**A SURVEY ON SOC COMPLEMENTARITY FUNCTIONS AND** **SOLUTION METHODS FOR SOCPS AND SOCCPS**

Jein-Shan Chen* ^{∗}* and Shaohua Pan

^{†}*This paper is dedicated to the memory of Professor Paul Tseng.*

**Abstract: This paper makes a survey on SOC complementarity functions and related solution methods for**
the second-order cone programming (SOCP) and second-order cone complementarity problem (SOCCP).

Speciﬁcally, we discuss the properties of four classes of popular merit functions, and study the theoretical results of associated merit function methods and numerical behaviors in the solution of convex SOCPs.

Then, we present suitable nonsinguarity conditions for the B-subdiﬀerentials of the natural residual (NR) and Fischer-Burmeister (FB) nonsmooth system reformulations at a (locally) optimal solution, and test the numerical behavior of a globally convergent FB semismooth Newton method. Finally, we survey the properties of smoothing functions of the NR and FB SOC complementarity functions, and provide numerical comparisons of the smoothing Newton methods based on them. The theoretical results and numerical experience of this paper provide a comprehensive view on the development of this ﬁeld in the past ten years.

**Key words: second-order cone, complementarity functions, merit functions, smoothing function, nons-***mooth Newton methods, smoothing Newton methods*

**Mathematics Subject Classification: 26B05, 26B35, 90C33, 65K05**

**1 Introduction**

The second-order cone (SOC) in IR^{n}*(n≥ 1), also called the Lorentz cone, is deﬁned as*
*K** ^{n}*:={

*(x*_{1}*, x*_{2})*∈ IR × IR*^{n}^{−1}*| x*1*≥ ∥x*2*∥*}
*,*

where*∥·∥ denotes the Euclidean norm. If n = 1, then K** ^{n}*is the set of nonnegative reals IR+.
We are interested in optimization and complementarity problems whose constraints involve
the direct product of SOCs. In particular, we are interested in the SOC complementarity

*system which is to ﬁnd vectors x, y∈ IR*

^{n}*and ζ∈ IR*

*satisfying*

^{l}*x∈ K, y ∈ K, ⟨x, y⟩ = 0, E(x, y, ζ) = 0,* (1.1)
where *⟨·, ·⟩ denotes the Euclidean inner product, E : IR*^{n}*× IR*^{n}*× IR*^{l}*→ IR*^{n}*× IR** ^{l}* is a
continuously diﬀerentiable mapping, and

*K is the direct product of SOCs given by*

*K = K*^{n}^{1}*× K*^{n}^{2}*× · · · × K*^{n}* ^{m}* (1.2)

*∗*Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Oﬃce. The author’s
work is partially supported by National Science Council of Taiwan.

*†*The author’s work is supported by National Young Natural Science Foundation (No. 10901058), Guang-
dong Natural Science Foundation (No. 9251802902000001) and the Fundamental Research Funds for the
Central Universities (SCUT).

*with m, n*1*, . . . , n**m**≥ 1, and n*1+*· · · + n**m* *= n. Throughout this paper, corresponding to*
the structure of*K, we write x = (x*1*, . . . , x**m**), y = (y*1*, . . . , y**m**) with x**i**, y**i* *∈ IR*^{n}* ^{i}*.

A special case of (1.1) is the generalized second-order cone complementarity problem
*(SOCCP) which, for given two continuously diﬀerentiable mappings F = (F*_{1}*, . . . , F** _{m}*) and

*G = (G*

_{1}

*, . . . , G*

_{m}*) with F*

_{i}*, G*

*: IR*

_{i}

^{n}*→ IR*

^{n}

^{i}*, is to ﬁnd a vector ζ∈ IR*

^{n}*such that*

*F (ζ)∈ K, G(ζ) ∈ K, ⟨F (ζ), G(ζ)⟩ = 0.* (1.3)
*When G becomes an identity one, (1.3) reduces to ﬁnding a vector ζ∈ IR** ^{n}* such that

*ζ∈ K, F (ζ) ∈ K, ⟨ζ, F (ζ)⟩ = 0,* (1.4)
which is a direct extension of the NCPs studied well in the past 30 years (see [22, 24]).

Another special case of (1.1) is the KKT conditions of the second-order cone programming
minimize *f (x)*

*subject to Ax = b, x∈ K* (1.5)

*where f : IR*^{n}*→ IR is a twice continuously diﬀerentiable function, A is an m × n matrix with*
*full row rank, and b∈ IR*^{m}*. When f is linear, (1.5) becomes the standard linear SOCP that*
has wide applications in engineering design, control, ﬁnance, management science, and so
on; see [1, 35] and the references therein. In addition, system (1.1) arises directly from some
engineering and practical problems; for example, the three-dimensional frictional contact
problems [34] and the robust Nash equilibria [28].

During the past ten years, there appeared active research for SOCPs and SOCCPs,
and various methods had been proposed which include the interior-point methods [1, 35,
41, 63, 51], the smoothing Newton methods [18, 25, 27], the semismooth Newton methods
[32, 43], and the merit function methods [10, 12]. Among others, the last three kinds
of methods are typically developed by an SOC complementarity function. Recall that a
*mapping ϕ : IR*^{n}*× IR*^{n}*→ IR*^{n}*is an SOC complementarity function associated withK** ^{n}* if

*ϕ(x, y) = 0* *⇐⇒ x ∈ K*^{n}*, y∈ K*^{n}*,* *⟨x, y⟩ = 0.* (1.6)
However, there are lack of comprehensive studies for the properties of SOC complementarity
functions and the numerical behavior of related solution methods. In this work, we give a
survey for popular SOC complementarity functions and the related merit function methods,
semismooth Newton methods and smoothing Newton methods.

The squared norm of SOC complementarity functions gives a merit function associated
with*K*^{n}*, where ψ : IR*^{n}*× IR*^{n}*→ IR*+is called a merit function associated with *K** ^{n}* if

*ψ(x, y) = 0* *⇐⇒ x ∈ K*^{n}*, y∈ K*^{n}*,* *⟨x, y⟩ = 0.* (1.7)
Apart from this, there are other ways to construct merit functions; for example, the LT
*merit function in Subsection 3.3. Here we are interested in those smooth ψ so that the*
SOCCP (1.3) can be reformulated as an unconstrained smooth minimization problem

min

*ζ**∈IR*^{n}*Ψ(ζ) :=*

∑*m*
*i=1*

*ψ (F**i**(ζ), G**i**(ζ)) ,* (1.8)

*in the sense that ζ** ^{∗}*is a solution to (1.3) if and only if it solves (1.8) with zero optimal value.

*This is the so-called merit function approach. Note that with a smooth merit function ψ,*
system (1.1) can also be reformulated as a smooth minization problem

min

*(x,y,ζ)**∈IR*^{2n+l}*∥E(x, y, ζ)∥*^{2}*+ ψ(x, y),*

but the reformulation is not eﬀective for the solution of (1.1) due to the conﬂict between
the feasibility and the decrease of complementarity gap involved in the objective. So, in this
paper we consider the merit function methods for the SOCCP (1.3). In Section 3, we survey
and compare the properties of four classes of popular smooth merit functions associated
with*K** ^{n}*. In Section 4, we focus on the theoretical results of corresponding merit function
methods, and their numerical performance in the solution of linear SOCPs from DIMACS
[52] and nonlinear convex SOCPs generated randomly.

*With an SOC complementarity function ϕ associated withK** ^{n}*, we can rewrite (1.1) as

*Φ(z) = Φ(x, y, ζ) :=*

*E(x, y, ζ)*
*ϕ(x*_{1}*, y*_{1})

...
*ϕ(x**m**, y**m*)

*= 0,* (1.9)

By [22, Prop. 9.1.1], system (1.9) is eﬀective only for those nondiﬀerentiable but (strongly)
*semismooth ϕ. Two popular such ϕ are the vector-valued natural residual (NR) function*
*ϕ*_{NR}: IR^{n}*× IR*^{n}*→ IR*^{n}*and Fischer-Burmeister (FB) function ϕ*_{FB} : IR^{n}*× IR*^{n}*→ IR** ^{n}*:

*ϕ*_{NR}*(x, y) := x− (x − y)*+ (1.10)

and

*ϕ*_{FB}*(x, y) := (x + y)− (x*^{2}*+ y*^{2})^{1/2}*,* (1.11)
where (*·)*+ denotes the Euclidean projection onto *K*^{n}*, x*^{2} *means the Jordan product of x*
*and itself, and x*^{1/2}*with x∈ K*^{n}*is the unique square root of x such that x*^{1/2}*◦ x*^{1/2}*= x.*

*The two nondiﬀerentiable functions are strongly semismooth, where the proof for ϕ*_{NR} can
*be found in [18, Prop. 4.3], [9, Prop. 7] or [27, Prop. 4.5], and the proof for ϕ*_{FB} is given by
Sun and Sun [58] and Chen [11] by using diﬀerent techniques. In Section 5, we review the
nonsingularity conditions for the B-subdiﬀerentials of Φ at a solution of (1.1) without strict
complementarity, and test the behavior of a global FB nonsmooth Newton method.

*Let θ : IR*^{n}*× IR*^{n}*× IR → IR** ^{n}* be a continuously diﬀerentiable on IR

^{n}*× IR*

^{n}*× IR*++ with

*θ(·, ·, 0) ≡ ϕ(·, ·) for ϕ = ϕ*NR

*or ϕ*

_{FB}. Then (1.1) is also equivalent to the augmented system

*Θ(ω) = Θ(ε, x, y, ζ) :=*

*ε*
*E(x, y, ζ)*
*θ(x*1*, y*1*, ε)*

...
*θ(x*_{m}*, y*_{m}*, ε)*

*= 0,* (1.12)

which is continuously diﬀerentiable in IR++*×IR*^{n}*×IR*^{n}*×IR** ^{l}*. In the past several years, some
smoothing Newton methods have been proposed for (1.1) by solving a sequence of smooth
systems or a single augmented system (see, e.g., [25, 18, 27]), but there is no comprehensive
study for their numerical performance. Motivated by the eﬃciency of the smoothing Newton

method [54], we in Section 6 apply it for the system (1.12) involving the CHKS smoothing
*function and the squared smoothing function of ϕ*_{NR}, and the FB smoothing function, re-
spectively, and compare their numerical behaviors. Similar to the NR and FB nonsmooth
Newton methods, the locally superlinear (quadratic) convergence of these smoothing meth-
ods does not require the strict complementarity of solutions. So, these nonsmooth and
smoothing methods are superior to interior point methods in theory since singular Jaco-
bians will occur to the latter if strict complementarity is not satisﬁed.

*Throughout this paper, I means an identity matrix of appropriate dimension, IR*^{n}*(n≥ 1)*
*denotes the space of n-dimensional real column vectors, and IR*^{n}^{1}*×· · ·×IR*^{n}* ^{m}*is identiﬁed with
IR

^{n}^{1}

^{+}

^{···+n}

^{m}*. For a given set S, we denote int(S) and bd(S) by the interior and boundary of*

*S, respectively. For any x∈ IR*

^{n}*, we write x≽*

_{Kn}*0 (respectively, x≻*

_{Kn}*0) to mean x∈ K*

^{n}*(respectively, x∈ int(K*

^{n}*)). For any diﬀerentiable F : IR*

^{n}*→ IR*

^{l}*, we denote F*

^{′}*(x)*

*∈ IR*

^{l}

^{×n}*by the Jacobian of F at x, and*

*∇F (x) by the transposed Jacobian of F at x. A square*

*matrix B∈ IR*

^{n}*is said to be positive deﬁnite if*

^{×n}*⟨u, Bu⟩ > 0 for all nonzero u ∈ IR*

*, and*

^{n}*B is said to be positive semideﬁnite if⟨u, Bu⟩ ≥ 0 for all u ∈ IR*

*.*

^{n}**2 Preliminaries**

This section recalls some background materials that are needed in the subsequent sections.

*For any x = (x*1*, x*2*), y = (y*1*, y*2)*∈ IR × IR*^{n}* ^{−1}*, their Jordan product [23] is deﬁned by

*x◦ y := (⟨x, y⟩, y*1

*x*

_{2}

*+ x*

_{1}

*y*

_{2}

*).*

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 under this
*product is e := (1, 0, . . . , 0)*^{T}*∈ IR*^{n}*. For any given x = (x*1*, x*2)*∈ IR × IR*^{n}* ^{−1}*, the matrix

*L**x*:=

[ *x*1 *x*^{T}_{2}
*x*2 *x*1*I*

]

will be used, which can be viewed as a linear mapping from IR* ^{n}* to IR

^{n}*given by L*

*x*

*y = x◦ y.*

*For each x = (x*_{1}*, x*_{2})*∈ IR × IR*^{n}^{−1}*, let λ*_{1}*(x), λ*_{2}*(x) and u*^{(1)}*x* *, u*^{(2)}*x* be the spectral values
*and the corresponding spectral vectors of x, respectively, given by*

*λ**i**(x) := x*1+ (*−1)*^{i}*∥x*2*∥ and u*^{(i)}*x* := 1
2
(

*1, (−1)*^{i}*x*¯2

)

*,* *i = 1, 2*

with ¯*x*_{2}*= x*_{2}*/∥x*2*∥ if x*2*̸= 0, and otherwise ¯x*2being any vector in IR^{n}* ^{−1}*satisfying

*∥¯x*2

*∥ = 1.*

*Then x admits a spectral factorization [23] associated withK** ^{n}* in the form of

*x = λ*1

*(x)u*

^{(1)}

_{x}*+ λ*2

*(x)u*

^{(2)}

_{x}*.*

*When x*_{2}*̸= 0, the spectral factorization is unique. The following lemma states the relation*
*between the spectral factorization of x and the eigenvalue decomposition of L** _{x}*.

**Lemma 2.1 ([25]). For any given x**∈ IR^{n}*, let λ*_{1}*(x), λ*_{2}*(x) be the spectral values of x, and*
*u*^{(1)}*x* *, u*^{(2)}*x* *be the associated spectral vectors. Then, L**x* *has the eigenvalue decomposition*

*L*_{x}*= U (x)diag (λ*_{2}*(x), x*_{1}*,· · · , x*1*, λ*_{1}*(x)) U (x)*^{T}

*where*

*U (x) =(√*

*2u*^{(2)}_{x}*, u*^{(3)}_{x}*,· · · , u*^{(n)}*x* *,√*
*2u*^{(1)}_{x}

)*∈ IR*^{n}^{×n}

*is an orthogonal matrix, and u*^{(i)}*x* *for i = 3, . . . , n have the form of (0, ¯u**i**) with ¯u*3*, . . . , ¯u**n*

*being any unit vectors in IR*^{n}^{−1}*that span the linear subspace orthogonal to x*2*.*

*By using Lemma 2.1, it is not hard to calculate the inverse of L** _{x}*whenever it exists:

*L*^{−1}* _{x}* = 1

*det(x)*

*x*1 *−x** ^{T}*2

*−x*2

*det(x)*
*x*1

*I +* 1
*x*1

*x*2*x*^{T}_{2}

(2.1)

*where det(x) := x*^{2}_{1}*− ∥x*2*∥*^{2}*denotes the determinant of x.*

*By the spectral factorization above, for any given scalar function g : IR* *→ J ⊆ IR, we*
*may deﬁne the associated vector-valued function g*^{soc}: IR^{n}*→ S ⊆ IR** ^{n}* by

*g*^{soc}*(x) := g(λ*1*(x))u*^{(1)}_{x}*+ g(λ*2*(x))u*^{(2)}_{x}*.* (2.2)
*For example, taking g(t) =* *√*

*t for t* *≥ 0, we have that g*^{soc}*(x) = x*^{1/2}*with x* *∈ K** ^{n}*. The

*vector-valued g*

^{soc}

*inherits many desirable properties from g (see [9]). The following lemma*

*provides the formulas to compute the Jacobian of g*

^{soc}and its inverse.

**Lemma 2.2. Let g : IR**→ J ⊆ IR be a given scalar function, and g^{soc}: IR^{n}*→ S ⊆ IR*^{n}*be*
*deﬁned by (2.2). If g is diﬀerentiable on int(J), then g*^{soc} *is diﬀerentiable in int(S) with*

*∇g*^{soc}*(x) =*

*g*^{′}*(x*_{1}*)I* *if x*_{2}*= 0,*

*b(x)* *c(x)* *x*^{T}_{2}

*∥x*2*∥*
*c(x)* *x*2

*∥x*2*∥* *a(x)I + (b(x)− a(x))x*2*x*^{T}_{2}

*∥x*2*∥*^{2}

* if x*^{2}*̸= 0*

*for any x = (x*1*, x*2)*∈ int(S), where*
*a(x) =* *g(λ*2*(x))− g(λ*1*(x))*

*λ*_{2}*(x)− λ*1*(x)* *, b(x) =* *g*^{′}*(λ*2*(x)) + g*^{′}*(λ*1*(x))*

2 *, c(x) =* *g*^{′}*(λ*2*(x))− g*^{′}*(λ*1*(x))*

2 *.*

*If∇g*^{soc}(*·) is invertible at x ∈ int(S), then letting d(x) = b*^{2}*(x)− c*^{2}*(x), we have that*

(*∇g*^{soc}*(x))** ^{−1}*=

*(g*^{′}*(x*1))^{−1}*I* *if x*2*= 0,*

*b(x)*

*d(x)* *−c(x)*

*d(x)*
*x*^{T}_{2}

*∥x*2*∥*

*−c(x)*
*d(x)*

*x*2

*∥x*2*∥*
1
*a(x)I +*

(*b(x)*
*d(x)−* 1

*a(x)*
) *x*2*x*^{T}_{2}

*∥x*2*∥*^{2}

* if x*^{2}*̸= 0.*

*Proof. The ﬁrst part is direct by Prop. 5.2 of [25] or Prop. 5 of [9]. For the second part,*
it suﬃces to calculate the inverse of*∇g*^{soc}*(x) when x*_{2} *̸= 0. By the expression of ∇g*^{soc},
*it is easy to verify that b(x) + c(x) and b(x)− c(x) are the eigenvalues of ∇g*^{soc}*(x) with*
*(1,*_{∥x}^{x}^{2}

2*∥**) and (1,−*_{∥x}^{x}^{2}_{2}_{∥}*) being the corresponding eigenvectors, and a(x) is the eigenvalue of*
*multiplicity n− 2 with corresponding eigenvectors of the form (0, ¯v**i*), where ¯*v*1*, . . . , ¯v**n**−2*

are any unit vectors in IR^{n}^{−1}*that span the subspace orthogonal to x*2. By this, using an
elementary calculation yields the formula of (*∇g*^{soc}*(x))** ^{−1}*.

We next recall some joint properties of two mappings which are the direct extensions
*of the uniform Cartesian P -property [17], the uniform Jordan P -property [60], the weak*
*coerciveness [67], and the R*0-property [6], respectively.

* Definition 2.3. The mappings F = (F*1

*, . . . , F*

*m*

*) and G = (G*1

*, . . . , G*

*m*) are said to have

**(i) joint uniform Cartesian P -property if there exists a constant ϱ > 0 such that, for***every ζ, ξ∈ IR*^{n}*, there exists an index ν∈ {1, 2, . . . , m} such that*

*⟨F**ν**(ζ)− F**ν**(ξ)), G**ν**(ζ)− G**ν**(ξ)⟩ ≥ ϱ∥ζ − ξ∥*^{2}*.*

**(ii) joint uniform Jordan P -property if there exists a constant ϱ > 0 such that, for***every ζ, ξ∈ IR** ^{n}*,

*λ*2*[(F (ζ)− F (ξ)) ◦ (G(ζ) − G(ξ))] ≥ ϱ∥ζ − ξ∥*^{2}*.*

**(iii) joint Cartesian weak coerciveness if there is an element ξ**∈ IR* ^{n}* such that

*∥ζ∥→∞*lim max

1*≤i≤m*

*⟨G**i**(ζ)− G**i**(ξ), F**i**(ζ)⟩*

*∥ζ − ξ∥* = +*∞.*

**(iv) joint Cartesian strong coerciveness if the last equation holds for all ξ***∈ IR** ^{n}*.

**(v) joint Cartesian R**_{0}

^{w}**-property if, for any sequence**

*{ζ*

^{k}*} ⊆ IR*

*satisfying*

^{n}*∥ζ*^{k}*∥ → +∞, lim sup*

*k**→∞* *∥(F (ζ** ^{k}*))

_{−}*∥ < +∞, lim sup*

*k**→∞* *∥(G(ζ** ^{k}*))

_{−}*∥ < +∞,*there holds that

lim sup

*k**→∞* max

1*≤i≤m*

⟨*F*_{i}*(ζ*^{k}*), G*_{i}*(ζ** ^{k}*)⟩

= +*∞.*

*It is easy to see that the joint uniform Cartesian P -property implies the joint Cartesian*
strong coerciveness. From the arguments in [47], it follows that the joint uniform Carte-
*sian P -property implies the joint uniform Jordan P -property, and the joint Cartesian weak*
*coerciveness with respect to an element ξ with G(ξ)∈ K implies the joint Cartesian R** ^{w}*0-

*property. Now we are not clear whether the joint uniform Jordan P -property implies the*joint Cartesian weak coerciveness. Note that the above several properties do not imply the

*joint monotonicity of F and G, but the joint monotonicity of F and G with some additional*

*conditions may imply their joint Cartesian R*

^{w}_{0}-property; see the remarks after Prop. 4.2.

The following deﬁnition recalls the concept of linear growth of a mapping, which is weaker than the global Lipschitz continuity.

**Definition 2.4. A mapping F : IR**^{n}*→ IR*^{n}**is said to have linear growth if there exists a**
*constant C > 0 such that∥F (ζ)∥ ≤ ∥F (0)∥ + C∥ζ∥ for any ζ ∈ IR** ^{n}*.

*We next introduce the Cartesian (strict) column monotonicity of matrices M and N ,*
which is weaker than the (strict) column monotonicity introduced in [22, page 1014] and
*[37, page 222]. Particularly, when N is invertible, this property reduces to the Cartesian P*0

*(P )-property of the matrix−N*^{−1}*M introduced by Chen and Qi [17].*

**Definition 2.5. The matrices M, N**∈ IR^{n}* ^{×n}*are said to be

**(i) Cartesian column monotone if for any u, v**∈ IR^{n}*with u̸= 0, v ̸= 0,*
*M u + N v = 0 =⇒ ∃ν ∈ {1, . . . , m} s.t. u**ν**̸= 0 and ⟨u**ν**, v**ν**⟩ ≥ 0.*

**(ii) Cartesian strictly column monotone if for any u, v**∈ IR^{n}*with (u, v)̸= (0, 0),*
*M u + N v = 0 =⇒ ∃ν ∈ {1, . . . , m} s.t. ⟨u**ν**, v**ν**⟩ > 0.*

To close this section, we recall the concept of B-subdiﬀerential for a locally Lipschitz
*continuous mapping. If H : IR*^{n}*→ IR** ^{m}* is locally Lipschitz continuous, then the set

*∂**B**H(z) :=*{

*V* *∈ IR*^{m}^{×n}*| ∃{z*^{k}*} ⊆ D**H* *: z*^{k}*→ z, H*^{′}*(z** ^{k}*)

*→ V*}

*is nonempty and called the B-subdiﬀerential [55] of H at z, where D**H* *⊆ IR** ^{n}* is the set of

*points at which H is diﬀerentiable. The convex hull of ∂*

*B*

*H(z) is called the generalized*

*Jacobian of Clarke [20], i.e. ∂H(z) = conv∂*

*B*

*H(z). We assume that the reader is familiar*with the concept of (strong) semismoothness, and refer to [49, 55, 56] for the details.

*Unless otherwise stated, in the rest of this paper, we assume that F = (F*_{1}*, . . . , F** _{m}*) and

*G = (G*

_{1}

*, . . . , G*

_{m}*) with F*

_{i}*, G*

*: IR*

_{i}

^{n}*→ IR*

^{n}*are continuously diﬀerentiable. For a given*

^{i}*x∈ IR*

^{l}*for some l≥ 2, we write x = (x*1

*, x*2)

*∈ IR × IR*

^{l}

^{−1}*, where x*1 is the ﬁrst component

*of the vector x and x*2

*consists of the remaining l− 1 components of x.*

**3 Merit Functions Associated with** *K*

^{n}This section reviews four classes of smooth merit functions associated with *K** ^{n}* and their
properties related to the merit function approach. The nondiﬀerentiable NR function

*ψ*_{NR}*(x, y) :=∥x − (x − y)*+*∥*^{2} *∀x, y ∈ IR** ^{n}* (3.1)
is needed, which plays a crucial role in error bound estimations of other merit functions.

**3.1** **Implicit Lagrangian Function**

*The implicit Lagrangian ψ*_{MS}: IR^{n}*× IR*^{n}*→IR*+*, parameterized by α > 1, is deﬁned as*
*ψ*_{MS}*(x, y)* := max

*u,v**∈K*^{n}

{

*⟨x, y − v⟩ − ⟨y, u⟩ −* 1

*2α*(*∥x − u∥*^{2}+*∥y − v∥*^{2})
}

= *⟨x, y⟩ +* 1
*2α*

(*∥(x − αy)*+*∥*^{2}*− ∥x∥*^{2}+*∥(y − αx)*+*∥*^{2}*− ∥y∥*^{2})

*.* (3.2)
The function is introduced by Mangasarian and Solodov [38] for NCPs, and extended to
semideﬁnite complementarity problems (SDCPs) by Tseng [61] and general symmetric cone
*complementarity problems (SCCPs) by Kong et al. [33]. By Theorem 3.2(b) of [33], ψ*_{MS} is
a merit function induced by the trace of the SOC complementarity function

*ϕ*_{MS}*(x, y) := x◦ y +* 1
*2α*

[*(x− αy)*^{2}+*− x*^{2}*+ (y− αx)*^{2}+*− y*^{2}]

*∀x, y ∈ IR*^{n}*, α > 1.* (3.3)
The following results are extensions of known results, particularly [62, 65, 39], for NCPs.

**Lemma 3.1. For any ﬁxed α > 1 and all x, y**∈ IR^{n}*, we have the following results.*

**(a) ψ**_{MS}*(x, y) = 0* *⇐⇒ x ∈ K*^{n}*, y∈ K*^{n}*,* *⟨x, y⟩ = 0 ⇐⇒ ϕ*MS*(x, y) = 0.*

**(b) ϕ**_{MS} *and ψ*_{MS} *are continuously diﬀerentiable everywhere, with*

*∇**x**ψ*_{MS}*(x, y)* = *y + α*^{−1}*((x− αy)*+*− x) − (y − αx)*+*,*

*∇**y**ψ*_{MS}*(x, y)* = *x + α*^{−1}*((y− αx)*+*− y) − (x − αy)*+*.*

* (c) The gradient function∇ψ*MS

*is globally Lipschitz continuous.*

**(d)** *⟨x, ∇**x**ψ*_{MS}*(x, y)⟩ + ⟨y, ∇**y**ψ*_{MS}*(x, y)⟩ = 2ψ*MS*(x, y).*

**(e)** *⟨∇**x**ψ*_{MS}*(x, y),∇**y**ψ*_{MS}*(x, y)⟩ ≥ 0.*

**(f ) ψ**_{MS}*(x, y) = 0 if and only if∇**x**ψ*_{MS}*(x, y) = 0 and∇**y**ψ*_{MS}*(x, y) = 0.*

* (g) (α− 1)∥ϕ*NR

*(x, y)∥*

^{2}

*≥ ψ*MS

*(x, y)≥ (1 − α*

*)*

^{−1}*∥ϕ*NR

*(x, y)∥*

^{2}

*.*

**(h) α**^{−1}*(α− 1)*^{2}*ψ*_{MS}*(x, y)≤ ∥∇**x**ψ*_{MS}*(x, y) +∇**y**ψ*_{MS}*(x, y)∥*^{2}*≤ 2α(α − 1)ψ*MS*(x, y).*

*Proof. The proofs of parts (a)–(b) and (e)–(f) are given in [33]. Parts (c)–(d) are direct by*
*the expressions of ψ*_{MS} and *∇ψ*MS. Part (g) is a direct application of [62, Prop. 2.2] with

˜

*π =−ψ*MS. Part (h) is easily shown by [50, Theorem 4.2] and (b) and (g).

Analogous to the NCPs and SDCPs, the implicit Lagrangian has the most favorable properties among all projection merit functions. So, we do not review others in this class.

**3.2** **Fischer-Burmeister (FB) Merit Function**

*From [25], ϕ*_{FB} in (1.11) is an SOC complementarity function, and whence its squared norm
*ψ*_{FB}*(x, y) :=* 1

2*∥ϕ*FB*(x, y)∥*^{2}*.* (3.4)

is a merit function associated with *K*^{n}*. The function ψ*_{FB} was shown to be continuously
*diﬀerentiable everywhere with globally Lipschitz continuous gradient [10, 16], although ϕ*_{FB}
*itself is not diﬀerentiable. Recently, we extend these favorable properties of ψ*_{FB} to the
following one-parametric class of merit functions (see [14, 15]):

*ψ**τ**(x, y) :=* 1

2*∥ϕ**τ**(x, y)∥*^{2}*,* (3.5)

*where τ∈ (0, 4) is an arbitrary ﬁxed parameter and ϕ**τ*: IR^{n}*× IR*^{n}*→ IR** ^{n}* is deﬁned by

*ϕ*

*τ*

*(x, y) := (x + y)−*[

*(x− y)*^{2}*+ τ (x◦ y)*]*1/2*

*.* (3.6)

*Clearly, when τ = 2, ψ*_{τ}*becomes the FB merit function ψ*_{FB}. The one-parametric class of
functions was originally proposed by Kanzow and Kleinmichel [31] for NCPs, and was proved
to share all desirable properties of the FB NCP function. The following lemma summarizes
*those properties of ψ**τ* used in the merit function approach.

**Lemma 3.2. For any ﬁxed τ**∈ (0, 4) and all x, y ∈ IR^{n}*, we have the following results.*

**(a) ψ***τ**(x, y) = 0* *⇐⇒ ϕ**τ**(x, y) = 0* *⇐⇒ x ∈ K*^{n}*, y∈ K*^{n}*,* *⟨x, y⟩ = 0.*

**(b) ψ**_{τ}*is continuously diﬀerentiable everywhere with* *∇**x**ψ*_{τ}*(0, 0) =∇**y**ψ*_{τ}*(0, 0) = 0. Also,*
*if w = (x− y)*^{2}*+ τ (x◦ y) ∈ int(K*^{n}*), then*

*∇**x**ψ*_{τ}*(x, y)* =
(

*I− L**x+*^{τ−2}_{2} *y**L*^{−1}*√**w*

)

*ϕ*_{τ}*(x, y),*

*∇**y**ψ*_{τ}*(x, y)* =
(

*I− L*_{y+}^{τ−2}

2 *x**L*^{−1}*√**w*

)

*ϕ*_{τ}*(x, y);*

*and if (x− y)*^{2}*+ τ (x◦ y) ∈ bd(K*^{n}*) and (x, y)̸= (0, 0),*

*∇**x**ψ**τ**(x, y)* =
[

1*−* *x*1+^{τ}^{−2}_{2} *y*1

√*x*^{2}_{1}*+ y*_{1}^{2}*+ (τ− 2)x*1*y*1

]

*ϕ**τ**(x, y),*

*∇**y**ψ*_{τ}*(x, y)* =
[

1*−* *y*_{1}+^{τ}^{−2}_{2} *x*_{1}

√*x*^{2}_{1}*+ y*_{1}^{2}*+ (τ− 2)x*1*y*1

]

*ϕ*_{τ}*(x, y).*

**(c) The gradient function**∇ψ*τ* *is globally Lipschitz continuous.*

**(d)** *⟨x, ∇**x**ψ**τ**(x, y)⟩ + ⟨y, ∇**y**ψ**τ**(x, y)⟩ = 2ψ**τ**(x, y).*

**(e)** *⟨∇**x**ψ**τ**(x, y),∇**y**ψ**τ**(x, y)⟩ ≥ 0, with equality holding if and only if ψ**τ**(x, y) = 0.*

**(f ) ψ***τ**(x, y) = 0* *⇐⇒ ∇**x**ψ**τ**(x, y) = 0* *⇐⇒ ∇**y**ψ**τ**(x, y) = 0.*

**(g) There exist constant c**_{1}*> 0 and c*_{2}*> 0 independent on x, y such that*
*c*1*∥ϕ*NR*(x, y)∥ ≤ ∥ϕ**τ**(x, y)∥ ≤ c*2*∥ϕ*NR*(x, y)∥.*

* (h) There exist constants C*1

*> 0 and C*2

*> 0 only dependent on n, τ such that*

*C*

_{1}

*∥ϕ*

*τ*

*(x, y)∥ ≤ ∥∇*

*x*

*ψ*

_{τ}*(x, y) +∇*

*y*

*ψ*

_{τ}*(x, y)∥ ≤ C*2

*∥ϕ*

*τ*

*(x, y)∥.*

*Proof. The proofs of parts (a)–(b) and (d)–(e) can be found in [14]. Part (c) is proved in [15,*
Theorem 3.1]. Part (f) follows by parts (a), (b) and (e). Parts (g) and (h) are established
in [3].

*Comparing Lemma 3.2 with Lemma 3.1, we see that the functions ψ*_{FB} *and ψ*_{MS} share
*with similar favorable properties, but the properties (e)–(f) of ψ*_{FB} are stronger than those
*of ψ*_{MS}*, which make ψ*_{FB} require a weaker stationary point condition; see Prop. 4.1.

It should be pointed out that the squared norms of Evtushenko and Purtov [21] SOC
*complementarity functions ϕ**α*: IR^{n}*× IR*^{n}*→ IR*^{n}*and ϕ**β*: IR^{n}*× IR*^{n}*→ IR** ^{n}*, deﬁned as

*ϕ**α**(x, y)* := *−(x ◦ y) +* 1

*2α(x + y)*^{2}_{−}*0 < α≤ 1,*
*ϕ**β**(x, y)* := *−(x ◦ y) +* 1

*2β*

(*(x)*^{2}_{−}*+ (y)*^{2}* _{−}*)

*0 < β < 1,* (3.7)
*also provide the smooth merit functions ψ**α**and ψ**β* associated with*K** ^{n}*. But, since they do

*not enjoy the property (e) of ψ*

*τ*

*or the weaker property (e) of ψ*

_{MS}, it is hard to ﬁnd the conditions to guarantee that every stationary point of Ψ

*α*and Ψ

*β*is a solution of SOCCPs (see the proof of Prop. 4.1). In addition, unlike in the setting of NCPs, the squared norm of penalized FB SOC complementarity function is not smooth even nondiﬀerentiable. So, this paper does not include these functions.

**3.3** **Luo and Tseng (LT) Merit Function**

The third class of smooth merit functions is an extension of the class of functions introduced by Luo and Tseng [37] for NCPs, and subsequently extended to SDCPs in [61, 66]. In the setting of SOCs, this class of functions is deﬁned as

*ψ*_{LT}*(x, y) := ψ*_{0}(*⟨x, y⟩) + bψ(x, y),* *∀x, y ∈ IR** ^{n}* (3.8)

*where ψ*0: IR*→ IR*+ is an arbitrary smooth function satisfying

*ψ*0*(0) = 0,* *ψ*_{0}^{′}*(t) = 0* *∀t ≤ 0, and ψ** ^{′}*0

*(t) > 0*

*∀t > 0*(3.9) and b

*ψ : IR*

^{n}*× IR*

^{n}*→ IR*+ is an arbitrary smooth function such that

*ψ(x, y) = 0,*b *⟨x, y⟩ ≤ 0 ⇐⇒ x ∈ K*^{n}*, y∈ K*^{n}*,* *⟨x, y⟩ = 0.* (3.10)
*The requirment for ψ*0 is a little diﬀerent from the original LT merit functions [37]. There
*are many functions satisfying (3.9) such as the polynomial function q*^{−1}*max(0, t)*^{q}*(q≥ 2),*
*the exponential function exp(max(0, t)*^{2})*− 1, and logarithmic function ln(1 + max(0, t)*^{2}).

In addition, there are many choices for b*ψ such as ψ*_{MS}*, ψ**τ* and the following
*ψ*b1*(x, y) :=*1

2

(*∥(x)**−**∥*^{2}+*∥(y)**−**∥*^{2})

and b*ψ*2*(x, y) :=* 1

2*∥ϕ*FB*(x, y)*+*∥*^{2}*.* (3.11)
*In this paper, we are particularly interested in three subclasses of ψ*_{LT} with b*ψ chosen as ψ*_{FB},
*ψ*b1 and b*ψ*2*. Among others, ψ*_{LT} with b*ψ = ψ*_{FB} is an analog of the merit function studied by
*Yamashita and Fukushima [66] for SDCPs. In view of this, we write ψ*_{LT} with b*ψ = ψ*_{FB} as
*ψ*_{YF}*. We also write ψ*_{LT} with b*ψ = bψ*1 and b*ψ*2*as ψ*_{LT1} *and ψ*_{LT2}, respectively.

**Lemma 3.3. Let ψ be one of the functions ψ**_{YF}*, ψ*_{LT1} *and ψ*_{LT2}*. Then, for all x, y∈ IR*^{n}*,*
**(a) ψ(x, y) = 0***⇐⇒ x ∈ K*^{n}*, y∈ K*^{n}*,* *⟨x, y⟩ = 0.*

**(b) ψ is continuously diﬀerentiable everywhere. Furthermore,**

*∇**x**ψ*_{YF}*(x, y)* = *ψ*_{0}* ^{′}* (

*⟨x, y⟩) y + ∇*

*x*

*ψ*

_{FB}

*(x, y),*

*∇**y**ψ*_{YF}*(x, y)* = *ψ*_{0}* ^{′}* (

*⟨x, y⟩) x + ∇*

*y*

*ψ*

_{FB}

*(x, y),*

*where∇*

*x*

*ψ*

_{FB}

*and∇*

*y*

*ψ*

_{FB}

*are given by Lemma 3.2(c) with τ = 2;*

*∇**x**ψ*_{LT1}*(x, y) = ψ*_{0}* ^{′}*(

*⟨x, y⟩) y + (x)*

*−*

*,*

*∇*

*y*

*ψ*

_{LT1}

*(x, y) = ψ*

^{′}_{0}(

*⟨x, y⟩) x + (y)*

*−*;

*when ψ = ψ*

_{LT2}

*,∇*

*x*

*ψ*

_{LT2}

*(0, 0) =∇*

*y*

*ψ*

_{LT2}

*(0, 0) = 0, and if x*

^{2}

*+ y*

^{2}

*∈ int(K*

^{n}*),*

*∇**x**ψ*_{LT2}*(x, y)* = *ψ*_{0}* ^{′}* (

*⟨x, y⟩) y +*(

*I− L**x**L*^{−1}_{(x}_{2}_{+y}_{2}_{)}* _{1/2}*
)

*ϕ*_{FB}*(x, y)*_{+}*,*

*∇**y**ψ*_{LT2}*(x, y)* = *ψ*_{0}* ^{′}* (

*⟨x, y⟩) x +*(

*I− L**y**L*^{−1}_{(x}_{2}_{+y}_{2}_{)}* _{1/2}*
)

*ϕ*_{FB}*(x, y)*_{+}*,*
*and if x*^{2}*+ y*^{2}*∈ bd*^{+}(*K*^{n}*),*

*∇**x**ψ*_{LT2}*(x, y)* = *ψ*^{′}_{0}(*⟨x, y⟩) y +*
(

1*−* *x*1

√*x*^{2}_{1}*+ y*^{2}_{1}
)

*ϕ*_{FB}*(x, y)*+*,*

*∇**y**ψ*_{LT2}*(x, y)* = *ψ*^{′}_{0}(*⟨x, y⟩) x +*
(

1*−* *y*1

√*x*^{2}_{1}*+ y*_{1}^{2}
)

*ϕ*_{FB}*(x, y)*+*.*

**(c) The gradient**∇ψ is globally Lipschitz continuous on any bounded set of IR^{n}*× IR*^{n}*.*
**(d)** *⟨x, ∇**x**ψ(x, y)⟩ + ⟨y, ∇**y**ψ(x, y)⟩ ≥ 2ψ** ^{′}*0(

*⟨x, y⟩) ⟨x, y⟩ + 2ψ(x, y) ≥ 2 bψ(x, y).*

**(e)** *⟨∇**x**ψ(x, y),∇**y**ψ(x, y)⟩ ≥ 0, and when ψ = ψ*YF *and ψ*_{LT2}*,* *⟨∇**x**ψ(x, y),∇**y**ψ(x, y)⟩ = 0*
*if and only if ψ(x, y) = 0.*

**(f ) When ψ = ψ**_{YF} *and ψ*_{LT2}*, ψ(x, y) = 0⇐⇒ ∇**x**ψ(x, y) = 0⇐⇒ ∇**y**ψ(x, y) = 0; and when*
*ψ = ψ*_{LT1}*, ψ(x, y) = 0⇐⇒ ∇**x**ψ(x, y) = 0 and* *∇**y**ψ(x, y) = 0.*

* (g) If ψ*0

*is convex and nondecreasing in IR, then ψ*

_{LT1}

*is a convex function over IR*

^{n}*× IR*

^{n}*.*

*Proof. When ψ = ψ*

_{YF}

*, from the deﬁnition of ψ*

_{YF}and Lemma 3.2(a) and (c)–(d), we readily

*get parts (a)–(c); parts (d)–(e) are easily veriﬁed by using part (b), Lemma 3.2(e) with τ = 2*

*and equation (3.9). When ψ = ψ*

_{LT1}

*and ψ*

_{LT2}, parts (a)–(b) and (d)–(e) are established in

*Prop. 3.1 and Prop. 3.2 of [12] except the smoothness of ψ*

_{LT2}, which is implied by Lemma 1 of appendix. Part (c) is immediate by using the expressions of

*∇ψ*LT1and

*∇ψ*LT2and noting that

*∇ bψ is globally Lipschitz continuous on IR*

^{n}*× IR*

*.*

^{n}*When ψ = ψ*_{YF} *and ψ*_{LT2}*, part (f) follows by parts (b) and (e), and when ψ = ψ*_{LT1}, part
(f) follows by parts (b) and (d). By Prop. 3.1(b) of [12], b*ψ*1is convex over IR^{n}*× IR** ^{n}*. Since

*ψ*

_{0}

*is convex and nondecreasing in IR, it is easy to verify that ψ*

_{0}(

*⟨x, y⟩) is also convex over*IR

^{n}*× IR*

*. So, we obtain the result of part (g).*

^{n}*Comparing Lemma 3.3 with Lemmas 3.1 and 3.2, we observe that ψ*_{MS} *and ψ**τ* have two
remarkable advantages over the LT class of merit functions: one is the positive homogeneity
*of ψ*_{MS} *and ψ**τ*, which makes the corresponding merit functions for SOCCPs overcome the
bad-scaling of problems; the other is that their gradients have the same growth as the merit
function itself, which is the key to establish convergence rate of some descent algorithms.

It should be pointed out that although the LT class of merit functions does not possess the
*property (g) of ψ*_{MS}*and ψ*_{FB}, the corresponding merit functions for the SOCCPs may provide
a global error bound under a weaker condition (see Prop. 4.3), and moreover, Lemma 3.5
*below shows that they have faster growth than ψ*_{MS} *and ψ*_{FB}.

**3.4** **A Variant of LT Merit Function**

A variant of the LT merit functions is the function b*ψ*_{LT}: IR^{n}*× IR*^{n}*→ IR*+deﬁned by
*ψ*b_{LT}*(x, y) := ψ*0(*∥(x ◦ y)*+*∥*^{2}) + b*ψ(x, y)* *∀x, y ∈ IR*^{n}*,* (3.12)
*where ψ*0satisﬁes the ﬁrst and the third properties of (3.9) and b*ψ satisﬁes (3.10). This class*
of merit functions was considered by Chen [12]. In this work we are interested in b*ψ*_{LT} with
*ψ = ψ*b _{FB}*, bψ*1 and b*ψ*2, and write them as b*ψ*_{YF}, b*ψ*_{LT1} and b*ψ*_{LT2}, successively.

**Lemma 3.4. Let ψ be one of the functions b**ψ_{YF}*, bψ*_{LT1} *and bψ*_{LT2}*. Then, for all x, y∈ IR*^{n}*,*
**(a) ψ(x, y) = 0***⇐⇒ x ∈ K*^{n}*, y∈ K*^{n}*,* *⟨x, y⟩ = 0.*

**(b) ψ is continuously diﬀerentiable everywhere, with**

*∇**x**ψ(x, y)* = *2ψ*^{′}_{0}(

*∥(x ◦ y)*+*∥*^{2})

*L*_{y}*(x◦ y)*++*∇**x**ψ(x, y),*b

*∇**y**ψ(x, y)* = *2ψ*^{′}_{0}(

*∥(x ◦ y)*+*∥*^{2})

*L*_{x}*(x◦ y)*++*∇**y**ψ(x, y),*b
*where∇**x**ψ(x, y) and*b *∇**y**ψ(x, y) are same as in Lemma 3.3.*b

**(c) The gradient**∇ψ is globally Lipschitz continuous on any bounded set of IR^{n}*× IR*^{n}*.*

**(d)** *⟨x, ∇**x**ψ(x, y)⟩ + ⟨y, ∇**y**ψ(x, y)⟩ = 4ψ** ^{′}*0

(*∥(x ◦ y)*+*∥*^{2})

*∥(x ◦ y)*+*∥*^{2}+ 2 b*ψ(x, y).*

**(e) ψ(x, y) = 0***⇐⇒ ∇**x**ψ(x, y) = 0 and* *∇**y**ψ(x, y) = 0.*

*Proof. The proofs are same as that of Lemma 3.3, and we omit them.*

For the class of merit functions b*ψ*_{LT}, it is diﬃcult to establish the following inequality

*⟨∇**x**ψ*b_{LT}*(x, y),∇**y**ψ*b_{LT}*(x, y)⟩ ≥ 0 ∀x, y ∈ IR*^{n}

although numerical simulations show that they possess the property. The main diﬃculty is
to estimate the terms*⟨L**y**(x◦ y)*+*,∇**y**ψ(x, y)*b *⟩ and ⟨L**x**(x◦ y)*+*,∇**x**ψ(x, y)*b *⟩.*

To close this section, we characterize the growth of the above merit functions via a lemma, whose proof is direct by the arguments of [47, Sec. 4] and the remarks after it.

**Lemma 3.5. If the sequence**{(x^{k}*, y** ^{k}*)

*} ⊆ IR*

^{n}*× IR*

^{n}*satisﬁes one of the conditions:*

**(i) lim inf**_{k}_{→∞}*λ*_{1}*(x** ^{k}*) =

*−∞ or lim inf*

*k*

*→∞*

*λ*

_{1}

*(y*

*)*

^{k}*→ −∞;*

**(ii)** *{λ*1*(x** ^{k}*)

*} and {λ*1

*(y*

*)*

^{k}*} are bounded below, λ*2

*(x*

^{k}*), λ*

_{2}

*(y*

*)*

^{k}*→ +∞, and*

_{∥x}

^{x}

^{k}*k*

*∥*

*◦*

_{∥y}

^{y}

^{k}*k*

*∥*

*9 0;*

**(iii)** *{λ*1*(x** ^{k}*)

*} and {λ*1

*(y*

*)*

^{k}*} are bounded below, and lim sup*

*k*

*→∞*

*⟨x*

^{k}*, y*

^{k}*⟩ = +∞,*

*then lim sup*_{k}_{→∞}*ψ(x*^{k}*, y** ^{k}*)

*→ ∞ for ψ = ψ*YF

*, ψ*

_{LT1}

*, ψ*

_{LT2}

*, bψ*

_{YF}

*, bψ*

_{LT1}

*and bψ*

_{LT2}

*. If*

*{(x*

^{k}*, y*

*)*

^{k}*}*

*satisﬁes (i) or (ii), then lim sup*

_{k}

_{→∞}*ψ(x*

^{k}*, y*

*)*

^{k}*→ ∞ with ψ = ψ*NR

*, ψ*

_{MS}

*, ψ*

*τ*

*.*

The condition (ii) of Lemma 3.5 implies the condition (iii) since, when *{λ*1*(x** ^{k}*)

*} and*

*{λ*1

*(y*

*)*

^{k}*} are bounded below and λ*2

*(x*

^{k}*), λ*

_{2}

*(y*

*)*

^{k}*→ +∞, there must exist a vector d ∈ IR*

^{n}*such that x*

^{k}*− d ∈ K*

^{n}*and y*

^{k}*− d ∈ K*

*, which along with*

^{n}

_{∥x}

^{x}

^{k}

_{k}

_{∥}*◦*

_{∥y}

^{y}

^{k}*k*

*∥*9 0 yields that

*⟨x*^{k}*,y*^{k}*⟩*

*∥x*^{k}*∥∥y*^{k}*∥* *→ c > 0 (taking a subsequence if necessary), and lim sup**k**→∞**⟨x*^{k}*, y*^{k}*⟩ = +∞ then*
*follows. Hence, ψ*_{LT} and its variant b*ψ*_{LT} *have faster growth than ψ**τ* *and ψ*_{MS}.

**4 Merit Function Approach and Applications**

This section is devoted to the merit function methods for the generalized SOCCP (1.3),
*which yields a solution of (1.3) by solving an unconstrained minimization (1.8) with ψ being*
one of the merit functions introduced in last section. Throughout this section, we assume
that*K has the Cartesian structure of (1.2), and for any ζ ∈ IR** ^{n}*, write

*∇**x**ψ(F (ζ), G(ζ))* =

(*∇**x*1*ψ(F*1*(ζ), G*1*(ζ)), . . . ,∇**x**m**ψ(F**m**(ζ), G**m**(ζ))*
)

*,*

*∇**y**ψ(F (ζ), G(ζ))* =

(*∇**y*1*ψ(F*_{1}*(ζ), G*_{1}*(ζ)), . . . ,∇**y**m**ψ(F*_{m}*(ζ), G*_{m}*(ζ))*
)

*.*

When applying eﬀective gradient-type methods for the problem (1.8), we expect only a stationary point due to the nonconvexity of merit functions. Thus, it is necessary to know what conditions can guarantee every stationary point of Ψ to be a solution of (1.3). The following proposition provides a suitable condition for the ﬁrst three classes of functions.

**Proposition 4.1. Let Ψ be given by (1.8) with ψ being one of the previous merit functions.**

**(a) When ψ = ψ***τ**, ψ*_{YF} *and ψ*_{LT2}*, every stationary point of Ψ is a solution of (1.3) if∇F (ζ)*
*and−∇G(ζ) are Cartesian column monotone for any ζ ∈ IR*^{n}*.*

**(b) When ψ = ψ**_{MS} *or ψ*_{LT1}*, every stationary point of Ψ is a solution of (1.3) if∇F (ζ) and*

*−∇G(ζ) are Cartesian strictly column monotone for any ζ ∈ IR*^{n}*.*

*Proof. Since F and G are continuously diﬀerentiable, by Lemmas 3.1–3.3(b), the function*
Ψ is continuously diﬀerentiable with

*∇Ψ(ζ) = ∇F (ζ)∇**x**ψ(F (ζ), G(ζ)) +∇G(ζ)∇**y**ψ(F (ζ), G(ζ)).* (4.1)
*Let ζ∈ IR** ^{n}* be an arbitrary but ﬁxed stationary point of the function Ψ. Then,

*∇F (ζ)∇**x**ψ(F (ζ), G(ζ)) +∇G(ζ)∇**y**ψ(F (ζ), G(ζ)) = 0.* (4.2)
*Suppose that ζ is not a solution of (1.3). When ψ = ψ**τ**, ψ*_{YF} *and ψ*_{LT2}, we must have

*∇**x**ψ(F (ζ), G(ζ))* *̸= 0 and ∇**y**ψ(F (ζ), G(ζ))̸= 0 by Lemma 3.2– 3.3(f). Since ∇F (ζ) and*

*−∇G(ζ) are Cartesian column monotone, equality (4.2) implies that there exists an index*
*ν∈ {1, . . . , m} such that ∇**x**ν**ψ(F*_{ν}*(ζ), G*_{ν}*(ζ))̸= 0 and*

*⟨∇**x**ν**ψ(F*_{ν}*(ζ), G*_{ν}*(ζ)),∇**y**ν**ψ(F*_{ν}*(ζ), G*_{ν}*(ζ))⟩ ≤ 0.*

*Along with Lemma 3.2–3.3(e), we have ψ(F*_{ν}*(ζ), G*_{ν}*(ζ)) = 0. This, by Lemma 3.2–3.3(f),*
implies*∇**x**ν**ψ(F*_{ν}*(ζ), G*_{ν}*(ζ)) = 0, and then we get a contradiction. When ψ = ψ*_{MS} *or ψ*_{LT1},
by Lemma 3.1 and 3.3(f) we have (*∇**x**ψ(F (ζ), G(ζ)),∇**y**ψ(F (ζ), G(ζ))̸= (0, 0). Since ∇F (ζ)*
and*−∇G(ζ) are Cartesian strictly column monotone, (4.2) implies that there exists an index*
*ν∈ {1, . . . , m} such that ∇**x*_{ν}*ψ(F**ν**(ζ), G**ν**(ζ))̸= 0 and*

*⟨∇**x*_{ν}*ψ(F**ν**(ζ), G**ν**(ζ)),∇**y*_{ν}*ψ(F**ν**(ζ), G**ν**(ζ))⟩ < 0,*
which is impossible by Lemma 3.1 and 3.3(e). The proof is completed.

When *∇G(ζ) is invertible, since the Cartesian (strict) column monotonicity of ∇F (ζ)*
and*−∇G(ζ) is equivalent to the Cartesian (P ) P*0-property of*∇G(ζ)*^{−1}*∇F (ζ), Prop. 4.1*
*extends the results of [10, Prop. 3] and [12, Prop. 3.3] for ψ*_{YF} *and ψ*_{LT2}, respectively, as
*well as recovers the result of [44, Prop. 5.1]. When G is an identity mapping, in view of*
Lemmas 3.1– 3.3(e), using the same arguments as in [33, Theorem 5.3] can prove that every
regular stationary point of Ψ is a solution of (1.3). From [33], we know that the regularity is
*weaker than the Cartesian P -property of∇F , but it is not clear whether it is weaker than*
*the Cartesian P*0-property of*∇F .*

The property that *⟨∇**x**ψ(x, y),∇**y**ψ(x, y)⟩ ≥ 0 for all x, y ∈ IR** ^{n}* plays a crucial role in
the proof of Prop. 4.1. For the variant of LT functions bΨ

_{LT}, since the desirable property is not established, we can not provide suitable stationary point conditions for bΨ

_{LT}.

When solving the minimization problem (1.8), to guarantee that the iterative sequence
generated has a limit point, it is necessary to require that Ψ has bounded level sets, which is
implied by the coerciveness of Ψ, i.e., lim sup_{∥ζ∥→∞}*Ψ(ζ) = +∞. The following proposition*
provides the weakest coerciveness conditions for the previous merit functions.

**Proposition 4.2. Suppose the mappings F and G satisfy one of the following conditions:**

**(C.1) F and G have the joint uniform Jordan P -property and the linear growth.**