Global convergence of multidirectional algorithms for unconstrained optimization in normed spaces

Global convergence of multidirectional algorithms for unconstrained optimization in

normed spaces

J. A. Gomez and M. Romero

Center of Mathematics and Theoretical Physics.

ICIMAF, Havana, Cuba.

Abstract

Global convergence theorems for a class of descent methods for unconstrained optimization problems in normed spaces, using multi- directional search, are proved. Exact and inexact search are considered and the results allow to dene a globally convergent algorithm for an unconstrained optimal control problem which operates, at each step, on discrete approximations of the original continuous problem.

AMS Subject Classication: 49M10, 49M37, 65K10.

1 INTRODUCTION

Global convergence theorems of multidirectional search algorithms for nite dimensional optimization problems were derived in 11]. The general global convergence theorem of Zangwill (see 13]) was systematically used and ap- plications to numerical algorithms for unconstrained nite dimensional prob- lems were given. In nite dimensional optimization problems in Hilbert space were also considered in 11], but the heavy assumption about the closedness of the point-to-set map, which is essential in the Zangwill theorem, limited

1

the applications, in optimal control problems, only to algorithms de ned by point-to-set maps with range in a nite dimensional subspace.

In this paper we consider general in nite dimensional optimization prob- lem in a real normed space. Global convergence theorems are proved for a class of descent multidirectional algorithms, using the global convergence theorem of Polak (see 17]) without any nite dimensional assumption. Exact and inexact search are considered and in this last case we use the well known Wolfe conditions for global convergence, which is a common instrument in almost all the Quasi-Newton methods for unconstrained optimization (see for example 5] or 10]).

The results are used to design an algorithm for unconstrained optimal control problems which is globally convergent to local minima of the contin- uous problem but which operates, at each iteration, with nite dimensional approximation problems which are discretizations of the continuous one. In some sense the convergence result of this algorithm justi es the usual pro- cedure for solving unconstrained optimal control problems, which takes the optimal solution of a convenient nite dimensional discrete problem as a good approximation of the original one.

At each iteration, the algorithm nd a new point (i.e. a control func- tion) which satis es Wolfe's conditions for the continuous problem at the current (function) point, but this is performed nding a new point (i.e. a control sequence) which satis es Wolfe's conditions for the discrete prob- lem at the current (sequence) point. This point of view allows to use, in the implementation, all the common Quasi-Newton optimization methods and subroutines for the discrete problems. In addition, it also diers from the usual convergence results in common publications, in which it is always proved the convergence of theoptimal solution of the discrete problem to the optimal solution of the continuous problem (see for example 6]). Finally, the convergence results of the algorithm, given in the paper, are related with recent developments which study the inuence of the discretization step size in the algorithmic convergence (see 2]), since each iteration can be seen as an attempt to select not only a direction, but also a convenient step size to improve the current solution.

2

We consider the following problem:

min f(x)

x²X (1)

whereX is a real normed space, and we will study the global convergence of multidirectional descent algorithms, with exact and inexact search. We sup- pose f ² C¹ a continuously dierentiable function, and denote by ^rf(x)² X the Frechet derivative of f at x, where X is the topological dual space of X. In all the paper, the symbol ^hy xⁱ denotes the evaluation of the functionaly ²X at point x:

De nition 1

An algorithmic map in X is a point-to-set map A dened in X with values in the class ^P(X) of all subsets of X

.

A function c : X ^!<

is a stopping rule for the set ; X respect to the map A ⁱⁿ X ^{if for all} x =²; we have:

c(x⁰)< c(x) ⁸x^{0 2}A(x):

De nition 2

An algorithm ^A on X for nding points of a set ; X (shortly an algorithm for ;) is given by a sequence of pairs(Ai ci)i^2@ where Ai are algorithmic maps satisfying:

Ai(z)⁶= ⁸z =²; ⁸i^2@

and ci(:) are stopping rule functions for the set ;, and the following steps:

A1) Choose x^{0 2}X and set i = 0 A2) Compute Ai(xi)

A3) Choose yi ²Ai(xi)

A4) If ci(yi)ci(xi) stop and take xi as the last point of the sequence, If ci(yi)< ci(xi) go to step A5),

A5) Take xi⁺¹ =yi set i + 1 ^!i and go to step A2).

When both sequences are constant Ai = A ci = c ⁸i ^{2 @} we say that we have a uniform algorithm ^A = (A c):

De nition 3

A (nite or innite) sequence generated by an algorithm ^A= (An cn)n^2@, from the starting pointx²X is any sequence^fxn n = 1 2 :::^g

3

X which is obtained following the steps A1) - A5) in the last denition, with x⁰ = x:

An algorithm ^A = (An cn)n^2@ for ; X is called globally convergent to

; if for all x⁰²X and any sequence ^fxn^g generated by ^A from the starting point x⁰ we have that the last element of^fxn^gbelongs to ;, if ^fxn^g is nite, or that every accumulation point of ^fxn^g belongs to ; if ^fxn^g is innite.

Theorem 1

(Polak): Let X be a metric space and ^A = (A c) ^{a uniform} algorithm in X for ; X and suppose the maps A(:) and c(:) satisfy the following conditions:

i) c(:) is continuous or bounded below in X ii) ⁸z =²; ⁹" = "(z) > 0 ⁹ = (z) < 0 :

c(z⁰⁰)^;c(z⁰)< ⁸z⁰⁰²A(z⁰) ⁸z⁰²B (z ") (2) Then the algorithm ^A is globally convergent to ;:

For a proof see (17]).

De nition 4

Let be ² (0 1): We say the direction d ² X satises Wolfe's conditions respect to f ² C¹ at x ² X if the following inequalities hold:

f(x + d)f (x) + ^h5f (x) dⁱ (3)

h5f(x + d) d^{i h 5}f (x) dⁱ:

We call ^;condition (respectively ^;condition) the inequality corresponding to (respectively to ) in (3). We will also say that the point y = x + d satises the Wolfe conditions.

Lemma 1

^Let X be a normed space and f a Frechet dierentiable function.

Let's consider ² (0 1), > . Let be x ²X and d a descent direction in x, i.e.^{h 5}f (x) dⁱ < 0 . Suppose the set ^ff (x + d) 0^g is bounded below.

Then there existsy²X , with the formy = x+ d which strictly satises the Wolfe conditions:

f(y) < f (x) + ^h5f (x) dⁱ (4)

h5f(y) dⁱ> ^{h 5}f (x) dⁱ

and therefore, there exists an entire interval (^;" + ") where the Wolfe conditions hold.

4

Proof.

We have:

f (x + dl) = f(x) +^{h 5}f (x) dlⁱ+o(^jl^j)< f(x) + ^{h 5}f (x) dlⁱ (5) for small l > 0.

Since ^ff (x + dl) l^2<g is bounded below and l ^! f (x + dl) is contin- uous, there exists the least l⁰ > 0 such that:

f (x + dl⁰) =f(x) + ^h5f (x) dl⁰ⁱ:

In fact, for l near 0 we have (5) and for l^!+¹, the functionf (x + dl) is bounded and f(x) + ^h5f (x) dl^i!;1.

On the other hand, we have:

f (x + dl⁰)^;f(x) =^D5f(x + d^el) dl^0E for some ^el²(0 l⁰) therefore:

D

5f(x + d^el) dl^0E=^h5f(x) dl⁰ⁱ> ^{h 5}f (x) dl⁰ⁱ since > and ^h5f (x) dⁱ< 0, then:

D

5f(x + d^el) d^E> ^h5f (x) dⁱ:

Taking y = x + d~l and recalling that ^el < l⁰ and the de nition ofl⁰, we have:

f(x + d^el) = f(y) < f (x) + ^el^{h 5}f (x) dⁱ

D

rf(x + d^el) d^E = ^h5f(y) dⁱ> ^{h 5}f (x) dⁱ:

The existence of an interval is a consequence of the continuity of the functions l^!f (x + dl) and l^!5f(x + dl): ²

Lemma 2

Let X be a normed space, f a continuously Frechet dierentiable function and ²(0 1). Let be ^fxn^gn^2@X with xnn^!+1! x and ⁵f (x)⁶= 0and^fyn^gn^2@X withyn=xn+ndn, n 0 ^kdn^k= 1, such that, for all n ^2@ the inequalities:

h 5f (xn) dn^i;k5f (xn)^k

h5f(yn) dn^{ih 5}f (xn) dnⁱ

hold, and suppose ynn^!+1! y: Then x⁶=y. 5

Proof.

We have:

h⁵f(xn)^;5f(yn)] dn^ih5f (xn) dn^i;h5f (xn) dnⁱ=

= (1^;)^h5f (xn) dn^i;(1^;)^k5f (xn)^k< 0 (6) and

jh⁵f(xn)^;5f(yn)] dn^{i jk5}f(xn)^;5f(yn)^kkdn^k=^k5f(xn)^;5f(yn)^k: If we suppose x = y, then there exists the limit of the sequence:

jh⁵f(xn)^;5f(yn)] dn^ij

which, by continuity of ^rf satis es:

n^!+1lim ^jh5f(xn)^;5f(yn)] dn^{i j}= 0 (7) and therefore, the sequence of real numbers ^h5f(xn)^;5f(yn)] dnⁱ con- verges to 0 but on the other hand:

n^!+1lim ^h5f(xn)^;5f(yn)] dnⁱ

_nlim

!+1

;(1^;)^k5f(xn)^k=^;(1^;)^k5f (x)^k< 0:

We have a contradiction.²

Usually, a descent uniform algorithm ^A for the problem (1) is de ned through an algorithmic point-to-set mapA which is the composition of two maps A = SG, representing a "selector of directions" map and a "selector of new points" map respectively. We recall that the composition map of the point-to-set maps S and G is de ned by:

(SG)(x) =

y²G⁽x⁾S(y):

The stopping rule function c(:) of a descent uniform algorithm ^A for the problem (1), is almost always chosen as the objective function f(:): In de ning our rst algorithm, we will use this common point of view.

De nition 5

Let be p ^{2 @} and ² (0 1): The point-to-set map of -non orthogonal descent directions G^p(z) : X ^!X ^P(X^p) is dened by:

G^p(z) = ^f(z D)^2fz^gX^{p j9 2<p} :^kD ^k6= 0

hrf (z) D ^i;krf (z)^kkD ^{k g} (8) 6

where D is a p^;vector D = (D¹ ::: Dp), with Di ² X i = 1 ::: p and the productD is dened by the linear combination :D =^Ppi⁼¹ iDi:The norm in X^p is the usual product norm:

kD^k1= max_i

=1:::p^kDi^k:

De nition 6

^{Let be} p^2@. The point-to-set map of exact search S^p(z D) : X X^{p !}X is dened by:

S^p(z D) =y ²X ^j y = z + D f(y) = min

2<pf(z + D ): (9)

De nition 7

The descent uniform algorithm ^Ap = (Ap c) for the prob- lem (1), with multidirectional and exact search, is dened by the algorithmic point-to-set map Ap =S^pG^p and the stopping rule function c(x) = f(x):

In words, the algorithm^Apselects, at stepk, a p^;vector of directionsD = (d¹ ::: dp) belonging toX^p in such a way that the linear variety generated by those vectors contains a descent direction of the objective function f at the current pointxk: In the next step, a new point xk⁺¹is chosen as the minimum of the objective function f(z) in the linear variety xk +^Sd¹ ::: dp]:

Theorem 2

Let X be a normed space, f a continuously Frechet dieren- tiable and bounded below function, and ²(0 1). For anyp^2@ the descent uniform algorithm with multidirectional and exact search, ^Ap = (Ap f) ^is globally convergent to the set:

; =^fx²X ^j5f (x) = 0^g:

Proof.

If x =²; ^,rf(x)⁶= 0, there exists a "⁰ > 0 such that:

rf(x⁰)⁶= 0 ⁸x^{0 2}B(x ") ⁸"²(0 "⁰)

then a ^;descent direction always exist and therefore, Lemma (1) ensures that:

Ap(x⁰)⁶= ⁸x^{0 2}B(x ") ⁸"²(0 "⁰):

: We will verify the conditions (2) of Polak's global convergence theorem 1 for the set ;.

7

i) c(:) = f(:) is continuous.

ii) By contradiction, suppose it is not true. Then⁹x²X with⁵f (x)⁶= 0 such that⁸"x> 0 and ⁸x< 0, ⁹x^{0 2}B(x "x) and⁹x^{00 2}Ap(x⁰) such that x f(x⁰⁰)^;f(x⁰) < 0. Taking (x)n = ^;_n¹ and ("x)n = _n^{1 8}n ^{2 @}, we have ⁹ⁿx⁰n

o

n^2@ with x⁰n n^!+1! x and ⁹ⁿx⁰⁰n

o

n^2@ with x⁰⁰n ² Ap(x⁰n) such that f(x⁰⁰n)^;f(x⁰n)_n!+1^! 0. But, by de nition x⁰⁰n=x⁰n+Dn^{0 0}n , and denoting ⁰_n the vector of ^<p which satis es the ^;condition in (8), we will have:

f(x⁰⁰n)^;f(x⁰n) =f(x⁰n+D⁰n ⁰n)^;f(x⁰n)f(x⁰n+D⁰n(⁰_nl))^;f(x⁰n) ⁸l^2<

by optimality of x⁰⁰n ² S^p(x⁰n D⁰n). But for all n ^{2 @} D⁰n ⁰n

6= 0 and the vectors d⁰_n= k^DD⁰ⁿ⁰_n⁰ⁿ⁰_n^k , d⁰_n= 1 satisfy the inequalities:

D

5f(x⁰n) d⁰n

E

; ⁵f(x⁰n):

By Lema 1, for any ² (0 1), > and for all n ^{2 @} there exists z_n^{0 2}X , z_n⁰ =x⁰_n+d⁰_nl_n⁰ such that:

f(z⁰_n) f(x⁰_n) +^D5f(x⁰_n) d⁰_nl_n^0E

D

5f(zn⁰) d⁰n

E

^D5f(x⁰n) d⁰n

E:

Hence, there is not a subsequence l_n⁰_k converging to 0 since we would have: x⁰_{nk k} ^!

!+1

x ⁵f (x)⁶= 0 z⁰nk ²X zn⁰k =x⁰nk +d⁰nkl⁰nk

D

5f(x⁰n_k) d⁰n_k

E

;⁵f(x⁰n_k)

D

5f(z⁰n_k) d⁰n_k

E

^D5f(x⁰n_k) d⁰n_k

E

z⁰nk k^!+1! x

which is a contradiction with Lema 2. In addition, by the de nition of Ap

we have the inequalities:

f(x⁰⁰_n)^;f(x⁰_n)f(z_n⁰)^;f(x⁰_n)^D5f(x⁰_n) d⁰_nl⁰_n^E< 0 ⁸n^2@ 8

and then:

f(z⁰n)^;f(x⁰n)_n!+1^! 0^{) D5}f(x⁰n) d⁰nl⁰n

E

n^!+1! 0: (10) On the other hand,

n^!+1lim

D

5f(x⁰_n) d⁰_nl⁰_n^Elim sup_n

!+1

(^;)^rf(x⁰_n)l_n⁰ =^;krf(x)^klim inf_n!+1l_n⁰ < 0 which is a contradiction with (10). ²

The other descent uniform algorithm requires a non composite de nition:

De nition 8

Let be p^2@ and ²(0 1) > : The point-to-set map Ap :X ^!X ^{given by:}

Ap(z) =^fy²X ^{j 9 2<p} D ²X^p :D ⁶= 0 y = z + D

D

rf (z) D ^{E;k r}f (z)^kD f(z + D )D f (z) + ^D5f (z) D ^E

5f(z + D ) D ^ED5f (z) D ^{E o}

(11) and the stopping rule objective function c(:) = f(:)dene the descent uniform algorithm ^Apw with multidirectional inexact search.:

In words,^Apw nd, in the subspace generated by the directions belonging to D a linear combination D which satis es the ^{; ;}and ^;conditions at the same time. We can consider also an algorithm with variable number of directions, i.e. we use multidirectional search but in subspaces with dierent dimensions at each step:

De nition 9

Let ^fpn^g@ be a sequence of natural numbers. For ² (0 1) > let's consider the sequences of point-to-set maps Apn dened, for p = pn as in (11). The descent algorithm with variable multidirectional inexact search is dened by ^Afpn^gw = ( Apn c)n^2@ where the stopping rule sequence is constant, c(:) = f(:):

Theorem 3

Let X be a normed space and f a continuously Frechet dif- ferentiable and bounded below function. Let be ² (0 1), > ^{. For} any sequence ^fpn^g@ the descent uniform algorithm ^Afpn^gw with variable multidirectional and inexact search, is globally convergent to the set:

; =^fx²X ^j5f (x) = 0^g: 9

Proof.

Lemma (1) ensures again that:

8x =²; ⁹"x > 0 :⁸n ^2@ Apn(x⁰)⁶= ⁸x^{0 2}B(x ") ⁸"²(0 "x):

We will use similar arguments as in the preceding proof to verify the condi- tions (2) of the Polak's global convergence theorem.

i) c(:) = f(:) is continuous.

ii) By contradiction, suppose it is not true. Then⁹x²X with⁵f (x)⁶= 0 such that ⁸"x > 0 and ⁸x < 0, ⁹x^{0 2} B(x "x) and ⁹x^{00 2} Apn(x⁰) such that x f(x⁰⁰)^;f(x⁰)< 0. Taking (x)n =^;_n¹ and ("x)n = _n^{1 8}n^2@, we have⁹ⁿx⁰_n^o_n2@ with x⁰_nn!+1^! x and ⁹ⁿx⁰⁰_n^o_n2@ with x⁰⁰_n² Apn(x⁰_n) such that f(x⁰⁰n)^;f(x⁰n)_n!+1^! 0.

Butx⁰⁰n=x⁰n+l⁰nd⁰n, withd⁰n= ^kD_Dⁿ_nⁿ_n^k Dn²X^pn l⁰n=^kDn n^k

D

5f(x⁰n) d⁰n

E

;⁵f(x⁰_n),^D5f(x⁰⁰_n) d⁰_n^{E D5}f(x⁰_n) d⁰_n^E,f(x⁰⁰_n)f(x⁰_n)+^D5f(x⁰_n) d⁰_nl⁰_n^E,

d⁰n

= 1, l⁰n> 0 ⁸n^2@ then, there is not a subsequenceⁿl⁰n_k

o

k^2@such that ln⁰k k^!+1! 0, since we would havex⁰nk k^!+1! x and x⁰⁰nk =x⁰nk+ln⁰kd⁰nk k^!+1! x and this is a contradiction with Lema 2. Furthermore:

f(x⁰⁰n)^;f(x⁰n)^D5f(x⁰n) d⁰nln⁰

E< 0 and if we have f(x⁰⁰n)^;f(x⁰n)_n^!

!+1

0 we obtain:

^D5f(x⁰n) d⁰nl⁰n

E

n^!+1! 0:

On the other hand

^D5f(x⁰_n) d⁰_nl_n^0E;5f(x⁰_n)l_n⁰ and then:

0 = lim_n

!+1^D5f(x⁰n) d⁰nln⁰

E

limsup_n

!+1

;⁵f(x⁰n)ln⁰ =^;k5f(x)^kliminf_n

!+1 ln⁰ < 0 since there is no subsequence ⁿl⁰n_k

o

k^2@ converging to 0. This contradiction proves the theorem.²

.

10

PROBLEMS

3.1 Discretization of Unconstrained Optimal Control Problems

We consider the following optimal control problem:

minJ(u(:)) = 'x(t¹)] (12)

s:t: _x(t) = f(x(t) u(t)) a:a: t²t⁰:t¹] x(t⁰) = ^x⁰

u(t) ² U a:a: t²t⁰:t¹]

whereu(:)²L^m2 =L²(t⁰:t¹] ^<m) the set of square integrable functions with image in ^<m andx(:)^2C =Ca(t⁰:t¹] ^<n) the set of absolutely continuous functions with image in ^<n.

The maps' :^{<n !<}andf :^{<n<m !<n} are supposed continuously dierentiable respect to its arguments, the functionalJ(u) is assumed to be Frechet dierentiable respect to the L^{m2 ;}norm and the set U ^<m for our purposes, will be considered the whole space ^<m.

If we apply Euler's integration scheme we obtain a discrete approxima- tion of the problem (12) by the following nite dimensional optimal control problem:

minJN(uN) = 'xN] (13)

s:t: xi⁺¹ = xi+hf(xi ui) i = 0 1 ::: N^;1 x⁰ = ^x⁰

uN = ^fu⁰ u¹ ::: uN^;1g

ui ² U i = 0 1 ::: N^;1:

where h = ^(t1_N^;t0) is the integration step, which de nes a partition ^fi i = 0 N^g of t⁰ t¹] inN subintervals:

i =t⁰+ih i = 0 ::: N

which in turn allows us to de ne, from a given feasible control sequence uN = ^fu⁰ u¹ ::: uN^{;1g 2 L}mN of the discrete problem (13), a feasible con- trol function uN(:) ² L^m2 of the continuous problem (12), with the classical

11

piecewise constant form:

uN(t) = uj fort ²j j⁺¹) j = 0 1 ::: N ^;1: (14) We can makethis de nition from any given vector sequence de ned on any partition ^fi^g. The piecewise constant function uN(:) de ned in (14), from a vector sequence uN =^fu⁰ u¹ ::: uN^;1ggiven on a partition^fi i = 0 N^g, will be called "the constant canonical extension" of the sequence to a function in L^m2 :

>From the corresponding discrete trajectory, xN = ^fx⁰ x¹ ::: xN^{g 2}

LnN⁺¹ we can de ne the continuous function:

xNt] = xj+ (t^;j)f(xj uj) t² j j⁺¹] (15) which is the classical polygonal Euler approximation of the solution xN(:) of the dierential equation system of (12), corresponding to the constant canonical extension uN(:), and satisfying:

x_N(t) = f(xN(t) uN(t)) t²t⁰:t¹] xN(t⁰) = ^x⁰:

The continuous function de ned in (15), from a vector sequence xN given in a partition ^fi^g will be called "the polygonal canonical extension" of the sequence to a function in ^C:

We can de ne, reciprocally, from any piecewise continuous function z(t) on t⁰ t¹] with image in ^<k a vector sequence zN = ^fz⁰ z¹ ::: zN^{;1g 2L}kN

de ned on a partition ^fi i = 0 N^g of t⁰ t¹] in the trivial way:

zi =z(i) i = 0 ::: N^;1 (16) and we will call (16) "the sequential canonical reduction" ofz(:) to a sequence in ^LkN:

As nal remarks about notation, we are denoting by ^L_kN the set of nite sequence ofN vectors in^<k which is isomorphic to^<kNthe vector sequence associated with a partition^fi i = 1 ::: N^g with elements^fzi i = 1 ::: N^g is denoted by zN for the constant canonical extension to L^m2 we add paren- thesis zN(:) and for the polygonal canonical extension to ^C we add square brackets zN:] nally, for the sequential canonical reduction of z(:) to ^LkN, we suppress the parenthesis, add an over bar to z and a subindex with the number of vectors de ned in the partition: zN:

12

We note also that a vector sequence uN ^{2 <}mN given on the partition

fi^g can be considered de ned in any partition ^fi^0g of t⁰ t¹] which con- tains ^fi^g (with N⁰ intervals, N⁰ > N) de ning rst the constant canonical extension uN(:) of uN to L^m2 and then taking the sequential canonical reduc- tion u_N⁰ of uN(:) to ^fi^0g. It's easy to see that we also have equality for the constant canonical extension of uN and u_N⁰:

uN(t) = u_N⁰(t) ⁸t ²t⁰:t¹] ⁸N⁰ N

therefore, from now on we will identify uN(:) with uN⁰(:) for any N⁰ > N which corresponds to a partition ^f_i^0g containing^fi^g:

There are many publications about the convergence of the optimal solu- tion of the discrete problem (13) to an optimal solution of the continuous problem (12) when N ^! +¹: There are even quantitative results in the speed of convergence of error estimates of optimal trajectories, controls and adjoint variables, and also many generalized results in several directions. As examples, it can be mentioned the works of Alt 1], Daniel 4], Dontchev 6], Evtuschenko 8], Hager 12], Malanowski 14], Mordukhovich 16], Teo 20]

and many others.

In our opinion, all these results are more in the "stability of optimal so- lution" framework than in the "convergence of an algorithm" context. We haven't seen that the concepts of "descent and feasible directions" or "in- exact line search", appearing naturally in the context of nite dimensional optimization algorithm (see for example 5]), have been su ciently exploited in the design of " nite dimensional approximation" algorithms for optimal control problems and in the proof of they global convergence. In this paper we will present an example of this other point of view.

To any control sequence uN = ^fu⁰ u¹ ::: uN^{;1g 2 L}mN which is feasible to the problem (13), corresponds a piecewise constant function uN(:)²L^m2 through the constant canonical extension, which is feasible to the problem (12) and reciprocally.

>From a current point u_kN ^{2 <mN} the k + 1^;step in a classical descent algorithm for the discrete optimization problem (13) consists of nding a di- rection of decrease⁴ukN in ukN and then performing an inexact line search to nd a step length k and a new point u^kN⁺¹ = u_kN+ k⁴u_kN which satis es the ^; and ^;global convergence conditions of Wolfe. This k +1^;step can be viewed as one step of a single-direction optimization algorithm for the contin- uous problem (12), identifying uN and ⁴ukN with their canonical extensions

13

to L^m2 : If any of the Wolfe conditions fails to hold or even when they both hold, we can perform several iteration now in a non classical algorithm for the discrete problem and this can be viewed as one step of a multidirectional optimization algorithm for the continuous problem. Increasing of N implies an increment of the number of variables in the discrete problem and a reduc- tion of the integration step in Euler's formula for the continuous problem.

It also can be considered as the start of searching in a new direction of a multidirectional optimization algorithm for (12).

The number of iteration made by the optimization algorithm in each approximating discrete problem until a satisfactory point was found would be the number of directions that we take at that step of the algorithm for the continuous problem. This number can be the same or can be varied in dierent iteration during computation, but in practice it is always bounded.

Therefore, since we have global convergence theorems to local minima for unconstrained multidirectional descent methods with inexact line search (using Wolfe conditions) and with possible variable number of direction at each iteration, we have the conditions to model an algorithm for the contin- uous problem which is based on iterations in the discrete one. Hence, we will examine the following questions:

a) When ^;non orthogonal conditions in the discrete problems implies the same condition in the continuous problem?

b) When the global convergence ^; and ^;Wolfe conditions in the dis- crete problem implies the same conditions in the continuous problem?

c) Is it possible to design a global convergence algorithm for the contin- uous problem, only ensuring the (possible varying) Wolfe conditions in the discrete problems?

In the next section we will answer these questions positively, and the idea is quite simple:

1) The descent ^;condition for the direction ⁴ukN(:) depends on the gradient of J(:) at the current point ukN(:) :

D

rJ(u_kN(:)) ⁴u_kN(:)^E_Lm

2

;^rJ(u_kN(:))_Lm

2

4u_kN(:)_Lm

2

therefore we should have the discrete gradient^rJN(ukN) close, in some sense, to the continuous gradient^rJ(u_kN(:)) at the current point,

2) The ^;condition for the direction⁴ukN(:) depends on the gradient of 14

J(:) at the current point ukN(:) and at the new point vkN(:) = ukN(:)+⁴ukN(:) :

D

rJ(vkN(:)) ⁴ukN(:)^E_Lm

2

^DrJ(ukN(:)) ⁴ukN(:)^E_Lm

2

therefore we should also have the discrete gradient ^rJN(vkN) close, in the same sense as before, to the continuous gradient^rJ(v_kN(:)) at the new point, 3) The ^;condition for the direction ⁴ukN(:) depends on the increment of J(:) at the new point respect to the current one, and on the gradient of J(:) at the current point :

J(vkN(:))^;J(ukN(:))^D5J(ukN(:)) ⁴ukN(:)^E_Lm

2

therefore we should have also the discrete incrementJN(vkN)^;JN(ukN) close to the continuous incrementJ(v_kN(:))^;J(u_kN(:)):

We can't expect that the direction ⁴ukN verify both the discrete and the continuous global convergence conditions for the same parameters and . Then, the main di culties are:

- rst, to nd conditions on the parameters and on the closeness of the required quantities in such a way that the corresponding ^{; ;} or ^;condition is satis ed at each problem,

- second, to prove that it is possible to choose the parameter values in such a way that they satis es all the conditions together, and

- third, to design a globally convergent algorithm for the continuous prob- lem, using the above results.

3.2 Relations between Wolfe's Conditions

We need rst to point out some relations between the scalar products and the norm of the sequence of controls and their canonical extensions to L^m2 :

In any feasible control sequence uN = ^fu⁰ u¹ ::: uN^{;1g 2 L}mN of (13), each uj is a vector in^<m with euclidean norm:

kuj^km =^qhuj uj^i<m =^qkuj^k2m=

v

u

u

t

m

X

i⁼¹u²ji

and hence, we can de ne the norm of the sequence uN as the `^2;norm:

kuN^k`² = ^qhuN uN^iL_mN =^qhuN uN^i<mN =

v

u

u

u

t

N^X;1 j⁼⁰

huj uj^i<m = 15

=

v

u

u

u

t

N^X;1

j^{=0 k}uj^k2m =

v

u

u

u

t

N^X;1 j⁼⁰

m

X

i⁼¹u²_ji =^kuN^kmN

i.e. the euclidean norm of the vector (u⁰¹ ::: u⁰m ::: uN^;11 ::: uN^;1:m) ²

<mN:

For any uN ^2LmN the L^{m2 ;}norm of the function uN(:) can be calculated:

kuN(:)^kL^m2 =

s

Z t¹

t^{0 k}uN(t)^k2mdt =

v

u

u

u

t

N^X;1 j⁼⁰

Z j⁺¹

j ^kuj^k2mdt =^ph^kuN^kmN

and for any v(:)²L^m2 we have:

hv(:) uN(:)ⁱ_L^m2 =^Z_t^t1

0

v^T(t)uN(t)dt = ^NX;1

j⁼⁰

Z j⁺¹

j v^T(t)dt

!

uj:

If, for example, v(:) is the function of L^m2 representing the gradient

rJ(uN(:)) of the objective function of the continuous problem (12) at the point uN(:), and since L^m1 is dense in L^m2 it should have the following well known formula: (see 17]):

rJ(uN(:))(t) = ^;_N^T(t)fu(xN(t) uN(t)) a:a: t²t⁰:t¹] where N(:) is the solution of the adjoint dierential system:

_N(t) = ^;fx^T(xN(t) uN(t))N(t) a:a: t²t⁰:t¹] N(t¹) = ^;'^Tx xN(t¹)]

and xN(:) is the continuous trajectory of the problem (12) corresponding to uN(:) then we have:

hrJ(uN(:))(:) uN(:)ⁱ_L^m2 =^;NX;1

j⁼⁰

Z j⁺¹

j ^TN(t)fu(xN(t) uN(t))dt

!

uj: Another important example is when v(:) is the constant canonical ex- tension to L^m2 of the vector sequence ^rJN(uN), i.e. the vector sequence of the gradient of the discrete problem objective function (13) at the point

16