza- tion method of Section 2.4, each method has two extensions, which differ in the treatment of nonconvexity. The methods use either subgradient locality measures, or subgradient deletion rules for localizing the past subgradient information. Advantages and drawbacks of the two approaches depend on specific properties of a given problem.
For simplicity, let us consider the unconstrained problem of minimizing a locally Lipschitz continuous function f , for which we can calculate the linearization
by evaluating
f
and its subgradient g a t each y. At the k-th iteration, several suchl j .
linearizations computed a t trial points y
,
] E J ~ , are used in the following polyhedral approximation to f around the current iterate x kp ( x ) = f ( ~ ~ ) + r n a x { - r r ~ ( x ~ , ~ ~ ) + < ~ ~ ( ~ ~ ) , x - x k
>:
j~J;},
(38) where the subgradient locality measuresa ,(xk,yj) = m a x
{
If(xk) - f ( x k , ~ j ) l , 7Jzk-~jI2}with a parameter 7 >O indicate how much the subgradient gj(yl) differs from being a subgradient of f a t xi;0bserve t h a t in the convex case with 7,=0 the approximation (38) reduces t o the previously used form (6) (cf. (4)). More generally, for ~ , > 0 the subgra- dients with relatively large locality measures cannot be active in
f
-k in the neighborhood of xk. Thus even in t h e nonconvex case f -k may be a good local approximation t of ;
pro- vided t h a t it is based on sufficiently local subgradients. This justifies the use off
^k in the search direction finding subproblems of the preceding sections (cf. (7), (16), (19), (21), (28), (37)).Ideally, the value of the locality parameter 7, should reflect the degree of noncon- vexity of f . Of course, for convex
f
the best value is 7,=0. Larger values of 7, decrease the influence of nonlocal subgradient information on the search direction findin This, fork
4
instance, prevents the algorithm from concluding t h a t x is optimal because
f
indicates t h a tf
has no descent direction a t xk. On the other hand, a large value of 7, may cause t h a t after a serious step all the past subgradients will be considered as nonlocal a t the search direction finding. Then the algorithm will be forced t o accumulate localK . Kiwiel, A . Stachurski
-
57-
Theoretical guide t o N O A 2 subgradients by performing many null steps with expensive line searches.In the strategy described so far the influence of a subgradient on
jk
decreases"smoothly" when this subgradient becomes less local. More drastic is the subgradient deletion strategy
,
which simply drops the nonlocal past subgradients fromf
^k.
In this case, we set ~ , = 0 in (39) and define the locality radiusof the ball around x k from which the past subgradients were collected. As before, the
^k k
approximation
f
is used t o generate a search direction d . A locality reset of the approxi- mation occurs ifldkl
5
ma a k,
(41)where ma is a positive parameter. This involves dropping from
Jf
k an index j with thek '
largest value of lz -yJ1, i.e. the most nonlocal subgradient is dropped so as t o decrease the locality radius a k . If the next d k satisfies (41), another reset is made, etc. Thus resets decrease the locality radius until it is comparable with the length of the search direction Idkl.
Dropping the j - t h subgradient corresponds to replacing cr ( x , y J ) k in (38) by a large number. Moreover, the frequency of resets is proportional to t
I!
e value of m, - in the test (41). Therefore, our preceding remarks on the choice of 7 , are relevant t o the selection of m a .In practice one may use y s = l and ma=O.l, increasing them t o r S = 1 0 and ma=0.5 for strong nonconvexities
Both strategies use line searches similar t o that of Section 2.1. Additionally, the subgradient resetting strategy requires t h a t a null step ( x k + ' = x k ) should produce a trial
k k k
point yk+' close t o x in the sense t h a t
1
y k + l - xI
is of order a . Since1
y k f l - x k ( = t i l d k l , the right stepsize t R k should be sufficiently small. This can be ensured either by testing progressively smaller initial trial stepsizes, or by introducing the direct requirementwhere c d ~ [ 0 . 1 , 0 . 5 ] is a parameter, e.g. Cd=m, References:
Clarke, F.H.(1983). Optimization and Nonsmooth Analysis. Wiley Interscience, New York.
Kiwiel, K.C.(1985a). Methods of Descent for Nondiflerentiable O p t i m i z a t i o n . Springer- Verlag, Berlin.
Kiwiel, K.C.(1985b). A linearization algorithm for nonsmooth minimization. Mathemat- ics of Operations Research 10, 185-194.
Kiwiel, K.C.(1985c). A n algorithm for linearly constrained convez nondiflerentiable m i n i m i z a t i o n problems. Journal of Mathematical Analysis and Applications 105, 452-465.
Kiwiel, K.C.(1985d). A n ezact penalty function algorithm for nonsmooth convez con- strained m i n i m i z a t i o n problems. IMA Journal of Numerical Analysis 5, 111-119.
K . Kiwiel, A . S t a c h u r s k i
-
58-
Theoretical guide t o N O A 2 Kiwiel, K.C.(1986a). A n aggregate subgradient method for n o n s m o o t h and n o n c o n v e zm i n i m i z a t i o n . Journal of Computational and Applied Mathematics 14, 391-400.
Kiwiel, K.C.(1986b). A method for solving certain quadratic programming problems aris- ing i n n o n s m o o t h optimization. IMA Journal of Numerical Analysis 6 , 137-152.
Kiwiel, K.C.(1986c). A method of linearizations for linearly constrained n o n c o n v e z n o n s m o o t h m i n i m i z a t i o n . Mathematical Programming 34, 175-187.
Kiwiel, K.C.(1987). A constraint linearization m e t h o d for nondiflerentiable c o n v e z m i n i m i z a t i o n . Numerische Mathematik ( t o appear).
Lemarechal, C.(1978). N o n s m o o t h optimization and descent methods. Report RR-78-4, International Institute for Applied Systems Analysis, Laxenburg, Austria.
Mifflin, R.(1982). A modification a n d a n e z t e n s i o n of Lemarechal's algorithm for n o n s m o o t h m i n i m i z a t i o n . Mathematical Programming Study 17, 77-90.
J . Majchrzak Implicit Utility Function . .