DOI 10.1007/s10957-007-9279-9
Conditions for Error Bounds and Bounded Level Sets of Some Merit Functions for the Second-Order Cone Complementarity Problem
J.-S. Chen
Published online: 26 October 2007
© Springer Science+Business Media, LLC 2007
Abstract Recently this author studied several merit functions systematically for the second-order cone complementarity problem. These merit functions were shown to enjoy some favorable properties, to provide error bounds under the condition of strong monotonicity, and to have bounded level sets under the conditions of monotonicity as well as strict feasibility. In this paper, we weaken the condition of strong monotonicity to the so-called uniform P∗-property, which is a new concept recently developed for linear and nonlinear transformations on Euclidean Jordan al- gebra. Moreover, we replace the monotonicity and strict feasibility by the so-called R01or R02-functions to keep the property of bounded level sets.
Keywords Error bounds· Jordan products · Level sets · Merit functions · Second-order cones· Spectral factorization
1 Introduction
The second-order cone complementarity problem (SOCCP), which is a natural ex- tension of nonlinear complementarity problem (NCP), is to find ζ∈ Rnsatisfying
F (ζ ), ζ = 0, F (ζ ) ∈ K, ζ ∈ K, (1) where·, · is the Euclidean inner product, F : Rn→ Rn is a continuous mapping, and K is the Cartesian product of second-order cones (SOC), also called Lorentz
This work is partially supported by National Science Council of Taiwan.
J.-S. Chen (
)Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan e-mail: [email protected]
J.-S. Chen
Mathematics Division, National Center for Theoretical Sciences, Taipei, Taiwan
cones [1]. In other words,
K = Kn1× · · · × Knm, (2)
where m, n1, . . . , nm≥ 1, n1+ · · · + nm= n, and
Kni:= {(x1, x2)∈ R × Rni−1| x2 ≤ x1}, (3) with · denoting the Euclidean norm and K1denoting the set of nonnegative reals R+. A special case of (2) isK = Rn+, the nonnegative orthant inRn, which corre- sponds to m= n and n1= · · · = nm= 1. If K = Rn+, then (1) reduces to the nonlinear complementarity problem. Throughout this paper, we assumeK = Knfor simplicity, i.e.,K is a single second-order cone (all the analysis can be easily carried over to the general case whereK has the direct product structure (2)).
Second-order cone programs (SOCP) are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with Cartesian product of SOCs. Linear programs, convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as SOCP problems. Many other problems from engineering, control, finance, and robust opti- mization can also be recast as SOCP problems [2,3]. It is well-known that the KKT optimality conditions of SOCP forms a SOCCP which is also a natural extension of nonlinear complementarity problems (NCP). Thus studying the SOCCP is very important from the above points of view.
There have been various methods proposed for solving SOCCP. They include interior-point methods [3–9], non-interior smoothing Newton methods [10–12]. Re- cently in the papers [13–15], the author studied an alternative approach based on reformulating SOCCP as an unconstrained smooth minimization problem. For this approach, it aims to find a smooth function ψ: Rn× Rn→ R+such that
ψ (x, y)= 0 ⇐⇒ x ∈ Kn, y∈ Kn,x, y = 0. (4) Then SOCCP can be expressed as an unconstrained smooth (global) minimization problem:
ζmin∈Rnf (ζ ):= ψ(F (ζ ), ζ ). (5) We call such a f a merit function for the SOCCP.
A popular choice of ψ is the squared norm of Fischer-Burmeister function, i.e., ψFB: Rn× Rn→ R+associated with second-order cone given by
ψFB(x, y)=1
2φFB(x, y)2, (6)
where φFB: Rn× Rn→ Rn is the well-known Fischer-Burmeister function (origi- nally proposed for NCP, see [16,17]) defined by
φFB(x, y)= (x2+ y2)1/2− x − y. (7)
More specifically, for any x= (x1, x2), y= (y1, y2)∈ R × Rn−1, we define their Jordan product associated withKnas
x◦ y := (x, y, y1x2+ x1y2). (8) The Jordan product◦, unlike scalar or matrix multiplication, is not associative, which is a main source on complication in the analysis of SOCCP. The identity element un- der this product is e:= (1, 0, . . . , 0)T ∈ Rn. We write x2to mean x◦x and write x +y to mean the usual componentwise addition of vectors. It is known that x2∈ Knfor all x∈ Rn. Moreover, if x∈ Kn, then there exists a unique vector inKn, denoted by x1/2, such that (x1/2)2= x1/2◦ x1/2= x. Thus, φFBdefined as (7) is well-defined for all (x, y)∈ Rn× Rnand mapsRn× RntoRn. It was shown in [11] that φFB(x, y)= 0 if and only if (x, y) satisfies (4). Therefore, ψFBdefined as (6) induces a merit function fFB:= ψFB(F (ζ ), ζ ))for the SOCCP.
The function ψFB given as in (6) was proved smooth with computable gradient formulas and enjoys several favorable properties, nonetheless, it does not have addi- tional bounded level-set and error bound properties (see [15]). To conquer this, four other functions associated with second-order cone were considered in [13–15]. The first one is ψYF: Rn× Rn→ R defined by
ψYF(x, y):= ψ0(x, y) + ψFB(x, y), (9) where ψ0: R → R+is any smooth function satisfying
ψ0(t )= 0 ∀t ≤ 0 and ψ0(t ) >0 ∀t > 0. (10) The function ψYFwas studied by Yamashita and Fukushima in [18] for SDCP (semi- definite complementarity problems) case and was extended to SOCCP case in [15].
An example of ψ0(t )is ψ0(t )=14(max{0, t})4. A slight modification of ψYFyields ψYF: Rn× Rn→ R defined by
ψYF(x, y):=1
2(x ◦ y)+2+ ψFB(x, y), (11) where (·)+means the orthogonal projection onto the second-order coneKn. The third function is ψLT: Rn× Rn→ R defined by
ψLT(x, y):= ψ0(x, y) + ˜ψ(x, y), (12) where ˜ψ: Rn× Rn→ R+satisfies
˜ψ(x, y) = 0, x,y ≤ 0 ⇐⇒ x ∈ Kn, y∈ Kn, x, y = 0. (13) The function ψ0is the same as the above (namely, it satisfies (10)) and examples of
˜ψ are
˜ψ1(x, y):=1 2
(−x)+2+ (−y)+2
and ˜ψ2(x, y):=1
2φFB(x, y)+2 (14)
which were recently investigated in [14]. The function ψLTwas proposed by Luo and Tseng for NCP case in [19] and was extended to the SDCP case by Tseng in [20].
The last function ψLT: Rn× Rn→ R, a slight variant of ψLT, is defined by ψLT(x, y):=1
2(x ◦ y)+2+ ˜ψ(x, y), (15) where ˜ψis given as in (13).
Each of the above functions naturally induces a merit function as follows:
fYF(ζ ):= ψYF(F (ζ ), ζ ), fYF(ζ ):= ψYF(F (ζ ), ζ ), fLT(ζ ):= ψLT(F (ζ ), ζ ),
fLT(ζ ):= ψLT(F (ζ ), ζ ). (16) It was shown that fYFprovides error bound [15, Prop. 5] if F is strongly monotone and fYFhas bounded level set [15, Prop. 6] if F is monotone as well as SOCCP is strictly feasible. The same results hold for fYF[13, Prop. 4.1 and Prop. 4.2], for fLT
[14, Prop. 4.1 and Prop. 4.3], and for fLT [14, Prop. 4.2 and Prop. 4.4]. The main purpose of this paper is to weaken the condition of strong monotonicity to so-called uniform P∗-property (will be introduced in Sect.2) which is a new concept recently developed for linear and nonlinear transformations on Euclidean Jordan Algebra [21, 22]. Moreover, we replace the monotonicity and strict feasibility by the so-called R01
(or R02)-functions (will be introduced in Sect. 2) to ensure that the level sets for fYF, fYF, fLT, fLTare still bounded.
2 Preliminaries
In this section, we review some definitions and preliminary materials that will be used in the subsequent analysis. First, we recall from [11] that each x= (x1, x2)∈ R × Rn−1admits a spectral factorization, associated withKn, of the form
x= λ1(x)· u(1)x + λ2(x)· u(2)x , (17) where λ1(x), λ2(x)and u(1)x , u(2)x are the spectral values and the associated spectral vectors of x given by
λi(x)= x1+ (−1)ix2,
u(i)x =
⎧⎪
⎪⎨
⎪⎪
⎩ 1 2
1, (−1)i x2
x2
, if x2= 0;
1 2
1, (−1)iw2
, if x2= 0,
(18) for i= 1, 2, with w2being any vector inRn−1satisfyingw2 = 1. If x2= 0, the factorization is unique. The set{u(1)x , u(2)x } is called a Jordan frame and possesses the following properties.
Property 2.1 For any x= (x1, x2)∈ R × Rn−1with the spectral values λ1(x), λ2(x) and spectral vectors u(1)x , u(2)x given as in (18), we have
(a) u(1)x and u(2)x are orthogonal under Jordan product and have length 1/√ 2 , i.e.,
u(1)x ◦ u(2)x = 0, u(1)x = u(2)x = 1
√2.
(b) u(1)x and u(2)x are idempotent under Jordan product, i.e., u(i)x ◦ u(i)x = u(i)x , i= 1, 2.
The above spectral factorization of x, as well as x2 and x1/2 have various in- teresting properties; see [11]. For instances, for any x= (x1, x2)∈ R × Rn−1, with spectral values λ1(x), λ2(x)and spectral vectors u(1)x , u(2)x , the following results hold:
(1) x2= λ1(x)2u(1)x + λ2(x)2u(2)x ∈ Kn. (2) If x∈ Kn, then 0≤ λ1(x)≤ λ2(x)and x1/2=√
λ1(x) u(1)x +√
λ2(x) u(2)x . It is also well-known that for any x= (x1, x2)∈ R × Rn−1, we have x∈ Knif and only if
Lx:=
x1 x2T x2 x1I
is positive semi-definite (see [11, p. 437] and [23]). If x∈ int(Kn), then 0 < λ1(x)≤ λ2(x), and Lxis invertible with
L−1x = 1 x12− x22
x1 −x2T
−x2
x12−x22
x1 I+x11x2x2T
.
In general, we have x◦ y = Lxyfor all y∈ Rn, and Lx 0 if and only if x ∈ int(Kn).
We say that x, y operator commute if Lxand Lycommute, i.e., LxLy= LyLx. From [1, Lemma X.2.2], we know that x and y operator commute if and only if x and y share a common Jordan frame in their spectral factorizations.
We now recall definitions of various monotonicities and P -properties of a contin- uous mapping which are needed for the assumptions of our main results later. To this end, we denote
x y := x − (x − y)+, x y := y + (x − y)+, (19) which will be used in the definitions of P -properties. As below, we state the def- initions of various P -properties associated with SOC. Indeed, such definitions are borrowed from [21,22,24], and are generalization of the familiar P -properties for matrices. Some of them may look slightly different from the original ones given in [21,22,24]; this is because ours are in SOCCP style.
Definition 2.1 Let xKny denote y− x ∈ Kn for any x, y∈ Rn. Then, for a con- tinuous mapping F : Rn→ Rn,
(a) F is monotone if
F (ζ ) − F (ξ), ζ − ξ ≥ 0 ∀ζ, ξ ∈ Rn; (b) F is strictly monotone if
F (ζ ) − F (ξ), ζ − ξ > 0 ∀ζ = ξ ∈ Rn; (c) F is strongly monotone if there exists ρ > 0 such that
F (ζ ) − F (ξ), ζ − ξ ≥ ρζ − ξ2 ∀ζ, ξ ∈ Rn; (d) F has Order P -property if
(ζ− ξ) (F (ζ ) − F (ξ)) Kn0Kn(ζ− ξ) (F (ζ ) − F (ξ)) ⇒ ζ = ξ;
(e) F has Jordan P -property if
(ζ− ξ) ◦ (F (ζ ) − F (ξ)) Kn0⇒ ζ = ξ, or equivalently,
ζ= ξ ⇒ λ2[(ζ− ξ) ◦ (F (ζ ) − F (ξ))] > 0;
(f) F has P -property if
ζ − ξ and F (ζ ) − F (ξ) operator commute (ζ− ξ) ◦ (F (ζ ) − F (ξ)) Kn0
⇒ ζ = ξ;
(g) F has uniform P∗-property if there exists ρ > 0 such that
i=1,2max(ζ − ξ) ◦ (F (ζ ) − F (ξ)) , u(i)ξ ≥ ρζ − ξ2 ∀ζ, ξ ∈ Rn,
where u(i)ξ , i= 1, 2, are the spectral vectors of ξ;
(h) F has uniform Jordan P -property if there exists ρ > 0 such that λ2[(ζ − ξ) ◦ (F (ζ ) − F (ξ))] ≥ ρζ − ξ2 ∀ζ, ξ ∈ Rn;
(i) F has uniform P -property if there exists ρ > 0 such that for any ζ, ξ ∈ Rn with ζ− ξ operator commuting with F (ζ ) − F (ξ), we have
λ2[(ζ − ξ) ◦ (F (ζ ) − F (ξ))] ≥ ρζ − ξ2; (j) F has P0-property if F (ζ )+ εζ has the P -property for all ε > 0.
As remarked in [22, Remark 3.1], when F is linear, strong monotonicity and strict monotonicity coincide; and uniform (Jordan) P -property and (Jordan) P -property also coincide. In addition, there have been established some inter-connections be- tween the above concepts, for instances, the following implications hold (see [22, 24]). For more details about P -properties, please refer to [21,22] and [25].
Property 2.2 For a continuous mapping F : Rn→ Rn,
(a) strong monotonicity⇒ strict monotonicity ⇒ Order P -property ⇒ Jordan P - property⇒ P -property ⇒ P0-property;
(b) strong monotonicity⇒ uniform P∗-property⇒ uniform Jordan P -property ⇒ uniform P -property⇒ P -property;
(c) monotonicity⇒ P0-property.
It is also worthy to point out that, when F is linear and self-adjoint, there have strongly monotonicity= Order P -property = Jordan P -property = P -property (see [21, Theorem 21]). Therefore, from Property2.2(a) and (b), strongly monotonicity, strictly monotonicity, Order P -property, uniform P∗-property, Jordan P -property, uniform Jordan P -property, uniform P -property, and P -property all coincide when F is linear and self-adjoint. This gives a rough direction to construct a counterexample that F has uniform P∗-property but is not strongly monotone function.
To close this section, we want to introduce some other concepts which will be used in analysis of boundedness of level sets. In fact, they are extensions of R0-property for NCP case. It is known that R0-property is used to prove the existence of solutions for P0-NCP. Such properties were recently studied for the following complementarity problems (see [22, Sects. 2, 3]): find x∈ V such that
x∈ K, F (x)+ q ∈ K, and x, F (x) + q = 0,
where V is a Euclidean Jordan algebra with the associated cone K and q ∈ V . We employ their definitions to prove the properties of bounded level sets for fYF, fYF, fLT, fLT.
Definition 2.2 For a mapping F : Rn→ Rn, it is called (a) R01-function if, for any sequence{ζk} such that
ζk → ∞, (−ζk)+
ζk → 0, (−F (ζk))+
ζk → 0, (20)
we have
lim inf
k→∞
ζk, F (ζk)
ζk2 >0; (21)
(b) R02-function if, for any sequence{ζk} such that (20) hold, we have
lim inf
k→∞
λ2(ζk◦ F (ζk))
ζk2 >0. (22)
The above concepts are taken from [24] and are extensions of the ones defined for NCP and SDCP settings. In particular, R02-property is equivalent to R0-property defined in Definition 3.2 of [22]; and hence is equivalent to R0-matrix when F is linear (note counterexample of R0-matrix but not monotone matrix can be found in Chap. 3 of [26]). It is easy to see that every R01-function is R02-function [24,
Lemma 4]. Also, from Lemma 5 of [24] or Proposition 3.2 of [22], if F has the uniform Jordan P -property then F is R02-function (see [24, Lemma 5]). Where are R01-function and R02-function located in Property2.2(a) and (b)? There is no answer yet, to the author’s best knowledge. However, when F is linear, a chart describes the relation between P -properties and R0-property is given in [27].
3 Conditions for Error Bounds
The error bound is an important concept that indicates how close an arbitrary point is to the solution set of SOCCP. Thus, an error bound may be used to provide stopping criterion for an iterative method. In this section, we study conditions under which the merit functions fYF, fLTdefined as in(16) provide error bounds for SOCCP. In fact, there have existing results: Proposition 5 of [15], Proposition 4.1 of [13], Proposi- tion 4.1 of [14], and Proposition 4.2 of [14], which indicate that fYF, fYF, fLT, fLT
provide error bounds for SOCCP, respectively, when F is strongly monotone. Our main work is to substitute the condition of strong monotonicity by a weaker con- dition, uniform P∗-property. We notice that this replacement can be done only for fYF, fLT, and it is not clear yet whether it is true for fYF, fLT or not. The reasons for it will be explained in the section of final remarks (Sect.5). We begin with the following technical lemmas to reach our claims.
Lemma 3.1 For any ζ∈ Rnand ξ∈ Kn, we haveζ, ξ ≤ (ζ )+, ξ.
Proof For any ζ∈ Rn, we can write ζ= (ζ )++ (ζ )−where (·)+, (·)−represent the projection ontoKnand−Kn, respectively. Since ξ∈ Knand (ζ )−∈ −Kn, we have
(ζ )−, ξ ≤ 0. Thus, ζ, ξ = (ζ )+, ξ + (ζ )−, ξ ≤ (ζ )+, ξ. In fact, the result is
true for any closed convex cone.
Lemma 3.2 ([15, Lemma 5.2]) Let ψFB, φFB be given by (6) and (7), respectively.
Then, for any (x, y)∈ Rn× Rn, we have
4ψFB(x, y)≥ 2φFB(x, y)+2≥(−x)+2+(−y)+2.
Lemma 3.3 Let ˜ψi, i= 1, 2, be given as in (14). Then, ˜ψi satisfies the following inequality:
˜ψi(x, y)≥ α
(−x)+2+ (−y)+2
∀(x, y) ∈ Rn× Rn, (23) for some positive constant α and i= 1, 2.
Proof For ˜ψ1, it is clear by definition (14) where α=12. For ψ2, the inequality is still
true, where α=14, due to Lemma3.2.
Proposition 3.1 Let fLTbe given as in (15) and (16) with ˜ψsatisfying (23). Suppose that F has uniform P∗-property and SOCCP has a solution ζ∗. Then, there exists a
scalar τ > 0 such that
τζ − ζ∗2≤ (F (ζ ) ◦ ζ )+ + (−F (ζ ))+ + (−ζ )+, ∀ζ ∈ Rn. (24) Moreover,
τζ − ζ∗2≤ √ 2+
√2
√α
fLT(ζ )1/2, ∀ζ ∈ Rn, (25)
where α is a positive constant.
Proof From the assumption of uniform P∗-property, there exists ρ > 0 such that ρζ − ζ∗2≤ max
i=1,2
(ζ− ζ∗)◦ (F (ζ ) − F (ζ∗)), u(i)ζ∗
, (26)
where u(i)ζ∗, i= 1, 2, are the spectral vectors of ζ∗. On the other hand, since ζ∗is a solution of SOCCP, we have ζ∗∈ Kn, F (ζ∗)∈ Kn, ζ∗◦ F (ζ∗)= 0. Thus,
(ζ− ζ∗)◦ (F (ζ ) − F (ζ∗))
= ζ ◦ F (ζ ) − ζ∗◦ F (ζ ) − ζ ◦ F (ζ∗)+ ζ∗◦ F (ζ∗)
= ζ ◦ F (ζ ) − ζ∗◦ F (ζ ) − ζ ◦ F (ζ∗).
Now, we express the spectral factorizations of ζ and F (ζ ) as below:
ζ = λ1(ζ )· u(1)ζ + λ2(ζ )· u(2)ζ ,
F (ζ )= λ1(F (ζ ))· u(1)F (ζ )+ λ2(F (ζ ))· u(2)F (ζ ).
We notice that ζ∗and F (ζ∗)operator commute due to ζ∗◦ F (ζ∗)= F (ζ∗)◦ ζ∗= 0, ζ∗∈ Kn, and F (ζ∗)∈ Kn. Hence, they share the same Jordan frame; indeed, we can express them as
ζ∗= λ1(ζ∗)· u(1)ζ∗ + λ2(ζ∗)· u(2)ζ∗ =
2 j=1
λj(ζ∗)· u(i)ζ∗,
F (ζ∗)= λ2(F (ζ∗))· u(1)ζ∗ + λ1(F (ζ∗))· u(2)ζ∗ =
2 j=1
λj∗(F (ζ∗))· u(i)ζ∗,
where j∗ denotes j∗= 2 for j = 1 and j∗= 1 for j = 2. It needs to point out that, when x and y share the same Jordan frame, it does not necessarily hold u(i)x = u(i)y
for i= 1, 2. In general, it holds that u(1)x = u(2)y and u(2)x = u(1)y . Then, for i= 1, 2.
we have
(ζ− ζ∗)◦ (F (ζ ) − F (ζ∗)), u(i)ζ∗
=
ζ◦ F (ζ ) − ζ∗◦ F (ζ ) − ζ ◦ F (ζ∗), u(i)ζ∗
=
ζ◦ F (ζ ), u(i)ζ∗
+
−ζ∗◦ F (ζ ), u(i)ζ∗
+
−ζ ◦ F (ζ∗), u(i)ζ∗
=
ζ◦ F (ζ ), u(i)ζ∗ +
−F (ζ ), ζ∗◦ u(i)ζ∗ +
−ζ, F (ζ∗)◦ u(i)ζ∗
=
ζ◦ F (ζ ), u(i)ζ∗ +
−F (ζ ),
2 j=1
λj(ζ∗)u(j )ζ∗ ◦ u(i)ζ∗
+
−ζ,
2 j=1
λj∗(F (ζ∗))u(j )ζ∗ ◦ u(i)ζ∗
=
(ζ◦ F (ζ )), u(i)ζ∗
+ λi(ζ∗)
−F (ζ ), u(i)ζ∗
+ λi∗(F (ζ∗))
−ζ, u(i)ζ∗
≤
(ζ◦ F (ζ ))+, u(i)ζ∗
+ λi(ζ∗)
(−F (ζ ))+, u(i)ζ∗
+ λi∗(F (ζ∗))
(−ζ )+, u(i)ζ∗
≤ (ζ ◦ F (ζ ))+ ·u(i)
ζ∗ +λi(ζ∗)(−F (ζ ))+
×u(i)ζ∗ +λi∗(F (ζ∗))(−ζ )+ ·u(i)ζ∗
= 1
√2
(ζ ◦ F (ζ ))+ + λi(ζ∗)(−F (ζ ))+ + λi∗(F (ζ∗))(−ζ )+
≤ max
1
√2,λi(ζ∗)
√2 ,λi∗(F (ζ∗))
√2
×
(ζ ◦ F (ζ ))+ + (−F (ζ ))+ + (−ζ )+
≤ max
1
√2,λ2(ζ∗)
√2 ,λ2(F (ζ∗))
√2
×
(ζ ◦ F (ζ ))+ + (−F (ζ ))+ + (−ζ )+ ,
where the last equality uses Property 2.1, the first inequality is from Lemma3.1, and the second inequality uses the fact that λi(ζ∗)≥ 0, λi∗(F (ζ∗))≥ 0 since ζ∗∈ Kn, F (ζ∗)∈ Kn. Also, note that i∗ denotes i∗= 2 for i = 1 and i∗= 1 for i = 2.
Now, let
τ:= ρ
max{√12,λ2√(ζ∗)
2 ,λ2(F (ζ√ ∗)) 2 }>0;
then the above and (26) give
τζ − ζ∗2≤ (F (ζ ) ◦ ζ )+ + (−F (ζ ))+ + (−ζ )+, ∀ζ ∈ Rn. Next, we come to the second part of the proposition. By fLT(ζ ) =
1
2(F (ζ ) ◦ ζ )+2+ ˜ψ(F (ζ ), ζ ), we have
(F (ζ ) ◦ ζ )+ ≤√
2fLT(ζ )1/2. In addition, we know that
(−F (ζ ))+ + (−ζ )+ ≤√
2((−F (ζ ))+2+ (−ζ )+2)1/2
≤
√2
√α ˜ψ(F (ζ), ζ)1/2
≤
√2
√αfLT(ζ )1/2,
where the second inequality is true by Lemma3.3. Thus,
(F (ζ ) ◦ ζ )+ + (−F (ζ ))+ + (−ζ )+ ≤ √ 2+
√2
√α
fLT(ζ )1/2.
This together with (24) yields (25).
Proposition 3.2 Let fYFbe given as in (11) and (16). Suppose that F has uniform P∗-property and SOCCP has a solution ζ∗. Then, there exists a scalar τ > 0 such that
τζ − ζ∗2≤ (F (ζ ) ◦ ζ )+ + (−F (ζ ))+ + (−ζ )+, ∀ζ ∈ Rn. (27) Moreover,
τζ − ζ∗2≤ 3√
2fYF(ζ )1/2, ∀ζ ∈ Rn. (28) Proof It follows totally the same arguments as in the proof for Proposition3.1to ob- tain (27). It remains to show the second part. Since by fYF(ζ )=12(F (ζ ) ◦ ζ )+2+ ψFB(F (ζ ), ζ ), we have
(F (ζ ) ◦ ζ )+ ≤√
2fYF(ζ )1/2. In addition, we know that
(−F (ζ ))+ + (−ζ )+ ≤√ 2
(−F (ζ ))+2+ (−ζ )+21/2
≤ 2√
2ψFB(F (ζ ), ζ )1/2
≤ 2√
2fYF(ζ )1/2, where the second inequality is true by Lemma3.2. Thus,
(F (ζ ) ◦ ζ )+ + (−F (ζ ))+ + (−ζ )+ ≤ 3√
2fYF(ζ )1/2.
This together with (27) yield (28).
4 Conditions for Bounded Level Sets
The boundedness of level sets of a merit function is also important since it ensures that the sequences generated by a descent method has at least one accumulation point.
In particular, there have existing results of bounded level sets for fYF, fYF, fLT, fLT,
respectively, for instances, Proposition 6 of [15], Proposition 4.2 of [13], Proposi- tion 4.3 of [14] and Proposition 4.4 of [14], which require that F is monotone and SOCCP is strict feasible. We note that the strict feasibility is necessary. For exam- ple, when F (ζ )≡ 0 every ζ ∈ Kn is a solution of SOCCP and hence the solution set is unbounded. In this section, we study another condition to replace this kind of
“strict” condition by F being R01-function for cases of fYF, fLT, while by F being R02-functions for cases of fLT, fYF.
We want to point it out that for fLTand fLTto possess property of bounded level sets, an additional condition is required (see Lemma4.1). In fact, the examples of ˜ψ1
and ˜ψ2given in (14) both satisfy this additional condition (Lemma4.1). It was also proved in Lemma 9 of [15] that this additional condition is satisfied with ψFBas well.
Lemma 4.1 ([14, Lemma 4.4]) For any {(xk, yk)}∞k=1⊆ Rn × Rn, let λ1(x)k ≤ λ2(x)k and μ1(y)k≤ μ2(y)k denote the spectral values of xk and yk, respectively.
Then, if λ1(x)k→ −∞ or μ1(y)k→ −∞, we have ˜ψi(xk, yk)→ ∞, for i = 1, 2.
Now, we come to another main work of this paper that is to claim the monotonicity of F plus the strict feasibility of SOCCP can be replaced by R01(or R02)-functions to ensure property of bounded level sets.
Proposition 4.1 (a) Let fYFbe given as in (9) and (16). Suppose that F is a R01- function. Then, the level set
L(γ ) := {ζ ∈ Rn| fYF(ζ )≤ γ } is bounded for all γ ≥ 0.
(b) Let fLTbe given as in (12) and (16) with ˜ψ satisfying Lemma4.1. Suppose that F is a R01-function. Then, the level set
L(γ ) := {ζ ∈ Rn| fLT(ζ )≤ γ } is bounded for all γ ≥ 0.
Proof (a) We will prove this result by contradiction. Suppose there exists an un- bounded sequence{ζk} ⊂ L(γ ) for some γ ≥ 0. It can be seen that the sequence of the smaller spectral values of{ζk} and {F (ζk)} are bounded below. In fact, if not, it follows from Lemma4.1(note ψFBalso satisfies this lemma, see Lemma 9 of [15]) that fYF(ζk)→ ∞, which contradicts {ζk} ⊂ L(γ ).
Therefore,{(−ζk)+} and {(−F (ζk))+} are bounded above, which says condition (20) is held. Then, by the assumption of R01-function, we have
lim inf
k→∞
ζk, F (ζk)
ζk2 >0.
This yieldsζk, F (ζk) → ∞, and hence fYF(ζk)→ ∞ by definition of fYFgiven as in (9) and (10). Thus, it is a contradiction to{ζk} ⊂ L(γ ).
(b) Same arguments as part (a).
Proposition 4.2 (a) Let fLTbe given as in (15) and (16) with ˜ψsatisfying Lemma4.1.
Suppose that F is a R02-function. Then, the level set L(γ ) := {ζ ∈ Rn| fLT(ζ )≤ γ } is bounded for all γ ≥ 0.
(b) Let fYFbe given as in (11) and (16). Suppose that F is a R02-function. Then, the level set
L(γ ) := {ζ ∈ Rn| fYF(ζ )≤ γ } is bounded for all γ ≥ 0.
Proof (a) Again, we will prove this result by contradiction. Suppose there exists an unbounded sequence{ζk} ⊂ L(γ ) for some γ ≥ 0. It can be seen that the sequence of the smaller spectral values of{ζk} and {F (ζk)} are bounded below. In fact, if not, it follows from Lemma4.1(note we assume ˜ψsatisfies this lemma) that fLT(ζk)→ ∞, which contradicts{ζk} ⊂ L(γ ).
Thus,{(−ζk)+} and {(−F (ζk))+} are bounded above, which says condition (20) is held. Then, by the assumption of R02-function, we have
lim inf
k→∞
λ2(ζk◦ F (ζk))
ζk2 >0.
This yields λ2(ζk◦ F (ζk))→ ∞, and hence (ζk◦ F (ζk))+ → ∞. This together with definition of fLTgiven as in (15) and (10) imply fLT(ζk)→ ∞. But, this con- tradicts{ζk} ⊂ L(γ ). Therefore, we complete the proof.
(b) Same arguments as part (a).
5 Final Remarks
In this paper, we have studied conditions for error bounds and bounded level sets of some merit functions, fYF, fYF, fLT, fLT given as in (16) for SOCCP. For property of bounded level sets, we propose a new condition, F being R01-function, to replace the traditional condition of monotonicity of F and strict feasibility of SOCCP in the cases of fYF, fLT. In the contrast, we propose another condition, F being R02- function, to replace the traditional condition of monotonicity of F and strict feasibil- ity of SOCCP in the cases of fLT, fYF. We notice that the condition of R02-function is even weaker than R01-function, which means we need a bit stronger condition in cases of fYF, fLTto obtain property of bounded level sets than in cases of fLT, fYFto do. This observation seems true for property of error bounds. More specifically, we have established the new condition of uniform P∗-property to ensure that fLT, fYF
provide error bounds (see Propositions3.1and3.2). Thus, due to this observation, we suspect that there needs a condition between strongly monotonicity and uniform P∗-property (see the implications in Property2.2(b)) to ensure fYF, fLT to provide error bounds for SOCCP. However, we still don’t know whether there is a condition
between strongly monotonicity and uniform P∗-property (see Property2.2(b)). That is the reason we don’t have similar results of error bounds for fYF, fLTyet.
We can elaborate more to explain the above reason from the other aspect. In fact, the existing results of error bounds in Proposition 5 of [15] and Proposition 4.1 of [14] (for fYFand fLT, respectively) say that there exists a scalar τ > 0 such that
τζ − ζ∗2≤ max{0, F (ζ ), ζ} + (−F (ζ ))+ + (−ζ )+, ∀ζ ∈ Rn. (29) Now, by the fact from Lemma 4.1 of [13],
x, y ≤√
2(x ◦ y)+, ∀x, y ∈ Rn, we can see that
max{0, F (ζ ), ζ} ≤√
2(F (ζ ) ◦ ζ )+.
In other words, if (29) is true then (24) is also held. But the converse is not guaranteed.
If we follow the same arguments as in Propositions3.1and3.2, we can obtain (24).
Nonetheless, (24) does not imply (29) as explained above. Thus, the uniform P∗- property does not guarantee the property of error bounds for fYF, fLT. Therefore, it is still worth of watching up on the issue of finding a weaker condition than strong monotonicity for fYF, fLTto provide error bounds for SOCCP.
Acknowledgements The author thanks the two referees for careful reading of the paper and helpful sug- gestions. In particular, the proof of Proposition3.1was improved due to one referee’s valuable comments.
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