J Glob Optim (2013) 55:337–347DOI 10.1007/s10898-011-9820-0

Convergence of Pham Dinh–Le Thi’s algorithmfor the trust-region subproblem

Hoang Ngoc Tuan · Nguyen Dong Yen

Received: 7 November 2010 / Accepted: 19 November 2011 / Published online: 21 December 2011© Springer Science+Business Media, LLC. 2011

Abstract It is proved that any DCA sequence constructed by Pham Dinh–Le Thi’s algo-rithm for the trust-region subproblem (Pham Dinh and Le Thi, in SIAM J. Optim. 8:476–505,1998) converges to a Karush–Kuhn–Tucker point of the problem. This result provides a com-plete solution for one open question raised by Le Thi et al. (J. Global Optim., Online First,doi:10.1007/s10898-011-9696-z, 2010).

Keywords Trust-region subproblem · d.c. Algorithm · DCA sequence · Convergence ·KKT point

1 Introduction

Let A ∈ Rn×n be a symmetric matrix, b ∈ R

n a given vector, and r > 0 a real number. Thetrust-region subproblem corresponding to the triple {A, b, r} is the optimization problem

min{

f (x) := 1

2xT Ax + bT x : ‖x‖2 ≤ r2

}, (1.1)

where ‖x‖ =( ∑n

i=1 x2i

)1/2denotes the Euclidean norm of x = (x1, . . . , xn)T ∈ R

n andT means the matrix transposition. Such problems are of frequent use in the development ofthe trust-region methods [1] in nonlinear programming. In fact, (1.1) is only a special caseof minimization problems under quadratic constraints (see for instance [4]).

H. N. TuanDepartment of Mathematics, Hanoi Pedagogical Institute No. 2, Xuan Hoa, Phuc Yen, Vinh Phuc, Vietname-mail: be2yes02@yahoo.com

N. D. Yen (B)Institute of Mathematics, Vietnamese Academy of Science and Technology, 18 Hoang Quoc Viet Road,10307 Hanoi, Vietname-mail: ndyen@math.ac.vn

123

338 J Glob Optim (2013) 55:337–347

Since the feasible region of (1.1) is a convex compact set with an infinite number ofextreme points, the structure of its solution set (resp., of its Karush–Kuhn–Tucker pointset) is quite different from that of quadratic programs with linear constraints; see [5] for aqualitative study and [2] for some applications of the latter problems.

The role of quadratic programming problems in the developments of the numerical opti-mization complexity theory and of the interior point methods can be seen in [14,15].

It is well-known [9,11] that if x ∈ E := {x ∈ Rn : ‖x‖ ≤ r} is a local minimum of

(1.1), then there exists a Lagrange multiplier λ ≥ 0 such that

(A + λI )x = −b, λ(‖x‖ − r) = 0, (1.2)

where I denotes the n × n unit matrix. If x ∈ E and there exists λ ≥ 0 satisfying (1.2), x issaid to be a Karush–Kuhn–Tucker point (or a KKT point) of (1.1). Let S (resp., S1 and M)stand for the solution set (resp., the Karush–Kuhn–Tucker point set and the local minimizerset) of (1.1). Observe that, in general, S1 \M �= ∅. If (x, λ) is a KKT pair, then x is a globalsolution of (1.1) if and only if A + λI is a positive semidefinite matrix (see [3,10]). It hasbeen established by Martinez [9] that (1.1) can have at most one local-nonglobal minimizer.Thus, either M = S, or M is the union of the compact set S and an isolated point. Accordingto Lucidi, Palagi and Roma [8], the number of distinct Lagrange multipliers correspondingto the KKT points of (1.1) is finite, and the number of distinct values of the objective functionat the KKT points is also finite. Structure of the KKT point set of (1.1) has been discussed in[6, Corollary 1] and [7, Proposition 4.1].

In order to find a global minimizer of (1.1), using their general theory of DCA(Difference-of-Convex-functions Algorithms), Pham Dinh and Le Thi [12] (see also [13])have suggested the following iterative algorithm:

• Choose ρ > 0 so that ρ I − A is a positive definite matrix.• Fix an initial point x0 ∈ R

n and a constant ε ≥ 0 (a tolerance). Set k = 0.• If

ρ−1‖(ρ I − A)xk − b‖ ≤ r (1.3)

then take

xk+1 = ρ−1[(ρ I − A)xk − b]. (1.4)

Otherwise, set

xk+1 = r‖(ρ I − A)xk − b‖−1[(ρ I − A)xk − b]. (1.5)

• If ‖xk+1 − xk‖ < ε, then terminate the computation. Otherwise, increase k by 1 andresume the test (1.3).

For ε = 0, the above algorithm generates a infinite sequence {xk}k≥0, called a DCAsequence. Basic properties of DCA sequences are recalled in the next statement.

Theorem 1.1 (See [13, Theorem 4.1]) For any k ≥ 1,

f (xk+1) ≤ f (xk) − 1

2(ρ + θ1)‖xk+1 − xk‖2,

where θ1 is the smallest eigenvalue of the matrix ρ I − A. If x0 ∈ E then the inequality holdsfor all k ≥ 0. It holds limk→∞ ‖xk+1 − xk‖ = 0 and f (xk) → β ≥ α as k → ∞, where α

is the optimal value of (1.1) and β is a constant depending on the choice of x0. In addition,every cluster point of the sequence {xk} is a KKT point.

123

J Glob Optim (2013) 55:337–347 339

According to Theorem 1.1, starting with a point x0 ∈ Rn one gets by the rules (1.3)–

(1.5) a KKT point x ∈ S1 which may not belong to the solution set S. To obtain a KKTpoint with a smaller value of the objective function, one can apply the restart procedure in[13, pp. 491–493] which yields a point x0 ∈ E with the property f (x0) < f (x). RestartingPham Dinh and Le Thi’s algorithm with the new initial point x0 := x0, one obtains a newDCA sequence, denoted again by {xk}. From Theorem (1.1) it follows that

f (xk+1) ≤ f (xk) ≤ · · · ≤ f (x0) < f (x) ∀k ≥ 1.

Hence, the value of f at any cluster point of this DCA sequence is strictly smaller than f (x).Since the number of distinct values of f on S1 is finite, after a finite number of restarts, oneobtains a global minimizer of (1.1).

The numerical results given in [13] showed that Pham Dinh and Le Thi’s algorithm worksefficiently for large-scale trust-region subproblems.

The following question was asked in [7]:Question. Under what conditions is the DCA sequence {xk} convergent?According to [7, Theorem 2.2], if A is a nonsingular matrix that has no multiple negative

eigenvalues, then any DCA sequence of (1.1) converges to a KKT point.This paper is aimed at solving completely the above Question. It will be proved that any

DCA sequence constructed by Pham Dinh–Le Thi’s algorithm for the trust-region subprob-lem (1.1) converges to a KKT point.

We organize the rest of the paper as follows: Sect. 2 presents some lemmas; Sect. 3 provesthe main result; Sect. 4 gives some illustrative examples.

2 Auxiliary results

In order to establish the convergence of DCA sequences, we have to rely on the followingauxiliary facts.

Lemma 2.1 Suppose that {αk} is a bounded sequence of real numbers satisfying the condi-tion limk→∞ |αk+1 − αk | = 0. Let m = lim infk→∞ αk and M = lim supk→∞ αk . Then, thecluster point set of the sequence {αk} is the whole segment [m, M].Proof If m = M , the the conclusion is obvious. Suppose that m < M . First, let us show that

(*) Every nonempty open interval (a, b) ⊂ (m, M), where a �= m and b �= M, mustcontain at least one element of the sequence {αk}.

Since limk→∞ |αk+1 − αk | = 0, there exists k0 ∈ N := {1, 2, . . . } such that

|αk+1 − αk | < b − a ∀k ≥ k0. (2.1)

Moreover, since a > m = lim infk→∞ αk ,

∃k1 > k0 : αk1 < a. (2.2)

Put A = {k ∈ N : k > k1, αk > a}. Since lim supk→∞ αk = M > b > a, we musthave A �= ∅. Denote the minimum element of A by �. Then α� > a. Since � > k1, thereare two possibilities: (i) � = k1 + 1, (ii) � > k1 + 1. If � = k1 + 1 then � − 1 = k1, soα�−1 = αk1 < a by (2.2). If � > k1 + 1 then � − 1 > k1, so α�−1 ≤ a (otherwise we get� − 1 ∈ A, a contradiction to the choice of �). Therefore,

α�−1 ≤ a < α�. (2.3)

123

340 J Glob Optim (2013) 55:337–347

Since � − 1 ≥ k1 > k0, by (2.1) we have

α� − α�−1 < b − a. (2.4)

From (2.3) and (2.4) it follows that a < α� < b. We have thus proved (*).Next, take an arbitrary element α ∈ (m, M) and a positive sequence εi ↓ 0. In accor-

dance with (*), each interval (α, α + εi ) contains an element αki of the sequence {αk}. Wemay assume that ki+1 > ki for all i = 1, 2, . . .. Indeed, if one could find an index i0 withαk /∈ (α, α + εi0+1) for all k > ki0 , then the set of αk satisfying αk ∈ (α, α + εi0+1) wouldbe finite. If αm is the smallest element in that set, then the interval (α, αm) does not containany element of {αk}. This contradicts (*). Therefore, we can construct a subsequence {αki }of {αk} such that αki ∈ (α, α + εi ) for all i = 1, 2, . . .. Since limi→∞ εi = 0, we concludethat limi→∞ αki = α. Hence α is a cluster point of {αk}. As α ∈ (m, M) can be chosenarbitrarily and m, M belong to the cluster point set of {αk} by the definitions of lim inf andlim sup, we conclude that the cluster point set is the whole segment [m, M]. ��

The next claim follows directly from Lemma 2.1.

Corollary 2.1 If {αk} satisfies the assumption of Lemma 2.1 and has finitely many clusterpoints, then it is a convergent sequence.

For each i ∈ {1, 2. . . . , n}, we denote by S(i)1 ⊂ R the subset of real line that consists of

the i th coordinates of the points from the KKT point set S1 of (1.1).

Corollary 2.2 Let {xk} be a DCA sequence of (1.1). For any i ∈ {1, 2, . . . , n}, if the set S(i)1

is finite then the sequence {xki }k≥0 of the i th coordinates of {xk} is convergent.

Proof Assume that S(i)1 , where i ∈ {1, 2, . . . , n}, is a finite set. Since {xk} is bounded and

limk→∞ ||xk+1 − xk || = 0 by Theorem 1.1, {xki } is bounded and

limk→∞ |xk+1

i − xki | = 0.

Suppose that there exists a cluster point ui of {xki } that is not contained in S(i)

1 . Choose a

subsequence {xk′i } of {xk

i } such that limk′→∞ xk′i = ui . Since the vector sequence

{(xk′

1 , . . . , xk′i−1, xk′

i+1, . . . , xk′n

)}⊂ R

n−1

is bounded, it has a subsequence denoted by{(

xk′ ′1 , . . . , xk′ ′

i−1, xk′ ′i+1, . . . , xk′ ′

n

)}that converges

to(u1, . . . , ui−1, ui+1, . . . , un

) ∈ Rn−1.

Hence limk′ ′→∞ xk′ ′ = (u1, u2, . . . , un). This means that u := (u1, u2, . . . , un) is a clusterpoint of {xk}. Since ui /∈ S(i)

1 , u is not a KKT point of (1.1). This contradicts the fact thateach cluster point of {xk} is a KKT point (see Theorem 1.1). We have thus proved that thecluster point set of {xk

i } is a subset of S(i)1 .

As S(i)1 is finite, {xk

i } has finitely many cluster points. Applying Corollary 2.1, we canassert that {xk

i } is convergent.�

Suppose that λ1 ≤ λ2 ≤ · · · ≤ λn are the eigenvalues of A. Since A is symmetric, thereexists an orthogonal matrix Q such that

A = Q−1 AQ = QT AQ,

123

J Glob Optim (2013) 55:337–347 341

where A = diag{λ1, . . . , λn}.Lemma 2.2 For any ρ > 0, the matrix ρ I − A is positive definite if and only if ρ I − A ispositive definite.

Proof It is easy to see that the numbers ρ − λi , i = 1, . . . n, are the eigenvalues of ρ I − A.They form the main diagonal of the matrix ρ I − A. Hence, ρ I − A is positive definite if andonly if ρ I − A is positive definite. ��

Put

b = Qb, f (y) = 1

2yT Ay + bT y, yk = Qxk (k = 0, 1, . . .),

where {xk} is a DCA sequence of (1.1) constructed by the procedure (1.3)–(1.5).

Lemma 2.3 {yk} is a DCA sequence of the problem

min{ f (y) : ||y||2 ≤ r2} (2.5)

with the initial point y0 = Qx0 and the value ρ that appears in (1.3)–(1.5).

Proof According to Lemma 2.2, we can choose a common value ρ > 0 in the DCA algorithmfor the problems (1.1) and (2.5). Assume that {yk} is the DCA sequence of (2.5), with theinitial point y0 = Qx0.

By induction, we will show that yk = yk for all k. The equality yk = yk is obvious fork = 0. Suppose that it holds up to an index k. By the orthogonality of Q,

||(ρ I − A)yk − b|| = ||Q−1[(ρ I − A)yk − b]||= ||ρQ−1 Qxk − Q−1 AQxk − Q−1 Qb||= ||(ρ I − A)xk − b||.

(2.6)

Therefore, if ||(ρ I − A)xk − b|| ≤ ρr then

||(ρ I − A)yk − b|| = ||(ρ I − A)yk − b|| ≤ ρr.

Hence

xk+1 = ρ−1[(ρ I − A)xk − b]and

yk+1 = ρ−1[(ρ I − A)yk − b] = ρ−1[(ρ I − A)yk − b]= ρ−1 Q Q−1[(ρ I − A)Qxk − Qb]= ρ−1 Q[(ρ I − A)xk − b]= Q

{ρ−1[(ρ I − A)xk − b]}

= Qxk+1 = yk+1.

If ||(ρ I − A)xk − b|| > ρr , then ||(ρ I − A)yk − b|| > ρr by (2.6). As

(ρ I − A)yk − b = (ρ I − Q AQ−1

)Qxk − Qb

= Q[(ρ I − A)xk − b],this implies that

xk+1 = r(ρ I − A)xk − b

||(ρ I − A)xk − b||

123

342 J Glob Optim (2013) 55:337–347

and

yk+1 = r (ρ I− A)yk−b||(ρ I− A)yk−b|| = Q

[r (ρ I−A)xk−b

||(ρ I−A)xk−b||]

= Qxk+1 = yk+1.

We have obtained the desired equality yk+1 = yk+1. ��Lemma 2.4 The sequence {xk} is convergent if and only if {yk} is convergent.

Proof It suffices to note that yk = Qxk and the matrix Q is nonsingular. ��

3 Convergence theorem

The main result of this paper can be formulated as follows.

Theorem 3.1 For every initial point x0 ∈ Rn, the DCA sequence {xk}k≥0 for the trust-region

subproblem (1.1) constructed by Pham Dinh–Le Thi’s algorithm (1.3)–(1.5) converges to aKarush–Kuhn–Tucker point of (1.1).

Proof Let {xk}k≥0 be a DCA sequence constructed by the DCA algorithm for problem(1.1). According to Lemmas 2.1–2.4, we need to prove the theorem only in the case A =diag{λ1, . . . , λn}, where λ1 ≤ λ2 ≤ · · · ≤ λn . Take an arbitrary point x = (x1, . . . , xn) ∈ S1

and denote the Lagrange multiplier corresponding to x by λ. Then, by the definition of KKTpoint (see (1.2)) and by the diagonal form of A, the following is valid:

(λi + λ)xi = −bi (∀i ∈ {1, . . . , n}), λ(||x || − r) = 0. (3.1)

First, consider those indexes i with bi �= 0. The linear equation (λi + λ)ui = −bi , whereui is the unknown, has at most one solution. Indeed, if λi + λ �= 0 then the unique solutionis ui = −(λi + λ)−1bi . If λi + λ = 0 then the equation has no solution. Since the numberof the values λ satisfying the system (3.1) is finite, we can assert that S(i)

1 is finite. It followsfrom Corollary 2.2 that {xk

i } is convergent.Second, consider those indexes i with bi = 0. Since

xk+1 =⎧⎨⎩

(ρ I−A)xk−bρ

if ||(ρ I − A)xk − b|| ≤ ρr

r (ρ I−A)xk−b||(ρ I−A)xk−b|| if ||(ρ I − A)xk − b|| > ρr

and A = diag{λ1, . . . , λn}, we get

xk+1i =

{ρ−λi

ρxk

i if ||(ρ I − A)xk − b|| ≤ ρr

r ρ−λi||(ρ I−A)xk−b|| x

ki if ||(ρ I − A)xk − b|| > ρr.

(3.2)

If x0i = 0, then xk

i = 0 for all k. We see that {xki } is convergent. Let {i1, i2, . . . , ir } be the

set of the indexes i with the property x0i �= 0. (If there are no such indexes i , then {xk} is

convergent.) We may suppose that i1 ≤ i2 ≤ · · · ≤ ir . The sequences {xkj } , where j ∈

{1, . . . , n} \ {i1, . . . , ir }, are convergent. Suppose that the cluster points of these sequencesare u j , j ∈ {1, . . . , n} \ {i1, . . . , ir }.

We will show that {xki1} is convergent. Suppose on the contrary that this does not happen.

According to Lemma 2.1, the cluster point set of {xki1} is a segment [m, M] (m < M). Take

any element u ∈ [m, M]. Then there exists a subsequence {xkli1

} of {xki1} converging to u.

123

J Glob Optim (2013) 55:337–347 343

If r ≥ 2 then, for each k = 0, 1, 2, . . ., we get from (3.2) that

xk+1i j

xk+1i1

= ρ − λi j

ρ − λi1

·xk

i j

xki1

(∀ j ∈ {2, . . . , r}).

Consequently, for each k = 0, 1, 2, . . ., by recursion we obtain

xki j

xki1

=(

ρ − λi j

ρ − λi1

)k

·x0

i j

x0i1

(∀ j ∈ {2, . . . , r}).

Put α j = x0i j

x0i1

. Then we have

xki j

= α j ·(

ρ − λi j

ρ − λi1

)k

· xki1

(k = 0, 1, . . . ; j = 2, . . . , r). (3.3)

For each j = 2, . . . , r , setting

δ j ={

0 if λi j > λi1 ,

1 if λi j = λi1 ,

we have

limk→∞

(ρ − λi j

ρ − λi1

)k

= δ j . (3.4)

Since {xkli1

} converges to u, from (3.3) and (3.4) we can assert that {xkli j

} converges to α jδ j uas kl → ∞.

Thus, the point x = (u1, . . . , u, ui1+1, . . . , α2δ2u, ui2+1, . . . , un) is the limit of {xkl }. Inother words, x is a cluster point of {xk}. Hence x is a KKT point by Theorem 1.1. So, by thedefinition of KKT point (see (1.2)) there exists λ(u) ≥ 0 with

(A + λ(u)I )x = −b, λ(u)(||x || − λ) = 0. (3.5)

Observe that the equation√

u21 + · · · + u2 + u2

i1+1 + · · · + α22δ2

2u2 + u2i2+1 + · · · + u2

n − r = 0 (3.6)

(with the unknown u) has at most two solutions. Denote the solution set of that equation byU . Put

T = [m, M] \ U.

It is clear that T is an infinite set. For every

x = (u1, . . . , u, ui1+1, . . . , α2δ2u, ui2+1, . . . , un)

with u ∈ T , we have ||x ||−r �= 0. From the second equality of (3.5) it follows that λ(u) = 0for all u ∈ T . From the first equality of (3.5) we get Ax = −b for all

x = (u1, . . . , u, ui1+1, . . . , α2δ2u, ui2+1, . . . , un)

with u ∈ T . In particular, λi1 u = 0 for all u ∈ T . Since T is infinite, we infer that λi1 = 0.Consequently,

xk+1i1

={

xki1

if ||(ρ I − A)xk − b|| ≤ ρr,ρr

||(ρ I−A)xk−b|| xki1

if ||(ρ I − A)xk − b|| > ρr.

123

344 J Glob Optim (2013) 55:337–347

If x0i1

> 0, then {xki1} is a decreasing sequence. If x0

i1< 0, then {xk

i1} is an increasing sequence.

Hence the sequence {xki1} is monotone. Furthermore, since {xk

i1} is bounded, it is a conver-

gent sequence. This fact contradicts the initial assumption on {xki1}. Thus {xk

i1} is a convergent

sequence.We now suppose that r = 1. In this case, x = (u1, . . . , ui1−1, u, ui1+1, . . . , un) is a cluster

point of the sequence {xk}. Hence x is a KKT point. Similarly as in the case r ≥ 2, thereexists λ(u) ≥ 0 such that (x, λ(u)) satisfies (3.5). Then the equation (3.6) becomes

√u2

1 + · · · + u2i1−1 + u2 + u2

i1+1 + · · · + u2n − r = 0.

Clearly, this equation can have at most two solutions. Next, arguing similarly as in the pre-ceding case, we can show that the sequence {xk

i1} is convergent.

In the case r = 1 under consideration, we see at once that the sequence {xk} converges,because each of its coordinate components {xk

i } is convergent. As concerning the case wherer ≥ 2, the convergence of the sequence {xk

i1} together with the equalities (3.3) and (3.4)

implies that the sequences {xki j

}, j = 2, . . . , r , are also convergent. Therefore, in this case,

the coordinate components {xki }, i = 1, . . . , n, are convergent. This implies that the sequence

{xk} is convergent. ��

4 Illustrative examples

4.1 An example with n = 2

Consider the problem

min{

f (x) := −4x22 + x1 : x = (x1, x2)

T ∈ R2, x2

1 + x22 ≤ 1

}. (4.1)

Using the necessary optimality condition recalled in (1.2) we find the KKT point set

S1 ={

(−1, 0)T ,(

− 1

8,

√63

8

)T,(

− 1

8,−

√63

8

)T}

.

Then, by the criterion for global solutions recalled in Sect. 1 we get solution set

S ={(

− 1

8,

√63

8

)T,(

− 1

8,−

√63

8

)T}

.

Note that x := (−1, 0)T is the unique local-nonglobal solution of (4.1).Let {xk} be a DCA sequence of (4.1), with ρ = 1 and

x0 ∈ E :={

x = (x1, x2)T ∈ R

2 : x21 + x2

2 ≤ 1}

.

By Theorem 3.1, the sequence converges to a point of S1. For the purpose of illustration,we are going to show that: (a) If x0

2 = 0 then limk→∞ xk = (−1, 0)T ; (b) If x02 > 0 then

limk→∞ xk = (− 18 ,

√638 )T ; (c) If x0

2 < 0 then limk→∞ xk = (− 18 ,−

√638 )T .

From (1.3)–(1.5) it follows that

xk+11 =

⎧⎨⎩

xk1 − 1 if (xk

1 − 1)2 + 81(xk2 )2 ≤ 1

xk1−1√

(xk1−1)2+81(xk

2 )2if (xk

1 − 1)2 + 81(xk2 )2 > 1, (4.2)

123

J Glob Optim (2013) 55:337–347 345

and

xk+12 =

⎧⎨⎩

9xk2 if (xk

1 − 1)2 + 81(xk2 )2 ≤ 1

9xk2√

(xk1−1)2+81(xk

2 )2if (xk

1 − 1)2 + 81(xk2 )2 > 1. (4.3)

By (4.2) and by the condition x0 ∈ E , we have xk1 ≤ 0 for all k ≥ 1. Hence (xk

1 − 1)2 +81(xk

2 )2 > 1 for all k ≥ 1, unless xk1 = 0 and xk

2 = 0. If xk1 = 0 and xk

2 = 0, then (4.2) and(4.3) imply that xk+1

1 = −1 and xk+12 = 0. So the inequality (xk

1 − 1)2 + 81(xk2 )2 > 1 holds

with k being replaced by k + 1. Therefore, by (4.2) and (4.3) we get

xk+11 = xk

1 − 1√(xk

1 − 1)2 + 81(xk2 )2

(4.4)

and

xk+12 = 9xk

2√(xk

1 − 1)2 + 81(xk2 )2

(4.5)

for any index k ≥ 2.According to (4.5), if x0

2 = 0 then xk2 = 0 for all k. Combining this with (4.4) we can

assert that limk→∞ xk = (−1, 0)T . Assertion (a) has been proved.If x0

2 > 0, then by (4.3) we have xk2 > 0 for all k ≥ 1. Fix any number a > 0 with

a ≤ min{x22 ,

√63/8}. By induction, we are going to show that xk

2 ≥ a for every k ≥ 2.First, by the choice of a, the inequality xk

2 ≥ a is valid for k = 2. Next, suppose that theinequality xk

2 ≥ a holds up to an index k ≥ 2. By (4.3), xk1 ≤ 0 for all k ≥ 1. We have

(xk1 )2 + (xk

2 )2 ≤ 1 because xk ∈ E for for all k ≥ 0. Hence (xk1 )2 ≤ 1 − (xk

2 )2 ≤ 1 − a2. Itfollows that 0 ≥ xk

1 ≥ −√1 − a2. So, (xk

1 −1)2 ≤ (1+√1 − a)2. Since xk

2 ≥ a, combiningthis with (4.5) yields

xk+12 = 9xk

2√(xk

1 − 1)2 + 81(xk2 )2

= 1√(xk

1−1)2

81(xk2 )2 + 1

≥ 1√(1+√

1−a2)2

81a2 + 1≥ a.

(The last inequality is equivalent to 81(1 − a2) ≥ (1 + √1 − a2)2 which is true because

a ≤ √63/8.) Thus xk+1

2 ≥ a. By the induction principle we can assert that xk2 ≥ a for all

k ≥ 2. Since {xk} converges to a point of S1, and since (− 18 ,

√638 )T is the unique vector from

S1 with the ordinate greater or equal a, the last fact implies that limk→∞ xk = (− 18 ,

√638 )T .

So, the assertion (b) is valid.By symmetry, from the validity of (b) we get the assertion (c).

4.2 An example with n ≥ 3

Consider the problem

min{

f (x) := −4n∑

i=2

x2i + x1 : x = (x1, x2, . . . , xn)T ∈ R

n,

n∑i=1

x2i ≤ 1

}, (4.6)

123

346 J Glob Optim (2013) 55:337–347

where n ≥ 3. By using (1.2) and the criterion for global solutions recalled in Sect. 1, weobtain the KKT point set

S1 = {(−1, 0, . . . , 0)T } ∪{(

− 1

8, x2, . . . , xn

)T :n∑

i=2

x2i = 63

64

}

and the solution set

S ={(

− 1

8, x2, . . . , xn

):

n∑i=2

x2i = 63

64

}

of (4.6). It is easy to show that the problem has the unique local-nonglobal solution x :=(−1, 0, . . . , 0)T .

Let {xk} be a DCA sequence of (4.6), with ρ = 1 and

x0 ∈ E :={

x = (x1, x2, . . . , xn)T ∈ Rn :

n∑i=1

x2i ≤ 1

}.

Combining the analysis of the preceding example with the symmetry of (4.6) with respectto the variables x2, . . . , xn , we can assert that

• If x02 = · · · = x0

n = 0, then limk→∞ xk = x .

• If∑n

i=2(x0i )2 > 0 then limk→∞ xk = (− 1

8 , x2, . . . , xn)T

, where

(x2, . . . , xn

)T := μ(x0

2 , . . . , x0n

)T (4.7)

and

μ :=√

63

8

( n∑i=2

(x0i )2

)−1/2. (4.8)

Note that the limit point x := (− 18 , x2, . . . , xn

)Tof the sequence {xk} given by (4.7) and

(4.8) is contained in the 2-dimensional subspace generated by the vectors x0 and x . Moreover,it can be shown that the whole DCA sequence {xk} belongs to that 2-dimensional subspace.

Acknowledgements This research was supported by the National Foundation for Science and TechnologyDevelopment (NAFOSTED) of Vietnam. The authors are indebted to the two anonymous referees for theirinsightful comments and valuable suggestions

References

1. Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. MPS-SIAM Series on Optimization, Phil-adelphia (2000)

2. Exler, O., Schittkowski, K.: A trust region SQP algorithm for mixed-integer nonlinear programming.Optim. Lett. 1, 269–280 (2007)

3. Gay, D.M.: Computing optimal locally constrained steps. SIAM J. Sci. Stat. Comput. 2, 186–197 (1981)4. Jeyakumar, V., Srisatkunarajah, S.: Lagrange multiplier necessary conditions for global optimality for

non-convex minimization over a quadratic constraint via S-lemma. Optim. Lett. 3, 23–33 (2009)5. Lee, G.M., Tam, N.N., Yen, N.D.: Quadratic Programming and Affine Variational Inequalities: A Quali-

tative Study, Series: “Nonconvex Optimization and its Applications”, vol. 78. Springer, Berlin (2005)6. Le Thi, H.A., Pham Dinh, T.: A branch-and-bound method via DC optimization and ellipsoidal technique

for box constrained nonconvex quadratic programming problems. J. Global Optim. 13, 171–206 (1998)

123

J Glob Optim (2013) 55:337–347 347

7. Le Thi, H.A., Pham Dinh, T., Yen, N.D.: Behavior of DCA sequences for solving the trust-region sub-problem. J. Global Optim. Online First. doi:10.1007/s10898-011-9696-z (2010)

8. Lucidi, S., Palagi, L., Roma, M.: On some properties of quadratic programs with a convex quadraticconstraint. SIAM J. Optim. 8, 105–122 (1998)

9. Martinez, J.M.: Local minimizers of quadratic functions on Euclidean balls and spheres. SIAM J.Optim. 4, 159–176 (1994)

10. Moré, J.J., Sorensen, D.C.: Computing a trust region step. SIAM J. Sci. Stat. Comput. 4, 553–572 (1983)11. Pham Dinh, T., Le Thi, H.A.: Lagrangian stability and global optimality on nonconvex quadratic mini-

mization over Euclidean balls and spheres. J. Convex Anal. 2, 263–276 (1995)12. Pham Dinh, T., Le Thi, H.A.: Convex analysis approach to D.C. programming: theory, algorithms and

applications. Acta Math. Vietnam. 22, 289–355 (1997)13. Pham Dinh, T., Le Thi, H.A.: A d.c. optimization algorithm for solving the trust-region subproblem. SIAM

J. Optim. 8, 476–505 (1998)14. Pardalos, P.M. (ed.): Complexity in Numerical Optimization. World Scientific, River Edge (1993)15. Pardalos, P.M., Resende, M.G.C. : Interior point methods for global optimization problems. In: Terlaky,

T. (ed.) Interior Point Methods of Mathematical Programming, pp. 467–500. Kluwer Academic Publish-ers, Dordrecht (1996)

123