*Optimization Methods & Software*
Vol. 26, No. 1, February 2011, 1–22

**A least-square semismooth Newton method for the second-order** **cone complementarity problem**

Shaohua Pan^{a}and Jein-Shan Chen^{b}*

*a**Department of Mathematics, South China University of Technology, Guangzhou 510640,*
*People’s Republic of China;*^{b}*Department of Mathematics, National Taiwan Normal University,*

*Taipei 11677, Taiwan, Republic of China*

*(Received 11 March 2008; final version received 30 June 2009 )*

We present a nonlinear least-square formulation for the second-order cone complementarity problem
based on the Fischer–Burmeister (FB) function and the plus function. This formulation has two-fold
advantages. First, the operator involved in the over-determined system of equations inherits the favourable
properties of the FB function for local convergence, for example, the (strong) semi-smoothness; second,
the natural merit function of the over-determined system of equations share all the nice features of the
*class of merit functions f*YF*studied in [J.-S. Chen and P. Tseng, An unconstrained smooth minimization*
*reformulation of the second-order cone complementarity problem, Math. Program. 104 (2005), pp. 293–*

327] for global convergence. We propose a semi-smooth Levenberg–Marquardt method to solve the arising over-determined system of equations, and establish the global and local convergence results. Among others, the superlinear (quadratic) rate of convergence is obtained under strict complementarity of the solution and a local error bound assumption, respectively. Numerical results verify the advantages of the least-square reformulation for difficult problems.

**Keywords: second-order cone complementarity problem; Fischer–Burmeister function; semi-smooth;**

Levenberg–Marquardt method

**1.** **Introduction**

We consider the second-order cone complementarity problem (SOCCP), which is to find a vector
*ζ* ∈ R* ^{n}*such that

*F (ζ )∈ K, G(ζ) ∈ K, F (ζ ), G(ζ ) = 0,* (1)

where*·, · denotes the Euclidean inner product, F : R** ^{n}*→ R

^{n}*and G*: R

*→ R*

^{n}*are assumed to be continuously differentiable throughout this paper, and*

^{n}*K is the Cartesian product of second-*order cones (SOCs), also called Lorentz cones [11], i.e.

*K = K*^{n}^{1}*× K*^{n}^{2}*× · · · × K*^{n}^{q}*,* (2)

*with n*_{1}*+ · · · + n**q* *= n and K*^{n}^{i}*:= {(x**i1**, x*_{i2}*)*∈ R × R^{n}^{i}^{−1}*|x**i1**≥ ||x**i2*||}. In this paper, cor-
responding to the Cartesian structure of the cone *K, we will write F = (F*1*, . . . , F*_{q}*)* and
*G= (G*1*, . . . , G*_{q}*)with F*_{i}*, G** _{i}*: R

*→ R*

^{n}

^{n}*.*

^{i}*Corresponding author. Email: jschen@math.ntnu.edu.tw

*ISSN 1055-6788 print/ISSN 1029-4937 online*

© 2011 Taylor & Francis

*DOI: 10.1080/10556780903180366*
*http://www.informaworld.com*

*An important special case of problem (1) corresponds to G(ζ )≡ ζ, i.e.*

*F (ζ )∈ K, ζ ∈ K, F (ζ), ζ = 0.* (3)

This is a natural extension of the non-linear complementarity problem (NCP) [9,12], where
*K = R*^{n}_{+}, the non-negative orthant inR^{n}*, corresponds to n*_{1}*= · · · = n**q* *= 1 and q = n. Another*
important special case of (1) corresponds to the Karush–Kuhn–Tucker (KKT) conditions of the
convex second-order cone program (SOCP):

minimize *g(x)*

subject to *Ax* *= b, x ∈ K,* (4)

*where g*: R^{n}*→ R is a twice continuously differentiable convex function, A ∈ R*^{m}* ^{×n}*has full row

*rank, and b*∈ R

*. The KKT conditions of (4) can be rewritten as (1) with*

^{m}*F (ζ ):= ˆx + (I − A*^{T}*(AA*^{T}*)*^{−1}*A)ζ,* *G(ζ ):= ∇g(F (ζ)) − A*^{T}*(AA*^{T}*)*^{−1}*Aζ,* (5)
where*ˆx ∈ R*^{n}*satisfies Ax= b; see [5] for details. The convex SOCP arises in many applications*
from engineering design, finance, and robust optimization; see [1,20] and references therein.

*Motivated by Kanno et al. [17] where the three-dimensional quasi-static frictional contact was*
directly reformulated as a linear SOC complementarity problem, we believe that, besides these
applications, the SOCCP (1) will be found to have some applications in engineering which cannot
reduce to SOCPs.

Various methods have been proposed for solving convex SOCPs and SOCCPs, including the
interior point methods [1,2,20,21,28,30], the smoothing Newton methods [6,14,16], the merit
function method [5] and the semi-smooth Newton method [19]. Among others, the last three kinds
of methods are all based on an SOC complementarity function or a merit function. Specifically,
*φ*: R^{n}* ^{i}* × R

^{n}*→ R*

^{i}

^{n}

^{i}*is called an SOC complementarity function associated withK*

^{n}*if*

^{i}*φ (x*_{i}*, y*_{i}*)= 0 ⇐⇒ x ∈ K*^{n}^{i}*,* *y* *∈ K*^{n}^{i}*,* *x**i**, y*_{i}* = 0.* (6)
*Clearly, when n**i* = 1, an SOC complementarity function becomes an NCP function. A popular
*choice of φ is the Fischer–Burmeister (FB) function defined by*

*φ*_{FB}*(x*_{i}*, y*_{i}*):= (x**i*^{2}*+ y**i*^{2}*)*^{1/2}*− (x**i**+ y**i**)* *∀x**i**, y** _{i}* ∈ R

^{n}

^{i}*,*(7)

*where x*

_{i}^{2}

*= x*

*i*

*◦ x*

*i*

*means the Jordan product of x*

*i*with itself (the definition of Jordan product is

*given in Section 2), and (x*

*i*

*)*

*means a vector such that*

^{1/2}*[(x*

*i*

*)*

*]*

^{1/2}^{2}

*= x*

*i*

*. The function φ*FBis well- defined and satisfies (6); see [14]. Hence, the SOCCP (1) can be reformulated as the following non-smooth system

_{FB}*(ζ )*:=

⎛

⎜⎝

*φ*_{FB}*(F*_{1}*(ζ ), G*_{1}*(ζ ))*
*...*

*φ*FB*(F**q**(ζ ), G**q**(ζ ))*

⎞

⎟*⎠ = 0.* (8)

*The system (8) induces a natural merit function *FB: R* ^{n}*→ R+for (1), given by

_{FB}*(ζ )*:=1

2*||*FB*(ζ )*||^{2}=

*q*
*i*=1

*ψ*_{FB}*(F*_{i}*(ζ ), G*_{i}*(ζ ))* (9)
with

*ψ*FB*(x**i**, y**i**)*:=1

2*||φ*FB*(x**i**, y**i**)*||^{2}*.* (10)
*The function ψ*_{FB}was studied in [5] and used to develop a merit function method. Recently, we
*analysed in [22] that, to guarantee the boundedness of the level sets of the FB merit function *_{FB},

*it requires that the mapping F at least has the uniform Cartesian P -property. This means that φ*_{FB}
has some limitations in handling monotone SOCCPs.

Motivated by Kanzow and Petra [18] for the NCPs, we present a new reformulation for (1) in
*this paper to overcome the disadvantage of φ*_{FB}*. Let φ*_{0}: R^{n}* ^{i}*× R

^{n}*→ R*

^{i}_{+}be given by

*φ*_{0}*(x*_{i}*, y*_{i}*):= max{0, x**i**, y*_{i}*},* (11)
*and define the operator *: R* ^{n}*→ R

^{n}*as*

^{+q}*(ζ )*:=

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝

*ρ*_{1}*φ*_{FB}*(F*_{1}*(ζ ), G*_{1}*(ζ ))*
*...*

*ρ*1*φ*FB*(F**q**(ζ ), G**q**(ζ ))*
*ρ*2*φ*0*(F*1*(ζ ), G*1*(ζ ))*

*...*

*ρ*2*φ*0*(F**q**(ζ ), G**q**(ζ ))*

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠

*,* (12)

*where ρ*1*, ρ*2are arbitrary but fixed constants from (0,1) used as the weights for the first type of
*terms and the second one, respectively. In other words, we define by appending q components*
*to the mapping *_{FB}. These additional components, as will be shown later, play a crucial role in
*overcoming the disadvantage of *_{FB}mentioned above. Noting that

*ζ*^{∗}*solves (ζ )= 0 ⇐⇒ ζ*^{∗}*solves (1),* (13)
we have the following nonlinear least-square reformulation for the SOCCP (1)

min

*ζ*∈R^{n}*(ζ )*:=1

2*||(ζ)||*^{2}=

*q*
*i*=1

*ψ (F**i**(ζ ), G**i**(ζ )),* (14)

where

*ψ (x*_{i}*, y*_{i}*):= ρ*1^{2}*ψ*_{FB}*(x*_{i}*, y*_{i}*)*+1

2*ρ*_{2}^{2}*φ*_{0}*(x*_{i}*, y*_{i}*)*^{2}*.* (15)
*The reformulation has the following advantages: on the one hand, belongs to the class of merit*
*functions f*_{YF}*introduced in [5], which will be shown to have more desirable properties than *_{FB};
*on the other hand, inherits the semi-smoothness of *FB even strong semi-smoothness under
some conditions. By this, we propose a semi-smooth Levenberg–Marquardt type method for solv-
ing (14), and establish the superlinear (quadratic) rate of convergence under strict complementarity
and a local error bound assumption of the solution, respectively.

*Throughout this paper, I represents an identity matrix of suitable dimension,*|| · || denotes the
Euclidean norm,R^{n}*denotes the space of n-dimensional real column vectors, and*R^{n}^{1}× · · · × R^{n}* ^{q}*
is identified withR

^{n}^{1}

^{+···+n}

^{q}*. Thus, (x*1

*, . . . , x*

*q*

*)*∈ R

^{n}^{1}× · · · × R

^{n}*is viewed as a column vector inR*

^{q}

^{n}^{1}

^{+···+n}

^{q}*. For a differentiable mapping F*: R

*→ R*

^{n}*,*

^{m}*∇F (x) denotes the transpose of the*

*Jacobian F*

^{}

*(x). For a (not necessarily symmetric) square matrix A*∈ R

^{n}

^{×n}*, we write A*0

*(respectively, A 0) to mean A is positive semi-definite (respectively, positive definite). Given a*

*finite number of matrices Q*

_{1}

*, . . . , Q*

*, we denote the block diagonal matrix with these matrices as*

_{n}*block diagonals by diag (Q*

_{1}

*, . . . , Q*

_{n}*). IfJ and B are index sets such that J , B ⊆ {1, 2, . . . , q},*

*we denote P*

_{J B}*by the block matrix consisting of the sub-matrices P*

*∈ R*

_{j k}

^{n}

^{j}

^{×n}

^{k}*of P with j*

*∈ J*

*and k∈ B. We denote int(K*

^{n}*), bd(K*

^{n}*)*and bd

^{+}

*(K*

^{n}*)*by the interior, the boundary of

*K*

*, and the boundary of*

^{n}*K*

*excluding the origin, respectively.*

^{n}**2.** **Preliminaries**

This section recalls some background materials that will be used in the sequel. We start with the
*definition of the Jordan product. For any x= (x*1*, x*2*), y* *= (y*1*, y*2*)*∈ R × R^{n}^{−1}, we define their
*Jordan product [11] associated withK** ^{n}*as

*x◦ y := (x, y, x*1*y*2*+ y*1*x*2*).* (16)
The Jordan product ‘◦’, unlike the scalar or matrix multiplication, is not associative, which is a
main source on complication in the analysis of SOCCPs. The identity element under this product is
*e:= (1, 0, . . . , 0)*^{T}∈ R^{n}*. Given a vector x= (x*1*, x*2*)*∈ R × R^{n}^{−1}*, let L**x* :=

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

which
can be viewed as a linear mapping fromR* ^{n}* toR

^{n}*. It is easy to verify that L*

*x*

*y= x ◦ y and*

*L*

*x*

*+y*

*= L*

*x*

*+ L*

*y*

*for any x, y*∈ R

^{n}*. Furthermore, x∈ K*

^{n}*if and only if L*

*x*

*0, and x ∈ int(K*

^{n}*)*

*if and only if L*

*x*

*0. When x ∈ int(K*

^{n}*), the inverse of L*

*x*is given by

*L*^{−1}* _{x}* = 1

*det(x)*

⎡

⎣ *x*1 *−x*2^{T}

*−x*2

*det(x)*
*x*1

*I* + 1
*x*1

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

⎤

*⎦ ,* (17)

*where det(x) denotes the determinant of x defined by det(x):= x*1^{2}*− ||x*2||^{2}.

*From [11,14], we recall that each x= (x*1*, x*2*)*∈ R × R^{n}^{−1} admits a spectral factorization
associated with*K** ^{n}*, of the form

*x* *= λ*1*(x)· u*^{(1)}*x* *+ λ*2*(x)· u*^{(2)}*x* *,* (18)
*where λ*_{i}*(x)and u*^{(i)}_{x}*for i= 1, 2 are the spectral values and the associated spectral vectors of x,*
respectively, defined by

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

2*(1, (−1)*^{i}*¯x*2*),*

with*¯x*2 *= x*2*/||x*2*|| if x*2= 0 and otherwise being any vector in R^{n}^{−1}with*|| ¯x*2*|| = 1. If x*2 = 0, the
*factorization is unique. The spectral factorizations of x, x*^{2}*as well as x** ^{1/2}*have various interesting

*properties [14]; for example, x∈ K*

*if and only if 0*

^{n}*≤ λ*1

*(x)≤ λ*2

*(x), and x*

*∈ int(K*

^{n}*)*if and

*only if 0 < λ*

_{1}

*(x)≤ λ*2

*(x).*

*We next recall from Chen and Qi, [4] the definition of Cartesian P -property for a matrix and a*
nonlinear transformation.

Definition*2.1 A matrix M*∈ R^{n}^{×n}*is said to have*

*(a) the Cartesian P -property if for any non-zero ζ* *= (ζ*1*, . . . , ζ**q**)*∈ R^{n}*with ζ**i*∈ R^{n}^{i}*, there exists*
*an index ν∈ {1, 2, . . . , q} such that ζ**ν**, (Mζ )**ν** > 0;*

*(b) the Cartesian P*_{0}*-property if for any non-zero ζ* *= (ζ*1*, . . . , ζ*_{q}*)*∈ R^{n}*with ζ** _{i}* ∈ R

^{n}

^{i}*, there*

*exists an index ν∈ {1, 2, . . . , q} such that*

*ζ**ν* *= 0 and ζ**ν**, (Mζ )**ν** ≥ 0.*

Definition*2.2 The mappings F* *=(F*1*, . . . , F**q**)and G= (G*1*, . . . , G**q**)are said to have*
*(a) the jointly uniform Cartesian P -property if there exists a constant ρ > 0 such that, for any*

*ζ, ξ* ∈ R^{n}*, there exists ν∈ {1, 2, . . . , q} such that*

*F**ν**(ζ )− F**ν**(ξ ), G*_{ν}*(ζ )− G**ν**(ξ ) ≥ ρζ − ξ*^{2}*,*

*(b) the joint Cartesian P -property if for any ζ, ξ* ∈ R^{n}*with G(ζ )= G(ξ), there exists ν ∈*
*{1, 2, . . . , q} such that*

*F**ν**(ζ )− F**ν**(ξ ), G**ν**(ζ )− G**ν**(ξ ) > 0,*

*(c) the joint Cartesian P*0*-property if for any ζ, ξ* ∈ R^{n}*with G(ζ )= G(ξ), there exists ν ∈*
*{1, 2, . . . , q} such that*

*G**ν**(ζ )= G**ν**(ξ )* and *F**ν**(ζ )− F**ν**(ξ ), G**ν**(ζ )− G**ν**(ξ ) ≥ 0,*

*When G(ζ )≡ ζ, Definition 2.2 gives the Cartesian P -properties of F . Obviously, the uniform*
*Cartesian P -property⇒ the Cartesian P -property ⇒ the Cartesian P*0-property. Also, a contin-
*uously differentiable mapping has the Cartesian P*0-property if and only if its Jacobian matrix at
*every point has the Cartesian P*_{0}-property, and if the Jacobian matrix of a continuously differen-
*tiable mapping has the Cartesian P -property at every point, then the mapping has the Cartesian*
*P-property. From Definition 2.1, the positive semi-definitness implies the Cartesian P*_{0}-property.

*Given a mapping H* : R* ^{n}*→ R

^{m}*, if H is locally Lipschitz continuous, then*

*∂*_{B}*H (ζ ):= {V ∈ R*^{m}^{×n}*| ∃{ζ*^{k}*} ⊆ D**H* *: ζ*^{k}*→ ζ, H*^{}*(ζ*^{k}*)→ V }*

*is non-empty and called the B-subdifferential of H at ζ , where D** _{H}* ⊆ R

*denotes the set of points*

^{n}*at which H is differentiable. The convex hull ∂H (ζ ):= conv∂*B

*H (ζ )*is the generalized Jacobian

*of H at ζ in the sense of Clarke [4]. For the concepts of (strongly) semi-smooth functions, please*refer to [24,25] for details.

**3.** **Properties of the operator **

*To study the favourable properties of the operator , we first give two technical lemmas to*
*summarize some properties of φ*0*and φ*FB, respectively. The results of the first lemma are direct,
and the results of the second lemma can be found in [14, Prop. 4.2], [5, Prop. 2], [27, Cor. 3.3]

and [22, Prop. 3.1].

Lemma*3.1 Let φ*0: R* ^{n}*× R

*→ R+*

^{n}*be defined as in Equation (11). Then,*

*(a) the square of φ*0

*is continuously differentiable everywhere*;

*(b) φ*0*is strongly semi-smooth everywhere on*R* ^{n}*× R

*;*

^{n}*(c) the B-subdifferential ∂*B*φ*0*(x, y)of φ*0*at any (x, y)*∈ R* ^{n}*× R

^{n}*is given by*

*∂*B*φ*0*(x, y)= [∂*B*(x*^{T}*y)*_{+}*y*^{T} *∂*B*(x*^{T}*y)*_{+}*x*^{T}*],*
*where*

*∂*B*(x*^{T}*y)*_{+}=

⎧⎪

⎨

⎪⎩

{1} *if x*^{T}*y >0,*
*{1, 0} if x*^{T}*y* *= 0,*
{0} *if x*^{T}*y <0.*

Lemma*3.2 Let φ*_{FB}: R* ^{n}*× R

*→ R*

^{n}

^{n}*be defined as in Equation (7). Then, for any given x*=

*(x*

_{1}

*, x*

_{2}

*), y*

*= (y*1

*, y*

_{2}

*)*∈ R × R

^{n}^{−1}

*,the following results hold.*

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

*(b) φ*_{FB}*is strongly semismooth at (x, y).*

*(c) Each element[U**x**− I U**y**− I] of ∂*B*φ*FB*(x, y)has the following representation:*

*(c.1) If x*^{2}*+ y*^{2}*∈ int(K*^{n}*),then U*_{x}*= L*^{−1}* _{(x}*2

*+y*

^{2}

*)*

^{1/2}*L*

_{x}*and U*

_{y}*= L*

^{−1}

*2*

_{(x}*+y*

^{2}

*)*

^{1/2}*L*

_{y}*.*

*(c.2) If x*

^{2}

*+ y*

^{2}∈ bd

^{+}

*(K*

^{n}*),then[U*

*x*

*, U*

*y*

*] belongs to the set*

1 2√

*2w*_{1}

1 *¯w*^{T}_{2}

*¯w*2 *4I− 3 ¯w*2*¯w*2^{T}

*L**x*+1

2

1

*− ¯w*2

*u*^{T}*,*

1 2√

*2w*_{1}

1 *¯w*^{T}2

*¯w*2 *4I− 3 ¯w*2*¯w*2^{T}

*L**y*+1

2

1

*− ¯w*2

*v*^{T}

*u= (u*1*, u*2*),*

*v= (v*1*, v*_{2}*)*∈ R × R^{n}^{−1}satisfy*|u*1*| ≤ u*2* ≤ 1, |v*1*| ≤ v*2 ≤ 1

*,*

*where w= (w*1*, w*2*)= x*^{2}*+ y*^{2}*and w*2 *= w*2*/w*2*.*

*(c.3) If (x, y)= (0, 0), [U**x**, U**y**] belongs to {[L**ˆu**, L*_{ˆv}*]| ˆu*^{2}*+ ˆv*^{2}*= 1} or*

1 2

1

*¯w*2

*ξ*^{T}+1

2

1

*− ¯w*2

*u*^{T}+ 2

0 0

0 *(I− ¯w*2*¯w*^{T}2*)*

*L**s**,*

1 2

1

*¯w*2

*η*^{T}+1

2

1

*− ¯w*2

*v*^{T}+ 2

0 0

0 *(I− ¯w*2*¯w*2^{T}*)*

*L**ω*

*|| ¯w*2satisfies

* ¯w*2* = 1 and u = (u*1*, u*2*), v= (v*1*, v*2*), ξ* *= (ξ*1*, ξ*2*), η= (η*1*, η*2*),*
*s= (s*1*, s*2*), ω= (ω*1*, ω*2*)*∈ R × R^{n}^{−1}satisfy*|ξ*1*| ≤ ξ*2* ≤ 1,*

*|u*1*| ≤ u*2* ≤ 1, |η*1*| ≤ η*2* ≤ 1, |v*1*| ≤ v*2* ≤ 1, s*^{2}*+ ω*^{2} ≤ 1
2

*.*

*(d) The squared norm of φ*_{FB}*,i.e. *_{FB}*, is continuously differentiable at (x, y).*

*From Lemma 3.1 (b) and Lemma 3.2 (b), we obtain the semi-smoothness of .*

Proposition*3.3 The operator *: R* ^{n}*→ R

^{n}

^{+q}*defined by (12) is semi-smooth. If, in addition,*

*F*

^{}

*and G*

^{}

*are Lipschitz continuous, then is strongly semi-smooth.*

*Proof* *Let **i* *denote the ith component function of for i= 1, 2, . . . , 2q, i.e., **i**(ζ )*=
*φ*FB*(F**i**(ζ ), G**i**(ζ ))* *for i= 1, 2, . . . , q and **i**(ζ )= φ*0*(F**i**(ζ ), G**i**(ζ ))* *for i= q + 1, . . . , 2q.*

*Then, the mapping is (strongly) semi-smooth if every ** _{i}* is (strongly) semi-smooth. Note that

* _{i}* : R

*→ R*

^{n}

^{n}

^{i}*for i= 1, 2, . . . , q is the composite of the strongly semi-smooth function φ*FB

*and the smooth function ζ* *→ (F**i**(ζ ), G*_{i}*(ζ )), whereas *_{q}* _{+i}* : R

*→ R is the composite of the*

^{n}*strongly semi-smooth function φ*

_{0}

*and the function ζ*

*→ (F*

*i*

*(ζ ), G*

_{i}*(ζ )). Moreover, when F*

^{}and

*G*

^{}

*are Lipschitz continuous, ζ*

*→ (F*

*i*

*(ζ ), G*

_{i}*(ζ ))*is strongly semi-smooth. By [13, Theorem 19],

*we have that every component function of is semi-smooth, and strongly semi-smooth if, in*

*addition, F*^{}*and G*^{}are Lipschitz continuous.

*Next, we present an estimation for the B-subdifferential of at any ζ* ∈ R* ^{n}*.

Proposition*3.4 Let *: R* ^{n}*→ R

^{n}

^{+q}*be given by (12). Then, for any ζ*∈ R

^{n}*,*

*∂*_{B}*(ζ )*^{T}*⊆ ∇F (ζ)[ρ*1*(A(ζ )− I) ρ*2*C(ζ )] + ∇G(ζ )[ρ*1*(B(ζ )− I) ρ*2*D(ζ )*]
*where C(ζ )= diag(C*1*(ζ ), . . . , C**q**(ζ ))and D(ζ )= diag(D*1*(ζ ), . . . , D**q**(ζ ))with*

*C**i**(ζ )∈ G**i**(ζ )∂*B*(F**i**(ζ )*^{T}*G**i**(ζ ))*_{+} *and D**i**(ζ )∈ F**i**(ζ )∂*B*(F**i**(ζ )*^{T}*G**i**(ζ ))*_{+}*,*

*and A(ζ )= diag(A*1*(ζ ), . . . , A*_{q}*(ζ ))and B(ζ )= diag(B*1*(ζ ), . . . , B*_{q}*(ζ ))with the block diago-*
*nals A*_{i}*(ζ ), B*_{i}*(ζ )*∈ R^{n}^{i}^{×n}^{i}*having the following representation:*

*(a) If F**i**(ζ )*^{2}*+ G**i**(ζ )*^{2}*∈ int(K*^{n}^{i}*),then*
*A**i**(ζ )= L**F*_{i}*(ζ )**L*^{−1}_{[F}

*i**(ζ )*^{2}*+G**i**(ζ )*^{2}]^{1/2}*and B**i**(ζ )= L**G*_{i}*(ζ )**L*^{−1}_{[F}

*i**(ζ )*^{2}*+G**i**(ζ )*^{2}]^{1/2}*.*
*(b) If F**i**(ζ )*^{2}*+ G**i**(ζ )*^{2}∈ bd^{+}*(K*^{n}^{i}*), then[A**i**(ζ ), G**i**(ζ )] belongs to the set*

1

2√

*2w**i1**(ζ )L**F*_{i}*(ζ )*

1 *¯w**i2**(ζ )*^{T}

*¯w**i2**(ζ )* *4I− 3 ¯w**i2**(ζ )¯w**i2**(ζ )*^{T}

+1

2*u**i**(1,− ¯w**i2**(ζ )*^{T}*),*
1

2√

*2w**i1**(ζ )L**G*_{i}*(ζ )*

1 *¯w**i2**(ζ )*^{T}

*¯w**i2**(ζ )* *4I− 3 ¯w**i2**(ζ )¯w**i2**(ζ )*^{T}

+1

2*v**i**(1,− ¯w**i2**(ζ )*^{T}*)*
*u**i**= (u**i1**, u**i2**), v**i* *= (v**i1**, v**i2**)*satisfy*|u**i1**| ≤ u**i2** ≤ 1, |v**i1**| ≤ v**i2* ≤ 1

*,*
*where w(ζ )= (w**i1**(ζ ), w**i2**(ζ ))= F**i**(ζ )*^{2}*+ G**i**(ζ )*^{2}and *¯w**i2**(ζ )= w**i2**(ζ )/w**i2**(ζ ).*

*(c) If (F**i**(ζ ), G**i**(ζ ))= (0, 0), [A**i**(ζ ), B**i**(ζ )] ∈ {[L**ˆu**i**, L*_{ˆv}_{i}*] | ˆu**i*^{2}*+ˆv**i*^{2} *= 1} or*

1

2*ξ**i**(1,* *¯w**i2*^{T}*)*−1

2*u**i**(−1, ¯w**i2*^{T}*)+ 2L**s*_{i}

0 0

0 *(I− ¯w**i2**¯w*^{T}_{i2}*)*

*,*
1

2*η*_{i}*(1,* *¯w*^{T}_{i2}*)*−1

2*v*_{i}*(−1, ¯w*^{T}_{i2}*)+ 2L**ω**i*

0 0

0 *(I− ¯w**i2**¯w*_{i2}^{T}*)*

*| ¯w**i2*∈ R^{n}^{i}^{−1}
satisfies* ¯w**i2** = 1 and ξ**i* *= (ξ**i1**, ξ**i2**), u**i* *= (u**i1**, u**i2**), η**i* *= (η**i1**, η**i2**),*

*v**i* *= (v**i1**, v**i2**), s**i**= (s**i1**, s**i2**), ω**i* *= (ω**i1**, ω**i2**)*satisfy*|ξ**i1**| ≤ ξ**i2** ≤ 1,*

*|u**i1**| ≤ u**i2** ≤ 1, |η**i1**| ≤ η**i2** ≤ 1, |v**i1**| ≤ v**i2** ≤ 1, s**i*^{2}*+ ω**i*^{2}≤ 1
2

*.*
*Proof* *Let *_{i}*be the ith component function of , i.e. *_{i}*(ζ )= φ*FB*(F*_{i}*(ζ ), G*_{i}*(ζ ))* and

_{q}_{+i}*(ζ )= φ*0*(F*_{i}*(ζ ), G*_{i}*(ζ ))for i= 1, . . . , q. By the concept of B-subdifferential,*

*∂*B*(ζ )*^{T} *⊆ ∂*B1*(ζ )*^{T}*× ∂*B2*(ζ )*^{T}*× · · · × ∂*B*2q**(ζ )*^{T}*,* (19)
*where the latter means the set of all matrices whose (n**i*−1*+ 1)th to n**i*th columns belong to

*∂*B*i**(ζ )*^{T}*with n*0 *= 0, and (n + i)th column belongs to ∂*B*q**+i**(ζ )*^{T}. Note that

*∂*_{B}_{i}*(ζ )*^{T} *⊆ ρ*1*[∇F**i**(ζ )* *∇G**i**(ζ )] ∂*B*φ*_{FB}*(F*_{i}*(ζ ), G*_{i}*(ζ ))*^{T}*,*

*∂*B*q**+i**(ζ )*^{T} *⊆ ρ*2*[∇F**i**(ζ )* *∇G**i**(ζ )] ∂*B*φ*0*(F**i**(ζ ), G**i**(ζ ))*^{T}*.* (20)
Also, by Lemmas 3.1(c) and 3.2(c), each element in *∂*B*φ*FB*(F**i**(ζ ), G**i**(ζ ))*^{T} and

*∂*_{B}*φ*_{0}*(F*_{i}*(ζ ), G*_{i}*(ζ ))*^{T}has the form of

*A**i**(ζ )**−I*
*B*_{i}*(ζ )**−I*

and

*C**i**(ζ )*
*D*_{i}*(ζ )*

*, respectively, with A*_{i}*(ζ ), B*_{i}*(ζ )*and

*C**i**(ζ ), D**i**(ζ )for i= 1, . . . , q characterized as in the proposition. Combining with Equations (19)*

and (20), we obtain the desired result.

To prove the fast local convergence of non-smooth Levenberg–Marquardt methods, we need to
*know under what conditions every element H* *∈ ∂*B*(ζ*^{∗}*)has full rank n, where ζ*^{∗}is a solution
of the SOCCP (1). To the end, define the index sets

*I := {i ∈ {1, 2, . . . , q} | F**i**(ζ*^{∗}*)= 0, G**i**(ζ*^{∗}*)∈ int(K*^{n}^{i}*)},*

*B := {i ∈ {1, 2, . . . , q} | F**i**(ζ*^{∗}*)*∈ bd^{+}*(K*^{n}^{i}*), G**i**(ζ*^{∗}*)*∈ bd^{+}*(K*^{n}^{i}*)},*

*J := {i ∈ {1, 2, . . . , q} | F**i**(ζ*^{∗}*)∈ int(K*^{n}^{i}*), G**i**(ζ*^{∗}*)= 0}.* (21)
*If ζ*^{∗}*satisfies strict complementarity, i.e. F**i**(ζ*^{∗}*)+ G**i**(ζ*^{∗}*)∈ int(K*^{n}^{i}*)for all i, then{1, 2, . . . , q}*

can be partitioned as*I ∪ B ∪ J . Thus, if ∇G(ζ*^{∗}*)*is invertible, then by rearrangement the matrix
*P (ζ*^{∗}*)= ∇G(ζ*^{∗}*)*^{−1}*∇F (ζ*^{∗}*)*can be rewritten as

*P (ζ*^{∗}*)*=

⎛

⎝*P (ζ*^{∗}*)*_{II}*P (ζ*^{∗}*)*_{IB}*P (ζ*^{∗}*)*_{IJ}*P (ζ*^{∗}*)*_{BI}*P (ζ*^{∗}*)*_{BB}*P (ζ*^{∗}*)*_{BJ}*P (ζ*^{∗}*)*_{J I}*P (ζ*^{∗}*)*_{J B}*P (ζ*^{∗}*)*_{J J}

⎞

*⎠ .*

*Now we have the following results for the full rank of every element H* *∈ ∂*B*(ζ*^{∗}*).*

Theorem*3.5 Let ζ*^{∗}*be a strictly complementary solution of (1). Suppose that∇G(ζ*^{∗}*)is invert-*
*ible and let P (ζ*^{∗}*)= ∇G(ζ*^{∗}*)*^{−1}*∇F (ζ*^{∗}*). If P (ζ*^{∗}*)*_{II}*is non-singular and its Schur-complement*
*P (ζ*^{∗}*)*_{II}*:= P (ζ*^{∗}*)*_{BB}*− P (ζ*^{∗}*)*_{BI}*P (ζ*^{∗}*)*^{−1}_{II}*P (ζ*^{∗}*)*_{IB}*, in the matrix*

*P (ζ*^{∗}*)*_{II}*P (ζ*^{∗}*)*_{IB}*P (ζ*^{∗}*)*_{BI}*P (ζ*^{∗}*)*_{BB}

*has the*
*Cartesian P -property, then every element H in the B-subdifferential ∂*B*(ζ*^{∗}*)has full column*
*rank n.*

*Proof* *Let H* *∈ ∂*B*(ζ*^{∗}*). By Proposition 3.4, H*=

*ρ*_{1}*H*_{1}
*ρ*_{2}*H*_{2}

*with H*_{1}^{T}*from the set ∂*B1*(ζ*^{∗}*)*^{T}×

*· · · × ∂*B_{q}*(ζ*^{∗}*)*^{T}*. From Theorem 4.1 of [22], H*_{1}^{T}is non-singular under the given assumptions.

*This implies the desired result rank(H )= n.*

*The proof of Theorem 3.5 is based on the important property of the first block of H . Nevertheless,*
*when the first block H*_{1}*is singular, the second block H*_{2}may contribute something to guarantee
*that H has a full column rank n.*

To close this section, we give a technical lemma that will be used in Section 5.

Lemma*3.6 Let ζ*^{∗}*be a solution of (1) such that all elements in ∂*B*(ζ*^{∗}*)have full column rank.*

*Then, there exist constants ε > 0 and c > 0 such that(H*^{T}*H )*^{−1}* ≤ c for all ζ − ζ*^{∗}* < ε and*
*all H* *∈ ∂*B*(ζ ). Furthermore, for any given¯ν > 0, H*^{T}*H+ νI are uniformly positive definite*
*for all ν∈ [0, ¯ν] and H ∈ ∂*B*(ζ )withζ − ζ*^{∗}* < ε.*

*Proof* The proof is similar to [24, Lemma 2.6]. For completeness, we include it here. Suppose
that the claim of the lemma is not true. Then there exists a sequence*{ζ*^{k}*} converging to ζ*^{∗}and a
corresponding sequence of matrices*{H**k**} with H**k**∈ ∂*B*(ζ*^{k}*)for all k∈ IN such that either H**k*^{T}*H**k*

is singular or*(H*_{k}^{T}*H*_{k}*)*^{−1}* → +∞ on a subsequence. Noting that H*_{k}^{T}*H** _{k}* is symmetric positive
semi-definite, for the non-singular case, we have

*(H*

_{k}^{T}

*H*

_{k}*)*

^{−1}

*= 1/λ*min

*(H*

_{k}^{T}

*H*

_{k}*),*which implies that the condition

*(H*

_{k}^{T}

*H*

_{k}*)*

^{−1}

*→ +∞ is equivalent to λ*min

*(H*

_{k}^{T}

*H*

_{k}*)→ 0. Since ζ*

^{k}*→ ζ*

^{∗}and the

*mapping ζ*

*→ ∂*B

*(ζ )*is upper semi-continuous, it follows that the sequence

*{H*

*k*} is bounded, and

*hence it has a convergent subsequence. Let H*

_{∗}

*be a limit of such a sequence. Then λ*

_{min}

*(H*

_{∗}

^{T}

*H*

_{∗}

*)*= 0

*by the continuity of the minimum eigenvalue. This means that H*_{∗}^{T}*H*_{∗}is singular. However, from
*the fact that the mapping ζ* *→ ∂*B*(ζ )is closed, we have H*_{∗}*∈ ∂*B*(ζ*^{∗}*), which by the given*
*condition implies that H*_{∗}^{T}*H*_{∗}is non-singular. Thus, we obtain a contradiction, and the first part
*follows. By the result of the first part and the definition of matrix norm, there exist constants ε > 0*
*and c > 0 such that*

*[λ*min*(H*^{T}*H+ νI)]*^{−1}*= (H*^{T}*H+ νI)*^{−1}* ≤ c*
*for all ν∈ [0, ¯ν] and H ∈ ∂*B*(ζ )*with*ζ − ζ*^{∗}* < ε. This implies that*

*u*^{T}*(H*^{T}*H+ νI)u ≥ λ*min*(H*^{T}*H+ νI)u*^{2}≥ 1

*cu*^{2} *∀ u ∈ R*^{n}*.*

*Therefore, all the matrices H*^{T}*H+ νI are uniformly positive definite.*

**4.** **Properties of the merit function **

*This section is devoted to the favourable properties of defined by (14) and (15). To this end,*
*we need the following lemma which summarizes the properties of ψ .*

Lemma*4.1 Let ψ*: R* ^{n}*× R

*→ R*

^{n}_{+}

*be defined as in (15). Then, for any x, y*∈ R

^{n}*,*

*(a) ψ(x, y)= 0 ⇐⇒*FB

*(x, y)= 0 ⇐⇒ x ∈ K*

^{n}*, y∈ K*

^{n}*,*

*x, y = 0;*

*(b) ψ(x, y) is continuously differentiable;*

(c) *x, ∇**x**ψ (x, y) + y, ∇**y**ψ (x, y) ≥ 2ψ(x, y);*

*(d)* ∇*x**ψ (x, y),*∇*y**ψ (x, y) ≥ 0, and the equality holds if and only if ψ(x, y) = 0;*

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

*Proof* *Part (a) is direct by the definition of ψ , and part (b) is from Lemmas 3.1(a) and 3.2(d).*

*We next consider part (c). By the definition of ψ ,*

∇*x**ψ (x, y)= ρ*1^{2}∇*x**ψ*FB*(x, y)+ ρ*2^{2}*φ*0*(x, y)y,*

∇*y**ψ (x, y)= ρ*1^{2}∇*y**ψ*FB*(x, y)+ ρ*2^{2}*φ*0*(x, y)x.* (22)
*From Lemma 6 (a) of [5] and the definition of φ*_{0}*(x, y), it then follows that*

*x, ∇**x**ψ (x, y) + y, ∇**y**ψ (x, y)*

*= ρ*1^{2}*[x, ∇**x**ψ*FB*(x, y) + y, ∇**y**ψ*FB*(x, y)] + 2ρ*2^{2}*φ*0*(x, y)x*^{T}*y*

*= ρ*1^{2}*φ*FB*(x, y)*^{2}*+ 2ρ*2^{2}*φ*0*(x, y)*^{2}

= 2

*ρ*_{1}^{2}*ψ*FB*(x, y)*+1

2*ρ*_{2}^{2}*φ*0*(x, y)*^{2}

*+ ρ*^{2}2*φ*0*(x, y)*^{2}

*≥ 2ψ(x, y).*

(d) Using the formulas in (22) and [5, Lemma 6(a)], it follows that

∇*x**ψ (x, y),*∇*y**ψ (x, y) = ρ*1^{4}∇*x**ψ*FB*(x, y),*∇*y**ψ*FB*(x, y) + ρ*2^{4}*x*^{T}*yφ*0*(x, y)*^{2}
*+ ρ*1^{2}*ρ*_{2}^{2}*φ*0*(x, y)[x, ∇**x**ψ*FB*(x, y) + y, ∇**y**ψ*FB*(x, y)*]

*= ρ*1^{4}∇*x**ψ*_{FB}*(x, y),*∇*y**ψ*_{FB}*(x, y) + ρ*2^{4}*φ*_{0}*(x, y)*^{3}

*+ 2ρ*1^{2}*ρ*_{2}^{2}*φ*0*(x, y)ψ*FB*(x, y).* (23)

The first term on the right-hand side of (23) is non-negative by [5, Lemma 6(b)], and the
last two terms are also non-negative. Therefore, ∇*x**ψ (x, y),*∇*y**ψ (x, y)* ≥ 0, and moreover,

∇*x**ψ (x, y),*∇*y**ψ (x, y)* = 0 if and only if

∇*x**ψ*_{FB}*(x, y),*∇*y**ψ*_{FB}*(x, y) = 0 and φ*0*(x, y)= 0,*
which, together with Lemma 6(b) of [5], implies the desired result.

*(e) If ψ (x, y)= 0, then from the definition of ψ it follows that φ*FB*(x, y)= 0 and φ*0*(x, y)*= 0.

From Proposition 1 of [5], we immediately obtain∇*x**ψ*FB*(x, y)*= ∇*y**ψ*FB*(x, y)*= 0, and conse-
quently∇*x**ψ (x, y)*= 0 and ∇*y**ψ (x, y)= 0 by (22). If ∇ψ(x, y) = 0, then by part (c) and the*
*non-negativity of ψ , we get ψ (x, y)*= 0. Thus we prove the first equivalence. For the second
equivalence, it suffices to prove the sufficiency. Suppose that ∇*x**ψ (x, y)*= 0. From part (d),
*we readily get ψ(x, y)= 0, which together with part (a) and (22) implies ∇ψ(x, y) = 0. Con-*
sequently, *∇ψ(x, y) = 0 ⇐⇒ ∇**x**ψ (x, y)= 0. Similarly, ∇ψ(x, y) = 0 ⇐⇒ ∇**y**ψ (x, y)*= 0.

This implies the last equivalence.

*Lemma 4.1(b) shows that is continuously differentiable. By Lemma 4.1(d), we can prove*
*every stationary point of is a solution of Equation (1) under mild conditions.*

Proposition*4.2 Let *: R* ^{n}*→ R+

*be given by (14) and (15). Then every stationary point of*

*is a solution of (1) under one of the following assumptions:*

*(a)* *∇F (ζ) and −∇G(ζ ) are column monotone*^{1}*for any ζ* ∈ R^{n}*.*

*(b) For any ζ* ∈ R* ^{n}*,

*∇G(ζ) is invertible and ∇G(ζ)*

^{−1}

*∇F (ζ) has Cartesian P*0

*-property.*

*Proof* When the assumption (a) is satisfied, using the same arguments as those of [5, Prop.

*3] yields the desired result. Now suppose that the assumption (b) holds. Let ¯ζ be an arbitrary*
*stationary point of and write*

∇*x**ψ (F (ζ ), G(ζ ))= (∇**x*_{1}*ψ (F*1*(ζ ), G*1*(ζ )), . . . ,*∇*x*_{q}*ψ (F**q**(ζ ), G**q**(ζ ))),*

∇*y**ψ (F (ζ ), G(ζ ))*=!

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

*.*
Then,

*∇(¯ζ) = ∇F (¯ζ)∇**x**ψ (F ( ¯ζ ), G( ¯ζ ))+ ∇G(¯ζ)∇**y**ψ (F ( ¯ζ ), G( ¯ζ ))= 0,*
which, by the invertibility of*∇G, can be rewritten as*

*∇G(¯ζ)*^{−1}*∇F (¯ζ)∇**x**ψ (F ( ¯ζ ), G( ¯ζ ))*+ ∇*y**ψ (F ( ¯ζ ), G( ¯ζ ))= 0.* (24)
*Suppose that ¯ζ is not the solution of Equation (1). By Lemma 4.1(e), we necessarily have*

∇*x**ψ (F ( ¯ζ ), G( ¯ζ ))= 0.*

*Using the Cartesian P*0-property of*∇G(¯ζ)*^{−1}*∇F (¯ζ), there must exist an index ν ∈ {1, 2, . . . , q}*

such that∇*x**ν**ψ (F**ν**( ¯ζ ), G**ν**( ¯ζ ))*= 0 and

∇*x**ν**ψ (F*_{ν}*( ¯ζ ), G*_{ν}*( ¯ζ )),[∇G(¯ζ)*^{−1}*∇F (¯ζ)∇**x**ψ (F ( ¯ζ ), G( ¯ζ ))*]*ν** ≥ 0.* (25)
In addition, note that (24) is equivalent to

*[∇G(¯ζ)*^{−1}*∇F (¯ζ)∇**x**ψ (F ( ¯ζ ), G( ¯ζ ))*]*i*+ ∇*y**i**ψ (F*_{i}*( ¯ζ ), G*_{i}*( ¯ζ ))= 0, i = 1, 2, . . . , q.*

Making the inner product with∇*x**ν**ψ (F ( ¯ζ ), G( ¯ζ ))for the νth equality, we obtain*

∇*x*_{ν}*ψ (F**ν**( ¯ζ ), G**ν**( ¯ζ )),[∇G(¯ζ)*^{−1}*∇F (¯ζ)∇**x**ψ (F ( ¯ζ ), G( ¯ζ ))*]*ν*
+ ∇*x*_{ν}*ψ (F**ν**( ¯ζ ), G**ν**( ¯ζ )),*∇*y*_{ν}*ψ (F**ν**( ¯ζ ), G**ν**( ¯ζ )) = 0.*

The first term on the left-hand side is non-negative by (25), whereas the second term is positive
*by Lemma 4.1(d) since ζ is not a solution of (1). This leads to a contradiction, and consequently*

*¯ζ must be a solution of (1).*

When*∇G(ζ) is invertible for any ζ ∈ R** ^{n}*, the assumption in (a) is equivalent to the positive
semi-definiteness of

*∇G(ζ)*

^{−1}

*∇F (ζ) at any ζ ∈ R*

^{n}*, which implies the Cartesian P*

_{0}-property of

*∇G(ζ)*^{−1}*∇F (ζ ). Thus, for the SOCCP (3), the assumption (a) is stronger than the assumption*
*(b) which is now equivalent to the Cartesian P*_{0}*-property of F .*

*Next we provide a condition to guarantee the boundedness of the level sets of *
*L**(γ )* *:= {ζ ∈ R*^{n}*| (ζ) ≤ γ }*

*for all γ* *≥ 0. This property is important since it guarantees that the descent sequence of must*
have a limit point, and the solution set of (1) is bounded if it is non-empty. It turns out that the
*following condition for F and G is sufficient.*

*Condition A. For any sequence{ζ*^{k}*} satisfying ζ** ^{k}* → +∞, whenever

lim sup*[−F (ζ*^{k}*)*]+* < +∞ and lim sup [−G(ζ*^{k}*)*]+* < +∞,* (26)
*there exists an index ν∈ {1, 2, . . . , q} such that lim supF**ν**(ζ*^{k}*), G**ν**(ζ*^{k}*)* = +∞.

Proposition*4.3 If the mappings F and G satisfy Condition A, then the level setsL**(γ )are*
*bounded for all γ* *≥ 0.*

*Proof* Assume that there is a unbounded sequence*{ζ*^{k}*} ⊆ L**(γ )for some γ* *≥ 0. Since (ζ*^{k}*)*≤
*γ* *for all k, the sequence{*FB*(ζ*^{k}*)*} is bounded. By Lemma 8 of [5],

lim sup*[−F**i**(x*^{k}*)*]+* < +∞ and lim sup [−G**i**(x*^{k}*)*]+* < +∞*

*hold for all i* *∈ {1, 2, . . . , q}. This shows that F and G satisfy Condition A, and hence there exists*
*an index ν such that lim supF**ν**(ζ*^{k}*), G**ν**(ζ*^{k}*) = +∞. From the definition of , it follows that*
the sequence*{(ζ*^{k}*)} is unbounded, which clearly contradicts the fact that {ζ*^{k}*} ⊆ L**(γ ). The*

proof is completed.

*Condition A is rather weak to guarantee that has bounded level sets since, as will be shown*
*below, the condition is implied by the joint monotonicity of F and G with the strict feasibility of*
*(1) used in [5] for f*_{YF}*, the jointly uniform Cartesian P -functions with a feasible point, and the*
joint ˜*R*01-property in the following sense.

Definition*4.4 The mappings F, G*: R* ^{n}*→ R

^{n}*are said to have the joint ˜R*

_{01}

*-property if for any*

*sequence{ζ*

^{k}*} with*

*ζ*^{k}* → +∞,* *[−G(ζ*^{k}*)*]+

*ζ*^{k}*→ 0,* *[−F (ζ*^{k}*)*]+

*ζ*^{k}*→ 0,* (27)

*there holds that*

lim inf

*k*→+∞

*F (ζ*^{k}*), G(ζ*^{k}*)*

*ζ*^{k}*>0.* (28)