Two globally convergent nonmonotone trust-region methods for unconstrained optimization

0.57 MB·English

by  Masoud Ahookhosh

Noname manuscript No.
(will be inserted by the editor)
Two globally convergent nonmonotone trust-region methods for
unconstrained optimization
Masoud Ahookhosh · Susan Ghaderi
5
1 Abstract This paper addresses some trust-region methods equipped with nonmonotone strategies for
0 solving nonlinear unconstrained optimization problems. More specifically, the importance of using non-
2
monotone techniques in nonlinear optimization is motivated, then two new nonmonotone terms are pro-

n posed,andtheircombinationsintothetraditionaltrust-regionframeworkarestudied.Theglobalconver-
a gence to first- and second-order stationary points and local superlinear and quadratic convergence rates
J
forbothalgorithmsareestablished.NumericalexperimentsontheCUTEsttestcollectionofunconstrained
9
problemsandsome highlynonlineartestfunctionsare reported, whereacomparisonamong state-of-the-

art nonmonotone trust-region methods show the efficiency of the proposed nonmonotne schemes.
]
C
Keywords Unconstrained optimization · Black-box oracle · Trust-region method · Nonmonotone
O
strategy · Global convergence
.
h
Mathematics Subject Classification (2000) 90C30 · 65K05
t
a
m
[ 1 Introduction

1
In this paper we consider the unconstrained minimization problem
v
1
minimize f(x)
1 (1)
0 subject to x∈Rn,
2
0 where f :Rn →R is a real-valued nonlinear function, which is bounded and continuously-differentiable.
. We suppose that first- or second-order black-box oracle of f is available.
1
0 Motivation & history. Trust-region methods, also called restricted step methods [21], are a class of
5 iterative schemes developed to solve convex or nonconvex optimization problems, see, for example, [13].
1 They also developed for nonsmooth problems, see [13, 16, 45, 23]. Trust-region methods have strong
:
v convergence properties, are reliable and robust in computation, and can handle ill-conditioned problems,
i cf.[34,35].Letx bethecurrentiteration.Intrust-regionframeworktheobjectivef isapproximatedby
X k
a simple model in a specific region around x such that it is an acceptable approximation of the original
k
r
a objective, which is called region of trust. Afterward, the model is minimized subject to the trust-region
constraint to find a new trial point d . Hence the simple model means that it can be minimized much
k
easier than the original objective function. If the founded model is an adequate approximation of the
objective function within the trust-region, then the point x =x +d is accepted by the trust-region
k+1 k k
method and the region can be expanded for the next iteration; conversely, if the approximation is poor,
the region is contracted and the model is minimized within the contracted region. This scheme will be
continued until finding an acceptable trial step d guaranteeing an acceptable agreement between the
k
model and the objective function.
M.Ahookhosh
FacultyofMathematics,UniversityofVienna,Oskar-Morgenstern-Platz1,1090Vienna,Austria
E-mail:[email protected]
S.Ghaderi
DepartmentofMathematics,FacultyofScience,RaziUniversity,Kermanshah,Iran
E-mail:[email protected]
2 MasoudAhookhosh,SusanGhaderi
Severalquadraticandnon-quadraticmodelshavebeenproposedtoapproximatetheobjectivefunction
inoptimization,see[14,22,36,39],however,theconicandquadraticmodelsaremorepopular,see[17,37].
If the approximated model is quadratic, i.e.,
1
q (d):=f +gTd+ dTB d, (2)
k k k 2 k
where f = f(x ), g = ∇f(x ), and B ≈ ∇2f(x ), the trust-region method can be considered as a
k k k k k k
globally convergent generalization on classical Newton’s method. Then the trust-region sunproblem is
defined by
minimize q (d),
k (3)
subject to (cid:107)d(cid:107)≤δ .
k
Hence the trust-region is commonly a norm ball C defined by
C :={d∈Rn |(cid:107)d(cid:107)≤δ },
k
where δ >0 is a real number called trust-region radius, and (cid:107)·(cid:107) is any norm in Rn, cf. [29]. Since C is
k
compactandthemodeliscontinuous,thetrust-regionsubproblemattainsitsminimizeronthesetC.The
most computational cost of trust-region methods relates to minimizing the model over the trust-region
C. Hence finding efficient schemes for solving (3) has received much attention during past few decades,
see [19, 20, 25, 31, 38]. Once the step d is computed, the quality of the model in the trust-region is
evaluated by a ratio of the actual reduction of objective, f −f(x +d), to the predicted reduction of
k k
model, q (0)−q (d), i.e.,
k k
f −f(x +d)
r = k k . (4)
k q (0)−q (d)
k k
Foraprescribedpositiveconstantµ ∈(0,1],ifr ≥µ ,themodelprovidesareasonableapproximation,
1 k 1
the step is accepted, i.e., x = x +d , and the trust-region C can be expanded for the next step.
k+1 k k
Otherwise, the trust-region C should be contracted by decreasing the radius δ and the subproblem (3)
k
is solved in the reduced region. This scheme is continued until that the step d accepted by trust-region
test r ≥µ . Our discussion can be summarized in the following algorithm:
k 1
Algorithm 1: TTR (traditional trust-region algorithm)
Input:x0∈Rn,B0∈Rn×n,kmax;0<µ1≤µ2≤1,0<ρ1≤1≤ρ2,ε>0;
Output:x ;f ;
b b
1 begin
2 δ0←(cid:107)g0(cid:107); k←0;
3 while(cid:107)gk(cid:107)≥ε & k≤kmax do
4 solvethesubproblem(3)tospecifydk;
5 x(cid:98)k←xk+dk; computef(x(cid:98)k);
6 determinerk using(4);
7 whilerk<µ1 do
8 δk←ρ1δk;
9 solvethesubproblem(3)tospecifydk;
10 x(cid:98)k←xk+dk; computef(x(cid:98)k)
11 determinerk using(4);
12 end
13 xk+1←x(cid:98)k;
14 if rk≥µ2 then
15 δk+1←ρ2δk;
16 end
17 updateBk+1; k←k+1;
18 end
19 xb←xk; fb←fk;
20 end
In Algorithm 1, it follows from r ≥µ and q (0)−q (d )>0 that
k 1 k k k
f −f ≥µ (q (0)−q (d ))>0,
k k+1 1 k k k
implying f ≤ f . This means that the sequence of function values {f } is monotonically decreasing,
k+1 k k
i.e.,thetraditionaltrust-regionmethodisalsocalledthemonotone trust-region method.Thisfeature
seems natural for minimization schemes, however, it slows down the convergence of TTR to a minimizer
if the objective involves a curved narrow valley, see [1, 27]. To observe the effect of nonmonotonicity on
TTR, we study the next example.
Twogloballyconvergentnonmonotonetrust-regionmethodsforunconstrainedoptimization 3
Example 1 Consider the two-dimensional Nesterov-Chebysheve-Rosenbrock function , cf. [28],
1
f(x ,x )= (x −1)2+(x −2x2+1)2,
1 2 4 1 2 1
where we solve the problem (1) by Newton’s method and TTR with the initial point x =(−0.61,−1). It
0
is clear that (1,1) is the optimizer. The implementation indicates that Newton’s method needs 7 iterations
and 8 function evaluations, while monotone trust-region method needs 22 iterations and 24 function
evaluations. We depict the contour plot of the objective and iterations as well as a diagram for function
values versus iteration attained by these two algorithms in Figure 1. Subfigure (a) of Figure 1 shows that
the iterations of TTR follow the bottom of the valley in contrast to those for Newton’s method that can
go up and down to reach the ε-solution with the accuracy parameter ε = 10−5. We see that Newton’s
method attains larger step compared with those of TTR. Subfigure (b) of Figure 1 illustrates function
values versus iterations for both algorithms showing that the related function values of TTR decreases
monotonically, while it is fluctuated nonmonotonically for Newton’s method.
102
1
0.5 100
0 10−2
−0−.15 function values10−4
10−6
−1.5
−2 10−8
Nes−Cheb−Rosen
TTR TTR
−2.5 Newton Newton
10−10
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 100 101 102
iterations
(a)Nes-Cheb-Rosencontourplot&iterations (b)functionvaluesversusiterations
Fig. 1: A comparison between Newton’s method and TTR: Subfigure (a) illustrates the contour plot
of the two-dimensional Nesterov-Chebysheve-Rosenbrock function and iterations of Newton and TTR;
Subfigure (b) shows the diagram of function values versus iterations.
In general the monotonicity may result to the slow iterative schemes for highly nonlinear or badly-
scaled problems. To avoiding this algorithmic limitation, the idea of nonmonotone strategies has been
proposedtracedbacktothewatch-dogtechniquetoovercometheMartoseffectforconstrainedoptimiza-
tion [12]. To improve the performance of Armijo’s line search, Grippo et al. in 1986 [27] proposed the
modified Armijo’s rule
f(x +α d )≤f +σα gTd , k =0,1,2,··· ,
k k k l(k) k k k
with the step-size α >0, σ ∈(0,1/2), and
k
f = max {f }, (5)
l(k) k−j
0≤j≤m(k)
where m(0) = 0, m(k) ≤ min{m(k − 1) + 1,N} for nonnegetive integer N. It was shown that the
associatedschemeisgloballyconvergent,andnumericalresultsreportedinGrippoetal.[27]andToint
[40]showedtheeffectivenessoftheproposedidea.Motivatedbytheseresults,thenonmonotonestrategies
has received much attention during past few decades. For example, in 2004, Zhang & Hager in [46]
proposed the nonmonotone term
(cid:26) (cid:26)
f if k =0, 1 if k =0,
C = 0 Q = (6)
k (η Q C +f(x ))/Q if k ≥1, k η Q +1 if k ≥1,
k−1 k−1 k−1 k k k−1 k−1
4 MasoudAhookhosh,SusanGhaderi
where 0≤η ≤η ≤η ≤1. Recently, Mo et al. in [30] and Ahookhosh et al. in [3] studied the
min k−1 max
nonmonotone term
(cid:26)
f if k =1,
D = k (7)
k η D +(1−η )f if k ≥2,
k k−1 k k
where η ∈ [η ,η ], η ∈ [0,1], η ∈ [η ,1]. More recently, Amini et al. in [7] proposed the
k min max min max min
nonmonotone term
R =η f +(1−η )f , (8)
k k l(k) k k
where0≤η ≤η ≤1andη ∈[η ,η ].Inallcasesitwasprovedthattheschemesareglobally
min max k min max
convergent and enjoy the better performance compared with monotone ones.
At the same importance of using monmonotone strategies for inexact line search techniques, the
combination of trust-region methods with nonmonotone strategies is interesting. Historically, the first
nonmonotone trust-region method was proposed in 1993 by Deng et al. in [15] for unconstrained opti-
mization.Undersomeclassicalassumptions,theglobalconvergenceandthelocalsuperlinearconvergence
rate were established. Nonmonotone trust-region methods were also studied by several authors such as
Toint [41], Xiao & Zho [43], Xiao & Chu [44], Zhou & Xiao [47], Ahookhosh & Amini [2], Amini
& Ahookhosh [6], and Mo et al. [30]. Recently, Ahookhosh & Amini in [1] and Ahookhosh et
al. in [4] proposed two nonmonotone trust-region methods using the nonmonotone term (8). Theoretical
resultswerereported,andnumericalresultsshowedtheefficiencyoftheproposednonmonotonemethods.
Content. In this paper we propose a trust-region method equipped with two novel nonmonotone terms.
More precisely, we first establish two nonmonotone terms and then combine them with Algorithm 1 to
construct two nonmonotone trust-region algorithms. If k ≥ N, the new nonmonotone terms are defined
by a convex combination of the last N successful function values, and if k < N, either a convex combi-
nation of k successful function values or f is used. The global convergence to first- and second-order
l(k)
stationary points is established on some classical assumptions. Moreover, local superlinear and quadratic
convergenceratesfortheproposedmethodsarestudied.Numericalresultsregardingexperimentsonsome
highly nonlinear problems and on 112 unconstrained test problems from the CUTEst test collection [24]
are reported indicating the efficiency of the proposed nonmonotone terms.
The remainder of paper is organized as follow. In Section 2 we propose new nonmonotone terms and
their combination with the trust-region framework. The global convergence of the proposed methods are
given in Section 3. Numerical results are reported in Section 4. Finally, some conclusions are given in
Section 5.
2 Novel nonmonotone terms and algorithm
In this section we first present two novel nonmonotone terms and then combine them into trust-region
frameworktointroducetwononmonotonetrust-regionalgorithmsforsolvingtheunconstrainedoptimiza-
tion problem (1).
We first assume that k denotes the current iteration and N ∈ N is a constant. The main idea is to
construct a nonmonotone term determined by a convex combination of the last k successful function
values if k < N and by a convex combination of the last N successful function values if k ≥ N. In the
other words, we construct new terms using function values collected in the set
(cid:26)
{f ,f ,··· ,f } if k <N,
F := 0 1 k (9)
k {f ,f ,··· ,f } if k ≥N,
k−N+1 k−N+2 k
which should be updated in each iteration. To this end, motivated by the term (40), we construct T
k
using the subsequent procedure

T =f if k =0,
TT012 ==((110−−ηη01))ff12++ηη01f(10−η0)f1+η1η0f0 iiff kk ==12,,
. . .
. . .
TT. N−1 ==. ((11−−ηηN−2))ffN−+1η+ηN−(12(−1η−ηN−)f3)fN−+2·+···+··η+ηN−·2···η··fη0f0 i.iff kk ==NN,−1,
N N−1 N N−1 N−2 N−1 N−1 0 0
Twogloballyconvergentnonmonotonetrust-regionmethodsforunconstrainedoptimization 5
where η ∈[0,1), for i=1,2,··· ,N, are some weight parameters. Hence the new term is generated by
i
(cid:26)
(1−η )f +η T if k <N,
T := k−1 k k−1 k−1 (10)
k (1−η ) f +η (1−η ) f +···+η ···η f if k ≥N,
k−1 k k−1 k−2 k−1 k−1 k−N k−N
whereT =f andη ∈[0,1)fori=1,2,··· ,k.ToshowthatT isaconvexcombinationofthecollected
0 0 i k
functionvaluesF ,itisenoughtoshowthatthesummationofmultipliersareequaltounity.Fork ≥N,
k
the definition for T implies
k
(1−η )+η (1−η )+···+η ···η (1−η )+η ···η =1 (11)
k−1 k−1 k−2 k−1 k−N−1 k−N k−1 k−N
For k <N, a similar summation of the last k multipliers is equal to one. Therefore, the generated term
T is a convex combination of the elements of F .
k k
The procedure of defining T clearly implies that the set F should be updated and saved in each
k k
iteration. Moreover, N(N +1)/2 multiplications is required to compute T . To avoid saving F and
k k
decrease the number of multiplications, we derive a recursive formula for (10). From the definition of T ,
k
for k ≥N, it follows that
T −η T =(1−η ) f +η (1−η ) f +···+η ···η f
k k−1 k−1 k−1 k k−1 k−2 k−1 k−1 k−N k−N
−η (1−η ) f −···−η ···(1−η ) f −η η ···η f
k−1 k−2 k−1 k−1 k−N−1 k−N k−1 k−2 k−N−1 k−N−1
=(1−η ) f +η η ···η (f −f )
k−1 k k−1 k−2 k−N−1 k−N k−N−1
=(1−η ) f +ξ (f −f )
k−1 k k k−N k−N−1
where ξ :=η η ···η . For k ≥N, this equation leads to
k k−1 k−2 k−N−1
T =(1−η ) f +η T +ξ (f −f ), (12)
k k−1 k k−1 k−1 k k−N k−N−1
whichrequirstosaveonlyf andf andonlyneedsthreemultiplications.Moreover,thedefinition
k−N k−N−1
of ξ implies
k
η η
ξ =η η ···η = k−1 η η ···η = k−1 ξ . (13)
k k−1 k−2 k−N−1 η k−2 k−3 k−N−2 η k−1
k−N−2 k−N−2
If ξ is recursively updated by (13), (10), and (12), a new nonmonotone term is defined by
k
(cid:26)
f +η (T −f ) if k <N,
Tk := mkax(cid:8)Tk−1,f (cid:9)k k if k ≥N, (14)
k k
where the max term is added to guarantee T ≥f .
k k
AsdiscussedinSection1,nonmonotoneschemesperformbetterwhentheyusestrongernonmonotone
termsfarawayfromtheoptimizerandweakeroneclosetoit.Thismotivateustoconsideranewversionof
thederivednonmonotonetermbyusingf incasesthatk <N.Moreprecisely,thesecondnonmonotone
l(k)
term is defined by
(cid:26)
f if k <N,
Tk = ml(akx)(cid:8)T ,f (cid:9) if k ≥N, (15)
k k
whereξ isdefinedby(13).Itisclearthatthenewtermusesastrongertermf definedby(5)forfirst
k l(k)
k <N iterations and then employs the relaxed convex term proposed above.
Now, to employ the proposed nonmonotone terms in the trust-region framework, it is enough to
replace the ratio r (4) by the nonmonotone ratio
k
T −f(x +d)
r = k k , (16)
(cid:98)k q (0)−q (d)
k k
where T is defined by (14) or (15). Hence in trust-region framework we replace (4) by (16). Notice that
k
if r ≥µ , the,
(cid:98)k 1
T −f ≥µ (q (0)−q (d ))≥0.
k k+1 1 k k k
This implies that f can be larger than f , however, the elements of {f } cannot arbitrarily increase,
k+1 k k
and the maximum increase is controlled by the nonmonotone term T . Moreover, the definitions (14)
k
and (15) imply that r ≥r increasing the possibility of attaining larger steps for nonmonotone schemes
(cid:98)k k
compared with monotone ones.
6 MasoudAhookhosh,SusanGhaderi
The above-mentioned discussion leads to the following nonmonotone trust-region algorithm:
Algorithm 2: NMTR (nonmonotone traditional trust-region algorithm)
Input: x ∈Rn, B ∈Rn×n, k ; 0<µ ≤µ ≥1, 0<ρ ≤1≤ρ ≥1, ε>0;
0 0 max 1 2 1 2
Output: x ; f ;
b b
1 begin
2 δ0 ←(cid:107)g0(cid:107); k ←0;
3 while (cid:107)gk(cid:107)≥ε & k ≤kmax do
4 solve the subproblem (3) to specify dk;
5 x(cid:98)k ←xk+dk; compute f(xˆk);
6 determine r(cid:98)k using (16);
7 if r(cid:98)k <µ1 while r(cid:98)k <µ1 do
8 δk ←ρ1δk;
9 solve the subproblem (3) to specify dk;
10 x(cid:98)k ←xk+dk; compute f(xˆk)
11 determine r(cid:98)k using (16);
12 end
13 xk+1 ←x(cid:98)k;
14 if r(cid:98)k ≥µ2 then
15 δk+1 ←ρ2δk;
16 end
17 update Bk+1; k ←k+1;
18 update Tk+1;
19 update ηk+1;
20 end
21 xb ←xk; fb ←fk;
22 end
In Algorithm 2, if r ≥µ (Line 7), it is called a successful iteration and if r ≥µ (Line 14), it is
(cid:98)k 1 (cid:98)k 2
called a very successful iteration. In addition, in the algorithm, the loop started from Line 3 to Line
20 is called the outer cycle, and the loop started from Line 7 to Line 12 is called the inner cycle.
3 Convergence analysis
This section concerns with the global convergence to first- and second-order stationary points of the
sequence {x } generated by Algorithm 2. More precisely, we intend to prove that all limit point x∗ of
k
the sequence {x } satisfy the condition g(x∗)=0, and there exists a point x∗ satisfying g(x∗)=0 where
k
H(x∗) is positive semidefinite. Furthermore, we show that Algorithm 2 is well-defined, which means that
theinnercycleofthealgorithmwillbeleavedafterafinitenumberinternaliterations,andthenproveits
global convergence. Moreover, local superlinear and quadratic convergence rates are investigated under
some classical assumptions.
To prove the global convergence of the sequence {x } generated by Algorithm 2, we require to make
k
the following assumptions:
(H1) The objective function f is continuously differentiable and has a lower bound on the upper level
set L(x )={x∈Rn |f(x)≤f(x )}.
0 0
(H2) The sequence {B } is uniformly bounded, i.e., there exists a constant M >0 such that
k
(cid:107)B (cid:107)≤M,
k
for all k ∈N.
(H3) There exists a constant c>0 such that the trial step d satisfies (cid:107)d (cid:107)≤c(cid:107)g (cid:107).
k k k
Twogloballyconvergentnonmonotonetrust-regionmethodsforunconstrainedoptimization 7
We also assume that the decrease on the model q is at least as much as a fraction of the decrease
k
obtained by the Cauchy point guaranteeing that there exists a constant β ∈(0,1) such that
(cid:26) (cid:27)
(cid:107)g (cid:107)
q (0)−q (d )≥β(cid:107)g (cid:107) min δ , k , (17)
k k k k k (cid:107)B (cid:107)
k
for all k. This condition is called the sufficient reduction condition. Inequality (17) implies that d (cid:54)= 0
k
whenever g (cid:54)= 0. It is noticeable that there are several schemes that can solve the the trust-region
k
subproblem (3) such that (18) is valid, see, for example, [13, 32].
Lemma 2 Suppose that sequence {x } is generated by Algorithm 2, then
k
|f −f(x +d )−(q (0)−q (d ))|≤O((cid:107)d (cid:107)2).
k k k k k k k
Proof The proof can be found in [12]. (cid:116)(cid:117)
Lemma 3 Suppose that the sequence {x } is generated by Algorithm 1, then we get
k
f ≤T ≤f , (18)
k k l(k)
for all k ∈N∪{0}.
Proof For k ≤ N, we consider two cases: (i) T is defined by (14); (ii) T is defined by (15). In Case (i)
k k
Lemma 2.1 in [3], f ≤f , for i=0,1,···k, and the fact that summation of multipliers in T equal to
i l(k) k
one give the result. Case (ii) is evident from (15).
For k ≥N, if T =f , the result is evident. Otherwise, since
k k
(1−η )+η (1−η )+···+η ···η (1−η )+η ···η =1, (19)
k−1 k−1 k−2 k−1 k−N−1 k−N k−1 k−N
the fact that f ≤f , for i=k−N +1,··· ,k, and (10) imply
i l(k)
f ≤T =(1−η ) f +η (1−η ) f +···+η ···η f
k k k−1 k k−1 k−2 k−1 k−1 k−N k−N
≤[(1−η )+η (1−η )+···+η ···η ]f =f ,
k−1 k−1 k−2 k−1 k−N l(k) l(k)
giving the result. (cid:116)(cid:117)
Lemma 4 Suppose that sequence {x } is generated by Algorithm 2, then the sequence {f } is decreas-
k l(k)
ing.
Proof The condition (18) implies that T ≤f . If x is accepted by Algorithm 2, then
k l(k) k+1
f −f(x +d ) T −f(x +d )
l(k) k k ≥ k k k ≥µ ,
q (0)−q (d ) q (0)−q (d ) 1
k k k k k k
leading to
f −f(x +d )≥µ (q (0)−q (d ))≥0, for all k ∈N,
l(k) k k 1 k k k
implying
f ≥f , for all k ∈N. (20)
l(k) k+1
Now, if k ≥N, by using m(k+1)≤m(k)+1 and (20), we get
f = max {f }≤ max {f }=max{f ,f }≤f .
l(k+1) k−j+1 k−j+1 l(k) k+1 l(k)
0≤j≤m(k+1) 0≤j≤m(k)+1
For k < N, it is obvious that m(k) = k. Since, for any k, f ≤ f , it is clear that f = f . Therefore,
k 0 l(k) 0
in both cases, the sequence {f } is decreasing. (cid:116)(cid:117)
l(k)
Lemma 5 Suppose that (H1) holds and the sequence {x } is generated by Algorithm 2, then L(x )
k 0
involves {x }.
k
Proof The definition of T indicates that T = f . By induction, we assume that x ∈ L(x ), for all
k 0 0 i 0
i=1,2,··· ,k, and then prove that x ∈L(x ). From (18), we get
k+1 0
f ≤T ≤f ≤f ≤f ,
k+1 k+1 l(k+1) l(k) 0
implying that L(x ) involves the sequence {x }. (cid:116)(cid:117)
0 k
8 MasoudAhookhosh,SusanGhaderi
Corollary 6 Suppose that (H1) holds and the sequence {x } is generated by Algorithm 2. Then the
k
sequence {f } is convergent.
l(k)
Proof The assumption (H1) and Lemma 4 imply that there exists a constant λ such that
λ≤f ≤f ≤···≤f ≤f ,
k+n l(k+n) l(k+1) l(k)
for all n∈N. This implies that the sequence {f } is convergent. (cid:116)(cid:117)
l(k)
Lemma 7 Suppose that (H1)-(H3) hold and the sequence {x } is generated by Algorithm 2, then
k
lim f(x )= lim f . (21)
l(k) k
k→∞ k→∞
Proof The condition (18) and Lemma 7 of [1] imply that the result is valid. (cid:116)(cid:117)
Corollary 8 Suppose (H1)-(H3) hold and the sequence {x } is generated by Algorithm 2, then we
k
lim T = lim f . (22)
k k
k→∞ k→∞
Proof From (18) and Lemma 7, the result is obtained. (cid:116)(cid:117)
Lemma 9 Suppose that (H1) and (H2) hold, and the sequence {x } is generated by Algorithm 2. Then
k
if (cid:107)g (cid:107)≥ε>0, we have
k
(i) The inner cycle of Algorithm 2 is well-defined;
(ii) For any k, there exists a nonnegative integer p such that x is a very successful iteration.
k+p+1
Proof (i) Let t denotes the internal iteration counter in step k, and dtk and δtk respectively show the
k k k
solutionofthesubproblem(3)andthecorrespondingtrust-regionradiusintheinternaliterationt .The
k
fact that (cid:107)g (cid:107)≥ε>0, (H2), and (17) imply
k
(cid:26) (cid:107)g (cid:107)(cid:27) (cid:110) ε (cid:111)
q (0)−q (dtk)≥β(cid:107)g (cid:107) min δtk, k ≥βε min δtk, . (23)
k k k k k (cid:107)B (cid:107) k M
k
Then Line 8 of Algorithm 2 implies
lim δtk =0.
tk→∞ k
From This, Lemma 2, and (24), we obtain
|rk−1|=(cid:12)(cid:12)(cid:12)(cid:12)fqk−(0)f−(xkq+(ddtktkk)) −1(cid:12)(cid:12)(cid:12)(cid:12)=(cid:12)(cid:12)(cid:12)(cid:12)fk−f(xk+q d(0tkk))−−q(q(kd(t0k))−qk(dtkk))(cid:12)(cid:12)(cid:12)(cid:12)
k k k k k k
O((cid:107)dtk(cid:107)2) O((δtk)2)
≤ k ≤ k →0 (t →∞),
βε min(cid:8)δtk,ε/M(cid:9) βε min(cid:8)δtk,ε/M(cid:9) k
k k
implying that there exists a positive integer k such that for k ≥k we have r ≥µ . This and (18) lead
0 0 k 1
to
T −f(x +dtk) f −f(x +dtk)
r = k k k ≥ k k k ≥µ ,
(cid:98)k q (0)−q (dtk) q (0)−q (dtk) 1
k k k k k k
implying that the inner cycle is well-defined.
(ii)Assumethatthereexistsapositiveintegerk suchthatforanarbitrarypositiveintegerpthepoint
x is not very successful. Hence, for any constant p=0,1,2,···, we get
k+p+1
r <µ .
(cid:98)k+p 2
The fact that (cid:107)g (cid:107)≥ε>0, (H2), and (17) imply
k
(cid:26) (cid:27)
(cid:107)g (cid:107)
T −f(x +d )≥µ (q (0)−q (d ))≥βµ (cid:107)g (cid:107) min δ , k+p
k+p k+p k+p 1 k+p k+p k+p 1 k+p k+p (cid:107)B (cid:107)
k+p (24)
(cid:110) ε (cid:111)
≥βµ ε min δ , .
1 k+p M
Twogloballyconvergentnonmonotonetrust-regionmethodsforunconstrainedoptimization 9
By using (22) and (24), we can write
lim δ =0. (25)
k+p
p→∞
From Lemma 2, (25), and (23), we obtain
(cid:12) (cid:12)
|rk+p−1|=(cid:12)(cid:12)(cid:12)f(xqk+p()0−)−f(qxk+p(d+dk)+p) −1(cid:12)(cid:12)(cid:12)
k+p k+p k+p
(cid:12) (cid:12)
=(cid:12)(cid:12)f(xk+p)−f(xk+p+dk+p)−(qk+p(0)−qk+p(dk+p))(cid:12)(cid:12)
(cid:12) q (0)−q (d ) (cid:12)
k+p k+p k+p
O((cid:107)d (cid:107)2) O(δ2 )
≤ k+p ≤ k+p →0 (p→∞).
βε min{δ ,ε/M} βε min{δ ,ε/M}
k+p k+p
Then, for a sufficiently large p, we get r ≥µ leading to
k+p 2
T −f(x +d ) f(x )−f(x +d )
k+p k+p k+p ≥ k+p k+p k+p ≥µ .
q (0)−q (d ) q (0)−q (d ) 2
k+p k+p k+p k+p k+p k+p
implying r ≥ µ , for a sufficiently large p. This contradicts with assumption r < µ giving the
(cid:98)k+p 2 (cid:98)k+p 2
result. (cid:116)(cid:117)
Lemma9(i)impliesthattheinnercyclewillbeleavedafterafinitenumberofinternaliterations,and
Lemma 9(ii) implies that if the current iteration is not a first-order stationary point, then at least there
exists a very successful iteration point, i.e., the trust-region radius δ can be enlarged. The next result
k
gives the global convergence of the sequence {x } of Algorithm 2.
k
Theorem 10 Supposethat(H1)and(H2)hold,andsupposethesequence{x }isgeneratedbyAlgorithm
k
2. Then
liminf(cid:107)g (cid:107)=0. (26)
k
k→∞
Proof We consider two cases: (i) Algorithm 2 has finitely many very successful iterations; (ii) Algorithm
2 has infinitely many very successful iterations.
In Case 1, we suppose that k be the largest index of very successful iterations. If (cid:107)g (cid:107)>0, then
0 k0+1
Lemma 9(ii) implies that there exist a very successful iteration with larger index than k . This is a
0
contradiction to the definition of k .
0
In Case 2, by contradiction, we assume that there exist constants ε>0 and K >0 such that
(cid:107)g (cid:107)≥ε, (27)
k
for all k ≥K. If x is a successful iteration and k ≥K, then by using (H2), (17), and (27), we get
k+1
T −f(x +d )≥µ (q (0)−q (d ))
k k k 1 k k k
(cid:26) (cid:107)g (cid:107)(cid:27) (cid:110) ε (cid:111) (28)
≥βµ (cid:107)g (cid:107) min δ , k ≥βµ ε min δ , ≥0.
1 k k (cid:107)B (cid:107) 1 k M
k
It follows from this inequality and (22) that
lim δ =0. (29)
k
k→∞
Since Algorithm 2 has infinitely many very successful iterations, then Lemma 9(ii) and (27) imply that
thesequence{x }involvesinfinitelymanyverysuccessfuliterationsinwhichthetrust-regionisenlarged,
k
which is a contradiction with (29). This implies the result is valid. (cid:116)(cid:117)
Theorem 11 Suppose that (H1) and (H2) hold, and the sequence {x } is generated by Algorithm 2.
k
Then
lim (cid:107)g (cid:107)=0. (30)
k
k→∞
Moreover, there is no limit point of the sequence {x } to be a local maximizer of f.
k
10 MasoudAhookhosh,SusanGhaderi
Proof Bycontradiction,weassumelim (cid:107)g (cid:107)=(cid:54) 0.Hencethereexistsε>0andaninfinitesubsequence
k→∞ k
of {x }, indexed by {t }, such that
k i
(cid:107)g (cid:107)≥2ε>0, (31)
ti
foralli∈N.Theorem10ensurestheexistence,foreacht ,afirstsuccessfuliterationr(t )>t suchthat
i i i
(cid:107)g (cid:107)<ε. We denote r =r(t ). Hence there exists another subsequence, indexed by {r }, such that
r(ti) i i i
(cid:107)g (cid:107)≥ε for t ≤k <r , (cid:107)g (cid:107)<ε. (32)
k i i ri
We now restrict our attention to the sequence of successful iterations whose indices are in the set
κ={k ∈N | t ≤k <r }.
i i
Using (32), for every k ∈κ, (28) holds. It follows from (22) and (28) that
lim δ =0, (33)
k
k→∞
for k ∈κ. Now, using (H2), (17), and (cid:107)g (cid:107)≥ε, the condition (23) holds, for k ∈κ. This, Lemma 2, and
k
(33) lead to
(cid:12) (cid:12) (cid:12) (cid:12)
|rk−1|=(cid:12)(cid:12)(cid:12)fqk−(0)f−(xkq+(ddk)) −1(cid:12)(cid:12)(cid:12)=(cid:12)(cid:12)(cid:12)fk−f(xkq+(d0k))−−q(q(kd(0))−qk(dk))(cid:12)(cid:12)(cid:12)
k k k k k k
O((cid:107)d (cid:107)2) O(δ2)
≤ k ≤ k →0 (k →∞, k ∈κ).
βε min{δ ,ε/M} βεδ
k k
Thus, for a sufficiently large k+1∈κ, we get
f −f(x +d )≥µ (q (0)−q (d ))
k k k 1 k k k
(cid:26) (cid:107)g (cid:107)(cid:27) (cid:110) ε (cid:111) (34)
≥βµ (cid:107)g (cid:107)min δ , k ≥βµ ε min δ , .
1 k k (cid:107)B (cid:107) 1 k M
k
The condition (33) implies that δ ≤ε/M. Hence, for a sufficiently large k ∈κ, we obtain
k
1
δ ≤ (f −f ). (35)
k βµ k k+1
1
Then (18) and (35) imply
r(cid:88)i−1 r(cid:88)i−1 1 1
(cid:107)x −x (cid:107)≤ (cid:107)x −x (cid:107)≤ δ ≤ (f −f )≤ (T −f ), (36)
ti ri j j+1 j βµ ti ri βµ ti ri
1 1
j∈κ,j=ti j∈κ,j=ti
for a sufficiently large i. Now, Corollary 8 implies
1
0≤ lim (cid:107)x −x (cid:107)≤ lim (T −f )=0,
i→∞ ti ri i→∞βµ1 ti ri
leading to
lim (cid:107)x −x (cid:107)=0.
i→∞ ti ri
Since the gradient is continuous, we get
lim (cid:107)g −g (cid:107)=0. (37)
i→∞ ti ri
In view of the definitions of {t } and {r }, it is impossible, guaranteeing (cid:107)g −g (cid:107)≥ε. Therefore, there
i i ti ri
is no subsequence that satisfies (31) giving the result.
To observe there is no limit point of the sequence {x } to be a local maximizer of f, see [27]. (cid:116)(cid:117)
k
ThenextresultgivestheglobalconvergenceofthesequencegeneratedbyAlgorithm2tosecond-order
stationary points. To this end, similar to [15], an additional assumption is needed:
(H4) If λ (B ) represents the smallest eigenvalue of the symmetric matrix B , then there exists a
min k k
positive scalar c such that
3
q (0)−q (d )≥c λ (B )δ2.
k k k 3 min k