see [1, 5, 8, 13, 19, 20, 24, 29–31, 34, 35] and references therein.

There is a spectral decomposition with respect to second-order cone *K** ^{n}* in IR

*, which plays a very important role in the study of second-order cone optimization problems. For*

^{n}*any vector x = (x*1

*, x*2)

*∈ IR × IR*

^{n}*, the spectral decomposition (or spectral factorization) with respect to*

^{−1}*K*

*is given by*

^{n}*x = λ*1*(x)u*^{(1)}_{x}*+ λ*2*(x)u*^{(2)}_{x}*,* (1.1)
*where λ*1*(x), λ*2*(x) and u*^{(1)}*x* *, u*^{(2)}*x* are called the spectral values and the spectral vectors of
*x, respectively, with their corresponding formulas as bellow:*

*λ**i**(x) = x*1+ (*−1)*^{i}*∥x*2*∥, i = 1, 2,* (1.2)

*u*^{(i)}* _{x}* =

1 2

[ 1

(*−1)*^{i x}_{∥x}^{2}_{2}* _{∥}*
]

*,* *if x*2*̸= 0,*

1 2

[ 1

(*−1)*^{i}*w*
]

*,* *if x*2*= 0,*

(1.3)

*for i = 1, 2 with w being any vector in IR*^{n}* ^{−1}* satisfying

*∥w∥ = 1. Moreover,*{

*u*^{(1)}*x* *, u*^{(2)}*x*

} is called a Jordan frame satisfying the following properties:

*u*^{(1)}_{x}*+ u*^{(2)}_{x}*= e,*

⟨

*u*^{(1)}_{x}*, u*^{(2)}_{x}

⟩

*= 0, u*^{(1)}_{x}*◦ u*^{(2)}*x* *= 0 and u*^{(i)}_{x}*◦ u*^{(i)}*x* *= u*^{(i)}_{x}*(i = 1, 2),*
*where e = (1, 0,· · · , 0)*^{T}*∈ IR*^{n}*is the unit element and Jordan product x◦ y is defined by*
*x◦ y := (⟨x, y⟩, x*1*y*2*+ y*1*x*2) *∈ IR × IR*^{n}^{−1}*for any x = (x*1*, x*2*), y = (y*1*, y*2)*∈ IR × IR*^{n}* ^{−1}*.
For more details about Jordan product, please refer to [11].

*In [5, 6], for any real-valued function f : IR→ IR and x = (x*1*, x*_{2})*∈ IR × IR*^{n}* ^{−1}*, based on

*the spectral factorization of x with respect toK*

*, a type of vector-valued function associated with*

^{n}*K*

^{n}*(also called SOC-function) is introduced. More specifically, if we apply f to the*

*spectral values of x in (1.1), then we obtain the function f*

^{soc}: IR

^{n}*→ IR*

*given by*

^{n}*f*^{soc}*(x) = f (λ*_{1}*(x))u*^{(1)}_{x}*+ f (λ*_{2}*(x))u*^{(2)}_{x}*.* (1.4)
*From the expression (1.4), it is clear that the SOC-function f*^{soc} is unambiguous whether
*x*_{2} *= 0 or x*_{2} *̸= 0. Further properties regarding f*^{soc} were discussed in [3–5, 7, 17, 32].

*It is also known that such SOC-functions f*^{soc} associated with second-order cone play a
crucial role in the theory and numerical algorithm for second-order cone programming,
see [1, 5, 8, 13, 19, 20, 24, 29–31, 34, 35] again.

*In this paper, in light of the definition of f*^{soc}, we define another type of SOC-function
Φ*µ* (see Section 2 for details). In particular, using the SOC-function Φ*µ*, we will solve the
following system of equalities and inequalities under the order induced by the second-order

cone: {

*f**I**(x)⪯**K*^{m}*0,*

*f**E**(x) = 0,* (1.5)

*where f**I**(x) = (f*1*(x),· · · , f**m**(x))*^{T}*, f**E**(x) = (f**m+1**(x),· · · , f**n**(x))*^{T}*, and “x⪯**K** ^{m}* 0” means

“*−x ∈ K*^{m}*”. Likewise, x⪰**K*^{m}*0 means x∈ K*^{m}*and x≻**K*^{m}*0 means x∈ int(K** ^{m}*) whereas

int(*K** ^{m}*) denotes the interior of

*K*

^{m}*. Throughout this paper, we assume that f*

*i*is continu-

*ously diﬀerentiable for any i∈ {1, 2, ..., n}. We also define*

*f (x) :=*

[ *f*_{I}*(x)*
*f*_{E}*(x)*

]

*and hence f is continuously diﬀerentiable. When* *K** ^{m}* = IR

^{m}_{+}, the system (1.5) reduces to the standard system of equalities and inequalities over IR

*. The corresponding standard system (1.5) has been studied extensively due to its various applications, and there are many methods for solving such problems, see [10, 27, 28, 33, 37]. For the setting of second-order cone, we know that the KKT conditions of the second-order cone constrained optimization problems can be expressed in form of (1.5), i.e., the system of equalities and inequalities under the order induced by second-order cones. For example, for the following second-order cone optimization problem:*

^{m}min *h(x)*

s.t. *−g(x) ∈ K*^{m}*,*
the KKT conditions of this problem is as follows

*∇h(x) + ∇g(x)λ* = *0,*
*λ*^{T}*g(x)* = *0,*

*−λ ⪯*_{K}^{m}*0,*
*g(x)* *⪯**K*^{m}*0,*
where*∇g(x) denotes the gradient matrix of g. Now, by denoting*

*f**I**(x, λ) :=*

[ *−λ*
*g(x)*

]

*and f**E**(x, λ) :=*

[ *∇h(x) + ∇g(x)λ*
*λ*^{T}*g(x)*

]
*,*

it is clear to see that the KKT conditions of the second-order cone optimization problem is in form of (1.5). From this view, the investigation of the system (1.5) provides a theoretical way for solving second-order cone optimization problems. Hence, the study of the system (1.5) is important and that is the main motivation for this paper.

So far, there are many kinds of numerical methods for solving the second-order cone optimization problems. Among which, there is a class of popular numerical method, the so-called smoothing-type algorithms. This kind of algorithm has also been a powerful tool for solving many other optimization problems, including symmetric cone complementarity problems [15, 16, 20–22], symmetric cone linear programming [23, 26], the system of inequal- ities under the order induced by symmetric cone [18, 25, 38], and so on. From these recent studies, most of the existing smoothing-type algorithms were designed on the basis of a monotone line search. In order to achieve better computational results, the nonmonotone line search technique is sometimes adopted in the numerical implementations of smoothing- type algorithms [15,36,37]. The main reason is that the nonmonotone line search scheme can improve the likelihood of finding a global optimal solution and convergence speed in cases where the function involved is highly nonconvex or has a valley in a small neighborhood of some point. In view of this, in this paper we also develop a nonmonotone smoothing-type algorithm for solving the system of equalities and inequalities under the order induced by second-order cones.

The remaining parts of this paper are organized as follows. In Section 2, some back-
ground concepts and preliminary results about the second-order cone are given. In Section
3, we reformulate (1.5) as a system of smoothing equations in which Φ*µ* is employed. In
Section 4, we propose a nonmonotone smoothing-type algorithm for solving (1.5), and show
that the algorithm is well defined. Moreover, we also discuss the global convergence and
locally quadratic convergence of the proposed algorithm. The preliminary numerical results
are reported to demonstrate that the proposed algorithm is eﬀective in Section 5. Some
numerical comparison in light of performance profiles is presented which indicates the dif-
ference of numerical performance when various smoothing functions are used.

**2 Preliminaries**

In this section, we briefly review some basic properties about the second-order cone and the vector-valued functions with respect to SOC, which will be extensively used in subsequent analysis. More details about the second-order cone and the vector-valued functions can be found in [3–5, 13, 14, 17].

*First, we review the projection of x∈ IR** ^{n}*onto the second-order cone

*K*

^{n}*⊂ IR*

*. For the second-order cone*

^{n}*K*

*, let (*

^{n}*K*

*)*

^{n}*denote its dual cone. Then, (*

^{∗}*K*

*)*

^{n}*is given by*

^{∗}(*K** ^{n}*)

*:={*

^{∗}*y = (y*1*, y*2)*∈ IR × IR*^{n}^{−1}*| ⟨x, y⟩ ≥ 0, ∀x ∈ K** ^{n}*}

*.*

Moreover, it is well known that the second-order cone*K** ^{n}*is a self-dual cone, i.e., (

*K*

*)*

^{n}*=*

^{∗}*K*

*.*

^{n}*Let x*+

*denote the projection of x∈ IR*

*onto the second-order cone*

^{n}*K*

^{n}*, and x*

*denote the projection of*

_{−}*−x onto the dual cone (K*

*)*

^{n}

^{∗}*. With these notations, for any x*

*∈ IR*

*, it is*

^{n}*not hard to verify that x = x*+

*− x*

*. In particular, due to the special structure of*

_{−}*K*

*, the*

^{n}*explicit formula of the projection of x∈ IR*

*onto*

^{n}*K*

*is obtained in [14] as below:*

^{n}*x*+=

*x* *if x∈ K*^{n}*,*

0 *if x∈ −(K** ^{n}*)

*=*

^{∗}*−K*

^{n}*,*

*u*

*otherwise,*

(2.1)

where

*u =*

*x*1+*∥x*2*∥*
( 2

*x*1+*∥x*2*∥*
2

) *x*2

*∥x*2*∥*

* .*

*In fact, according to the spectral decomposition of x, the expression of the projection x*_{+}
onto*K** ^{n}* can be alternatively expressed as (see [13, Prop. 3.3(b)])

*x*+*= ((λ*1*(x))*_{+}*u*^{(1)}_{x}*+ ((λ*2*(x))*_{+}*u*^{(2)}_{x}*,*
*where (α)*+= max*{0, α} for any α ∈ IR.*

From the definition (1.4) of the vector-valued function associated with *K** ^{n}*, we know

*that the projection x*+onto

*K*

*is a vector-valued function. Moreover, it is known that the*

^{n}*projection x*+

*and (α)*+

*for any α∈ IR have many the same properties, such as the continuity,*the directional diﬀerentiability and semismooth and so on. Indeed, these properties are established for general vector-valued functions associated with SOC. Among which, Chen,

*Chen and Tseng [5] have obtained that many properties of f*

^{soc}are inherited from the

*function f , which is presented in the following proposition.*

* Proposition 2.1. Suppose that x = (x*1

*, x*2)

*∈ IR × IR*

^{n}

^{−1}*has the spectral decomposition*

*given as in (1.1)-(1.3). For any the function f : IR→ IR and the vector-valued function f*

^{soc}

*defined by (1.4), the following hold.*

**(a) f**^{soc} *is continuous at x∈ IR*^{n}*with spectral values λ*_{1}*(x), λ*_{2}*(x)* *⇐⇒ f is continuous at*
*λ*_{1}*(x), λ*_{2}*(x);*

**(b) f**^{soc} *is directionally diﬀerentiable at x∈ IR*^{n}*with spectral values λ*1*(x), λ*2*(x)⇐⇒ f is*
*directionally diﬀerentiable at λ*1*(x), λ*2*(x);*

**(c) f**^{soc} *is diﬀerentiable at x∈ IR*^{n}*with spectral values λ*1*(x), λ*2*(x)⇐⇒ f is diﬀerentiable*
*at λ*1*(x), λ*2*(x);*

**(d) f**^{soc} *is strictly continuous at x∈ IR*^{n}*with spectral values λ*1*(x), λ*2*(x)⇐⇒ f is strictly*
*continuous at λ*1*(x), λ*2*(x);*

**(e) f**^{soc} *is semismooth at x∈ IR*^{n}*with spectral values λ*_{1}*(x), λ*_{2}*(x)* *⇐⇒ f is semismooth*
*at λ*_{1}*(x), λ*_{2}*(x);*

**(f ) f**^{soc} *is continuously diﬀerentiable at x∈ IR*^{n}*with spectral values λ*_{1}*(x), λ*_{2}*(x)* *⇐⇒ f is*
*continuously diﬀerentiable at λ*_{1}*(x), λ*_{2}*(x).*

*Note that the projection function x*+ onto*K** ^{n}* is not a smoothing function on the whole
space IR

*. From Proposition 2.1, we can make some smoothing functions for the projection*

^{n}*x*

_{+}onto

*K*

^{n}*if we smooth the functions f (λ*

_{i}*(x)) for i = 1, 2. More specifically, we consider*

*a family of smoothing functions ϕ(µ,·) : IR → IR with respect to (α)*+ satisfying

lim

*µ**↓0**ϕ(µ, α) = (α)*+ and 0*≤* *∂ϕ*

*∂α(µ, α)≤ 1.* (2.2)

*for all α* *∈ IR. Are there functions satisfying the above conditions? Yes, there are many.*

We illustrate three of them here:

*ϕ*1*(µ, α)* =

√*α*^{2}*+ 4µ*^{2}*+ α*

2 *,* *(µ > 0)*
*ϕ*2*(µ, α)* = *µ ln(e*^{α}^{µ}*+ 1),* *(µ > 0)*

*ϕ*_{3}*(µ, α)* =

*α,* *if α≥ µ,*

*(α+µ)*^{2}

*4µ* *,* if *− µ < α < µ,*
*0,* *if α≤ −µ.*

*(µ > 0)*

*In fact, the functions ϕ*1 *and ϕ*2 *were considered in [13, 17], while the function ϕ*3 was
*employed in [18, 37]. In addition, as for the function ϕ*3, there is a more general function
*ϕ**p**(µ,·) : IR → IR given by*

*ϕ**p**(µ, α) =*

*α* if *α≥*_{p}_{−1}^{µ}*,*

*µ*
*p**−1*

[*(p**−1)(α+µ)*
*pµ*

]*p*

if *−µ < α <* _{p}_{−1}^{µ}*,*

0 if *α≤ −µ,*

*where µ > 0 and p≥ 2. This function ϕ**p*is recently studied in [9] and it is not hard to verify
*that ϕ*_{p}*also satisfies the above conditions (2.2). All the functions ϕ*_{1}*, ϕ*_{2}*and ϕ*_{3}will play the

*role of smoothing functions as f (λ**i**(x)) in (1.4). In other words, based on these smoothing*
functions, we define a type of SOC-functions Φ*µ*(*·) on IR** ^{n}* associated with

*K*

^{n}*(n≥ 1) as*

Φ*µ**(x) := ϕ(µ, λ*1*(x))u*^{(1)}_{x}*+ ϕ(µ, λ*2*(x))u*^{(2)}_{x}*∀x = (x*1*, x*2)*∈ IR × IR*^{n}^{−1}*,* (2.3)
*where λ*1*(x), λ*2*(x) are given by (1.2) and u*^{(1)}*x* *, u*^{(2)}*x* are given by (1.3). In light of the
*properties of ϕ(µ, α), we show as below that the SOC-function Φ**µ**(x) becomes the smoothing*
*function for the projection function x*+ onto*K** ^{n}*.

*We depict the graphs of ϕ*_{i}*(µ, α) for i = 1, 2, 3, in Figure 1. From Figure 1, we see that*
*ϕ*_{3} *is the one which best approximates the function (α)*_{+} under the sense that it is closest
*to (α)*_{+} *among all ϕ*_{i}*(µ, α) for i = 1, 2, 3.*

*Figure 1: Graphs of max(0, t) and all three ϕ**i**(µ, t) with µ = 0.2.*

**Proposition 2.2. Suppose that x = (x**_{1}*, x*_{2})*∈ IR × IR*^{n}^{−1}*has the spectral decomposition*
*given as in (1.1)-(1.3), and that ϕ(µ,·) with µ > 0 is continuously diﬀerentiable function*
*satisfying (2.2). Then, the following hold.*

**(a) The function Φ**_{µ}*(x) : IR*^{n}*→ IR*^{n}*defined as in (2.3) is continuously diﬀerentiable. More-*
*over, its Jacobian matrix at x is described as*

*∂Φ**µ**(x)*

*∂x* =

*∂ϕ*

*∂λ**(µ, x*_{1}*)I* if *x*_{2}*= 0,*

[ *b* *cx*_{2}^{T}*/∥x*2*∥*

*cx*_{2}*/∥x*2*∥ aI + (b − a)x*2*x*_{2}^{T}*/∥x*2*∥*^{2}
]

if *x*_{2}*̸= 0,* (2.4)
*where*

*a* = ^{ϕ(µ,λ}_{λ}^{2}^{(x))}^{−ϕ(µ,λ}^{1}^{(x))}

2*(x)**−λ*1*(x)* *,*
*b* = ^{1}_{2}

(*∂ϕ*

*∂λ*_{2}*(µ, λ*2*(x)) +*_{∂λ}^{∂ϕ}

1*(µ, λ*1*(x))*
)

*,*
*c* = ^{1}_{2}

(*∂ϕ*

*∂λ*2*(µ, λ*2*(x))−*_{∂λ}^{∂ϕ}_{1}*(µ, λ*1*(x))*
)

;

(2.5)

**(b) Both**^{∂Φ}_{∂x}^{µ}^{(x)}*and I−*^{∂Φ}_{∂x}^{µ}^{(x)}*are positive semi-definite matrices;*

**(c) lim**

*µ**→0*Φ_{µ}*(x) = x*_{+}*= (λ*_{1}*(x))*_{+}*u*^{(1)}_{x}*+ (λ*_{2}*(x))*_{+}*u*^{(2)}_{x}*for i = 1, 2.*

*Proof. (a) From the expression (2.3) and the assumption of ϕ(µ,·) being continuously diﬀer-*
entiable, it is easy to verify that the function Φ*µ*is continuously diﬀerentiable. The Jacobian
matrix (2.4) of Φ*µ**(x) can be obtained by adopting the same arguments as in [13, Proposition*
5.2]. Hence, we omit the details here.

(b) First, we prove that the matrix ^{∂Φ}_{∂x}^{µ}^{(x)}*is positive semi-definite. For the case of x*2= 0,
we know that ^{∂Φ}_{∂x}^{µ}* ^{(x)}* =

^{∂ϕ}

_{∂λ}*(µ, x*1

*)I. Then, from 0*

*≤*

_{∂α}

^{∂ϕ}*(µ, α)*

*≤ 1, it is clear to see that*the matrix

^{∂Φ}

_{∂x}

^{µ}

^{(x)}*is positive semi-definite. For the case of x*2

*̸= 0, from*

_{∂α}

^{∂ϕ}*(µ, α)≥ 0 and*

*(2.5), we have b*

*≥ 0. In order to prove that the matrix*

^{∂Φ}

_{∂x}

^{µ}*is positive semi-definite,*

^{(x)}*we only need to verify that the Schur Complement of b with respect to*

^{∂Φ}

_{∂x}

^{µ}*is positive*

^{(x)}*semi-definite. Note that the Schur Complement of b has the form of*

*aI + (b− a)x*_{2}*x*^{T}_{2}

*∥x*2*∥*^{2} *−c*^{2}
*b*

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

*∥x*2*∥*^{2} *= a*
(

*I−* *x*_{2}*x*^{T}_{2}

*∥x*2*∥*^{2}
)

+*b*^{2}*− c*^{2}
*b*

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

*∥x*2*∥*^{2}*.*

Since _{∂α}^{∂ϕ}*(µ, α)* *≥ 0, we obtain that the function ϕ(µ, α) with respect to α is increasing,*
*which leads to a≥ 0. Besides, from (2.5), we observe that*

*b*^{2}*− c*^{2}= *∂ϕ*

*∂λ*_{2}*(µ, λ*2*(x))∂ϕ*

*∂λ*_{1}*(µ, λ*1*(x))≥ 0.*

*With this, it follows that the Schur Complement of b with respect to* ^{∂Φ}_{∂x}^{µ}* ^{(x)}* is a linear
non-negative combination of the matrices

_{∥x}

^{x}^{2}

^{x}

^{T}^{2}

2*∥*^{2} *and I* *−* _{∥x}^{x}^{2}_{2}^{x}_{∥}^{T}^{2}2. Thus, we show that the
*Schur Complement of b is positive semi-definite, which says the matrix* ^{∂Φ}_{∂x}^{µ}* ^{(x)}* is positive
semi-definite.

Combining with ^{∂ϕ}_{∂α}*(µ, α)≤ 1 and following similar arguments as above, we can also argue*
*that the matrix I−*^{∂Φ}_{∂x}^{µ}* ^{(x)}* is also positive semi-definite.

(c) By the definition of the function Φ*µ**(x), it can be verified directly.*

We point out that the definition of (2.3) includes the similar way to define smoothing
functions in [13, Section 4] as a special case, and hence [13, Prop. 4.1] is covered by Proposi-
tion 2.2. Indeed, Proposition 2.2 can be also verified by geometric views. More specifically,
*from Figures 2, 3 and 4, we see that when µ↓ 0, ϕ**i* *is getting closer to (α)*_{+}, which verifies
Proposition 2.2(c).

**3 Applying Φ**

_{µ}**to solve the system (1.5)**

In this section, in light of the smoothing vector-valued function Φ* _{µ}*, we reformulate (1.5) as
a system of smoothing equations. To this end, we need a partial order induced by SOC.

*More specifically, for any x∈ IR** ^{n}*, using the definition of the partial order “

*⪯*

*K*

*” and the*

^{m}*projection function x*

_{+}in (2.1), we have

*f*_{I}*(x)⪯**K** ^{m}*0

*⇐⇒ −f*

*I*

*(x)∈ K*

^{m}*⇐⇒ f*

*I*

*(x)∈ −K*

^{m}*⇐⇒ (f*

*I*

*(x))*

_{+}

*= 0.*

Hence, the system (1.5) is equivalent to the following system of equations:

{ *(f**I**(x))*+ = *0,*

*f**E**(x)* = *0.* (3.1)

*Figure 2: Graphs of ϕ*1*(µ, α) with µ = 0.01, 0.1, 0.3, 0.5.*

*Figure 3: Graphs of ϕ*2*(µ, α) with µ = 0.01, 0.1, 0.3, 0.5.*

*Note that the function (f**I*(*·))*+ in the above equation (3.1) is nonsmooth. Therefore, the
smoothing-type Newton methods cannot be directly applied to solve the equation (3.1).

To conquer this, we employ the smoothing function Φ*µ*(*·) defined in (2.3), and define the*
following function:

*F (µ, x, y) :=*

*f**I**(x)− y*
*f**E**(x)*
Φ*µ**(y)*

* .*

From Proposition 2.2(c), it follows that

*F (µ, x, y) = 0* and *µ = 0*

*⇐⇒ y = f**I**(x), f*_{E}*(x) = 0, Φ*_{µ}*(y) = 0 and µ = 0*

*⇐⇒ y = f**I**(x), f**E**(x) = 0 and y*+= 0

*⇐⇒ (f**I**(x))*+*= 0, f**E**(x) = 0*

*⇐⇒ f**I**(x)⪯**K*^{m}*0, f*_{E}*(x) = 0.*

*Figure 4: Graphs of ϕ*3*(µ, α) with µ = 0.01, 0.1, 0.3, 0.5.*

*In other words, as long as the system F (µ, x, y) = 0 and µ = 0 is solved, the corresponding*
*x is a solution to the original system (1.5). In view of Proposition 2.2(a), we can obtain*
the solution to the system (1.5) by applying smoothing-type Newton method for solving
*F (µ, x, y) = 0 and setting µ↓ 0 at the same time. To do this, for any z = (µ, x, y) ∈ IR*++*×*
IR^{n}*× IR*^{m}*, we further define a continuously diﬀerentiable function H : IR*_{++}*× IR*^{n}*× IR*^{m}*→*
IR++*× IR*^{n}*× IR** ^{m}* as follows:

*H(z) :=*

*µ*
*f**I**(x)− y + µx**I*

*f**E**(x) + µx**E*

Φ*µ**(y) + µy*

* ,* (3.2)

*where x**I* *:= (x*1*, x*2*, ..., x**m*)^{T}*∈ IR*^{m}*, x**E* *:= (x**m+1**, ..., x**n*)^{T}*∈ IR*^{n}^{−m}*, x := (x*^{T}_{I}*, x*^{T}* _{E}*)

^{T}*∈ IR*

^{n}*and y∈ IR*

^{m}*. Then, it is clear to see that when H(z) = 0, we have µ = 0 and x is a solution*

*to the system (1.5). Now, we let H*

^{′}*(z) denote the Jacobian matrix of the function H at z,*

*then for any z∈ IR*++

*× IR*

^{n}*× IR*

*, we obtain that*

^{m}*H*^{′}*(z) =*

1 0*n* 0*m*

*x**I* *f*_{I}^{′}*+ µU* *−I**m*

*x**E* *f*_{E}^{′}*+ µV* 0_{(n}_{−m)×m}

*∂Φ*_{µ}*(y)*

*∂µ* *+ y* 0*m**×n* *∂Φ*_{µ}*(y)*

*∂y* *+ µI**m*

* ,* (3.3)

*where U :=* [

*I** _{m}* 0

_{m}*]*

_{×(n−m)}*, V :=* [

0_{(n}_{−m)×m}*I*_{n}* _{−m}*]

*, 0*_{l}*denotes l dimensional zero*
vector, and 0_{l}_{×q}*denotes l× q zero matrix for any positive integer l and q. In summary,*
*we will apply smoothing-type Newton method to solve the smoothed equation H(z) = 0 at*
*each iteration and make µ > 0 as well as H(z)→ 0 to find a solution of the system (1.5).*

**4 A smoothing-type Newton algorithm**

Now, we consider a smoothing-type Newton algorithm with a nonmonotone line search, and
show that the algorithm is well defined. For convenience, we denote the merit function Ψ
*as Ψ(z) :=∥H(z)∥*^{2} *for any z∈ IR*++*× IR*^{n}*× IR** ^{m}*.

**Algorithm 4.1. (A smoothing-type Newton Algorithm)**

**Step 0 Choose γ***∈ (0, 1), ξ ∈ (0,*^{1}_{2}*). Take η > 0, σ* *∈ (0, 1) such that ση < 1. Let*
*µ*0 *= η and (x*^{0}*, y*^{0})*∈ IR*^{n}*× IR*^{m}*be an arbitrary vector. Set z*^{0} *= (µ*0*, x*^{0}*, y*^{0}*), e*^{0} :=

*(1, 0, ..., 0)∈ IR × IR*^{n}*× IR*^{m}*, G*0:=*∥H(z*^{0})*∥*^{2}*= Ψ(z*^{0}*) and S*0*:= 1. Choose β*min and
*β*maxsuch that 0*≤ β*min*≤ β*max*< 1. Set τ (z*^{0}*) := σ min{1, Ψ(z*^{0})*} and k := 0.*

**Step 1 If***∥H(z** ^{k}*)

*∥ = 0, stop. Otherwise, go to Step 2.*

**Step 2 Compute ∆z**^{k}*:= (∆µ*_{k}*, ∆x*^{k}*, ∆y** ^{k}*)

*∈ IR × IR*

^{n}*× IR*

*by*

^{m}*H*^{′}*(z*^{k}*)∆z** ^{k}* =

*−H(z*

^{k}*) + ητ (z*

^{k}*)e*

^{0}

*.*(4.1)

**Step 3 Let α**

_{k}*be the maximum of the values 1, γ, γ*

^{2}

*, ... such that*

*Ψ(z*^{k}*+ α*_{k}*∆z** ^{k}*)

*≤ [1 − 2ξ(1 − ση)α*

*k*

*] G*

_{k}*.*(4.2)

**Step 4 Set z**

^{k+1}*:= z*

^{k}*+ α*

*k*

*∆z*

*. If*

^{k}*∥H(z*

*)*

^{k+1}*∥ = 0, stop. Otherwise, go to Step 5.*

**Step 5 Choose β***k**∈ [β**min**, β**max*]. Set

*S**k+1* := *β**k**S**k**+ 1,*
*τ (z** ^{k+1}*) := min{

*σ, σΨ(z*^{k+1}*), τ (z** ^{k}*)}

*,*

*G*

*k+1*:= (

*β**k**S**k**G**k**+ Ψ(z** ^{k+1}*))

*/S*

*k+1*

*,*

(4.3)

*and set k := k + 1. Go to Step 2.*

The nonmonotone line search technique in Algorithm 4.1 was introduced in [36]. From
*the first and third equations in (4.3), we know that G*_{k+1}*is a convex combination of G*_{k}*and Ψ(z*^{k+1}*). In fact, G**k* *is expressed as a convex combination of Ψ(z*^{0}*), Ψ(z*^{1}*), ..., Ψ(z** ^{k}*).

*Moreover, the main role of β**k* *is to control the degree of non-monotonicity. If β**k* = 0 for
*every k, then the corresponding line search is the usual monotone Armijo line search.*

**Proposition 4.2. Suppose that the sequences***{z*^{k}*}, {µ**k**}, {G**k**}, {Ψ(z** ^{k}*)

*} and {τ(z*

*)*

^{k}*} are*

*generated by Algorithm 4.1. Then, the following hold.*

**(a) The sequence**{G*k**} is monotonically decreasing and Ψ(z** ^{k}*)

*≤ G*

*k*

*for all k∈ N;*

**(b) The sequence**{τ(z* ^{k}*)

*} is monotonically decreasing;*

**(c) ητ (z*** ^{k}*)

*≤ µ*

*k*

*for all k∈ N;*

**(d) The sequence**{µ*k**} is monotonically decreasing and µ**k**> 0 for all k∈ N.*

*Proof. The proof is similar to Remark 3.1 in [37], we omit the details.*

Next, we show that Algorithm 4.1 is well-defined and establish its local quadratic con-
*vergence. For simplicity, we denote the Jacobian matrix of the function f by*

*f*^{′}*(x) :=*

[ *f*_{I}^{′}*(x)*
*f*_{E}^{′}*(x)*

]

and use the following assumption.

**Assumption 4.1. f**^{′}*(x) + µI**n* *is invertible for any x∈ IR*^{n}*and µ∈ IR*++*.*

We point our that the Assumption 4.1 is only a mild condition and there are many
*functions satisfying the assumption. For example, if f is a monotone function, then f*^{′}*(x) is*
*a positive semi-definite matrix for any x∈ IR** ^{n}*. Thus, Assumption 4.1 is satisfied.

**Theorem 4.3. Suppose that f is a continuously diﬀerentiable function and Assumption 4.1***is satisfied. Then, Algorithm 4.1 is well-defined.*

*Proof. In order to show that Algorithm 4.1 is well-defined, we need to prove that Newton*
equation (4.1) is solvable, and the line search (4.2) is well-defined.

First, we prove that Newton equation (4.1) is solvable. By the expression of Jacobian matrix
*H*^{′}*(z) in (3.3), we see that the determinant det(H*^{′}*(z)) of H*^{′}*(z) satisfies*

*det(H*^{′}*(z)) = det (f*^{′}*(x) + µI**n*)*· det*

(*∂Φ*_{µ}*(y)*

*∂y* *+ µI**m*

)

*for any z∈ IR*++*× IR*^{n}*× IR** ^{m}*. Moreover, from Proposition 2.2(b), we know that

^{∂Φ}

_{∂y}

^{µ}*is*

^{(y)}*positive semi-definite for µ∈ IR*++. Hence, combing this with Assumption 4.1, we obtain

*that H*

^{′}*(z) is nonsingular for any z*

*∈ IR*++

*× IR*

^{n}*× IR*

^{m}*with µ > 0. Applying Proposition*4.2(d), it follows that Newton equation (4.1) is solvable.

Secondly, we prove that the line search (4.2) is well-defined. For notational convenience, we denote

*w*_{k}*(α) := Ψ*(

*z*^{k}*+ α∆z** ^{k}*)

*− Ψ*(
*z** ^{k}*)

*− αΨ** ^{′}*(

*z*

*)*

^{k}*∆z*^{k}*.*
From Newton equation (4.1) and the definition of Ψ, we have

Ψ(

*z*^{k}*+ α∆z** ^{k}*)

= *w**k**(α) + Ψ*(
*z** ^{k}*)

*+ αΨ** ^{′}*(

*z*

*)*

^{k}*∆z*^{k}

= *w**k**(α) + Ψ*(
*z** ^{k}*)

*+ 2αH*(
*z** ^{k}*)

*T*(

*−H(z*^{k}*) + ητ (z*^{k}*)e*^{0})

*≤ w**k**(α) + (1− 2α)Ψ*(
*z** ^{k}*)

*+ 2αητ (z** ^{k}*)

*H(z*

*)*

^{k}*.*

*If Ψ(z*

*)*

^{k}*≤ 1, then we have ∥H(z*

*)*

^{k}*∥ ≤ 1. Hence, it follows that*

*τ (z** ^{k}*)

*∥H(z*

*)*

^{k}*∥ ≤ σΨ(z*

*)*

^{k}*∥H(z*

*)*

^{k}*∥ ≤ σΨ(z*

^{k}*).*

*If Ψ(z*^{k}*) > 1, then we see that Ψ(z** ^{k}*) =

*∥H(z*

*)*

^{k}*∥*

^{2}

*≥ ∥H(z*

*)*

^{k}*∥, which yields*

*τ (z*

*)*

^{k}*∥H(z*

*)*

^{k}*∥ ≤ σ∥H(z*

*)*

^{k}*∥ ≤ σΨ(z*

^{k}*).*

Thus, from all the above, we obtain that Ψ(

*z*^{k}*+ α∆z** ^{k}*)

*≤ w**k**(α) + (1− 2α)Ψ(z*^{k}*) + 2αησΨ(z** ^{k}*)

= *w**k**(α) +*[

1*− 2(1 − ση)α*]

*Ψ(z** ^{k}*) (4.4)

*≤ w**k**(α) +*[

1*− 2(1 − ση)α*]
*G**k**.*

*Since the function H is continuous and diﬀerentiable for any z∈ IR*++*× IR*^{n}*× IR** ^{m}*, we have

*w*

*k*

*(α) = o(α) for all k∈ N. Combining with (4.4), this indicates that the line search (4.2)*is well-defined.

**Theorem 4.4. Suppose that f is a continuously diﬀerentiable function and Assumption***4.1 is satisfied. Then the sequence* *{z*^{k}*} generated by Algorithm 4.1 is bounded; and any*
*accumulation point of the sequence{x*^{k}*} is a solution of the system (1.5).*

*Proof. The proof is similar to [37, Theorem 4.1] and we omit it.*

In Theorem 4.4, we give the global convergence of Algorithm 4.1. Now, we analyze the
convergence rate for Algorithm 4.1. We start with introducing the following concepts. A
*locally Lipschitz function F : IR*^{n}*→ IR** ^{m}*is said to be semismooth (or strongly semismooth)

*at x∈ IR*

^{n}*if F is directionally diﬀerentiable at x and*

*F (x + h)− F (h) − V h = o(∥h∥) (or = O(∥h∥*^{2}))

*holds for any V* *∈ ∂F (x+h), where ∂F (x) is the generalized Jacobian matrix of the function*
*F at x∈ IR** ^{n}*in the sense of Clarke [2]. There are many functions being semismooth, such as
convex functions, smooth functions, piecewise linear functions and so on. In addition, it is
known that the composition of semismooth functions is still a semismooth function, and the
composition of strongly semismooth functions is still a strongly semismooth function [12].

From Proposition 2.2 (a), we know that Φ*µ**(x) defined by (2.3) is smooth on IR** ^{n}*.

*With the definition (3.2) of H, mimicking the arguments as in [37, Theorem 5.1], we*
have the local quadratic convergence of Algorithm 4.1.

**Theorem 4.5. Suppose that the conditions given in Theorem 4.4 are satisfied, and z*** ^{∗}* =

*(µ*

_{∗}*, x*

^{∗}*, y*

^{∗}*) is an accumulation point of sequence{z*

^{k}*} which is generated by Algorithm 4.1.*

**(a) If all V***∈ ∂H(z*^{∗}*) are nonsingular, then the sequence{z*^{k}*} converges to z*^{∗}*, and*

*∥z*^{k+1}*− z*^{k}*∥ = o(∥z*^{k}*− z*^{∗}*∥), µ**k+1**= o(µ**k*);

**(b) If the functions f and Φ***µ* *satisfy that f*^{′}*and Φ*^{′}_{µ}*are Lipschitz continuous on IR*^{n}*, then*

*∥z*^{k+1}*− z*^{k}*∥ = O(∥z*^{k}*− z*^{∗}*∥)*^{2} *and µ**k+1**= O(µ*^{2}_{k}*).*

**5 Numerical experiments**

In this section, we present some numerical examples to demonstrate the eﬃciency of Algo-
rithm 4.1 for solving the system (1.5). In our tests, all experiments are done on a PC with
*CPU of 1.9 GHz and RAM of 8.0 GB, and all the program codes are written in MATLAB*
*and run in MATLAB environment. We point out that if there are no n numbers in I∪ E,*
we can adopt a similar way to those given in [37], then the system (1.5) can be transformed
as a new problem and we can solve the new problem using Algorithm 4.1. By this approach,
a solution of the original problem can be found.

*Throughout the following experiments, we employ three functions ϕ*_{1}*, ϕ*_{2} *and ϕ*_{3} along
*with the proposed algorithm to implement each example. Note that, for the function ϕ*_{1}, its
corresponding SOC-function Φ* _{µ}* can be alternatively expressed as

eΦ*µ**(x) =* *x +*√

*x*^{2}*+ 4µ*^{2}*e*

2 with *e = (1, 0,· · · , 0)*^{T}*∈ K*^{n}*.*

This form is simpler than the Φ*µ**(x) induced from (2.3). Hence, we adopt it in our imple-*
mentation. Moreover, the parameters used in the algorithm are chosen as follows:

*γ = 0.3,* *ξ = 10*^{−4}*,* *η = 1.0,* *β*0*= 0.01,* *µ*0*= 1.0,* *S*0*= 1.0,*

*and the parameters c and σ are chosen according to the ones listed in Table 1 and Table*
4. In the implementation, the stopping rule is *∥H(z)∥ ≤ 10*^{−6}*, the step length ν* *≤ 10** ^{−6}*,
or the number of iteration is over 500; and the starting points are randomly generated from
the interval [

*−1, 1].*

Now, we present the test examples. We first consider two examples in which the system
*(1.5) only includes inequalities, i.e., m = n. Note that a similar way to construct the two*
examples was given in [25].

**Example 5.1. Consider the system (1.5) with inequalities only, where f (x) := M x+q**⪯_{K}* ^{n}*0
and

*K*

*:=*

^{n}*K*

^{n}^{1}

*× · · · × K*

^{n}

^{r}*. Here M is generated by M = BB*

^{T}*with B∈ IR*

^{n}*being a*

^{×n}*matrix whose every component is randomly chosen from the interval [0, 1] and q∈ IR*

*being a vector whose every component is 1.*

^{n}*For Example 5.1, the tested problems are generated with sizes n = 500, 1000, ..., 4500*
*and each n** _{i}*= 10. The random problems of each size are generated 10 times. Besides using
the three functions along with Algorithm 4.1 for solving Example 5.1, we have also tested it
by using the smoothing-type algorithm with the monotone line search which was introduced

*in [25] (for this case, we choose the function ϕ*

_{1}). Table 1 shows the numerical results where

“fun” denotes the three functions,

“suc” denotes the number that Algorithm 4.1 successfully solves every generated problem,

“iter” denotes the average iteration numbers,

“cpu” denotes the average CPU time in seconds,

“res” denotes the average residual norm*∥H(z)∥ for 9 test problems.*

The initial points are also randomly generated. In light of “iter” and “cpu” in Table 1, we can conclude that

*ϕ*_{3}*(µ, α) > ϕ*_{1}*(µ, α) > ϕ*_{2}*(µ, α)*

*where “>” means “better performance”. In Table 2, we compare Algorithm 4.1 with non-*
monotone line search and the smoothing-type algorithm with monotone line search studied
in [25]. Although the number that Algorithm 4.1 successfully solves every generated problem
is less than the one by the smoothing-type algorithm with monotone line search as afore-
mentioned in overall, the performance based on cpu time and iterations of our proposed
algorithm outperforms better than the other. This indicates that Algorithm 4.1 has some
advantages over the one with the monotone line search in [25].

*Another way to compare the performance of function ϕ*_{i}*(µ, α), i = 1, 2, 3, is via the so-*
called “performance profile”, which is introduced in [39]. In this means, we regard Algorithm
*4.1 corresponding to a smoothing function ϕ*_{i}*(µ, α), i = 1, 2, 3 as a solver, and assume that*
*there are n**s**solvers and n**p* test problems from the test set*P which is generated randomly.*

We are interested in using the iteration number as performance measure for Algorithm 4.1
*with diﬀerent ϕ**i**(µ, α). For each problem p and solver s, let*

*f*_{p,s}*= iteration number required to solve problem p by solver s.*

n fun suc iter cpu res

500 *ϕ*1 10 5.000 0.251 8.864e-09

500 *ϕ*2 10 7.800 1.496 2.600e-07

500 *ϕ*3 10 3.500 0.707 3.762e-07

1000 *ϕ*1 10 5.000 0.632 2.165e-08

1000 *ϕ*2 10 7.200 5.240 8.657e-08

1000 *ϕ*3 10 3.400 3.093 4.853e-07

1500 *ϕ*_{1} 9 5.000 1.224 1.537e-07

1500 *ϕ*_{2} 9 8.111 13.232 3.124e-07

1500 *ϕ*_{3} 9 4.222 8.781 2.706e-07

2000 *ϕ*1 10 5.000 2.145 1.599e-07

2000 *ϕ*2 10 7.700 24.130 2.234e-07

2000 *ϕ*3 10 4.200 16.925 1.923e-07

2500 *ϕ*_{1} 9 5.000 3.519 3.897e-08

2500 *ϕ*_{2} 9 6.889 34.849 2.016e-07

2500 *ϕ*3 9 4.000 27.870 1.479e-07

3000 *ϕ*1 10 5.000 5.161 9.769e-08

3000 *ϕ*2 10 8.300 69.723 1.714e-07

3000 *ϕ*3 10 4.100 45.891 1.608e-07

3500 *ϕ*1 7 5.000 7.415 2.226e-07

3500 *ϕ*2 7 7.857 102.272 4.037e-07

3500 *ϕ*3 7 4.429 75.068 2.334e-07

4000 *ϕ*_{1} 9 5.000 9.974 5.795e-08

4000 *ϕ*_{2} 9 6.444 106.850 3.132e-07

4000 *ϕ*_{3} 9 4.000 98.983 7.743e-08

4500 *ϕ*1 8 5.000 13.075 2.374e-07

4500 *ϕ*2 8 10.250 240.602 3.115e-07

4500 *ϕ*3 8 4.250 147.863 3.070e-07

*Table 1: Average performance of Algorithm4.1 for Example 5.1 (c = 0.01, σ = 10** ^{−5}*)

We employ the performance ratio

*r** _{p,s}*:=

*f*

_{p,s}min*{f**p,s**: s∈ S},*

where*S is the four solvers set. We assume that a parameter r**p,s**≤ r**M* *for all p, s are chosen,*
*and r**p,s* *= r**M* *if and only if solver s does not solve problem p. In order to obtain an overall*
assessment for each solver, we define

*ρ*_{s}*(τ ) :=* 1
*n**p*

size*{p ∈ P : r**p,s**≤ τ},*

*which is called the performance profile of the number of iteration for solver s. Then, ρ**s**(τ )*
*is the probability for solver s∈ S that a performance ratio f**p,s* *is within a factor τ* *∈ R of*
the best possible ratio.

We then need to test the three functions for Example 5.1. In particular, the random problems of each size are generated 50 times. In order to obtain an overall assessment for the

Non-monotone Monotone

n suc iter cpu res n suc iter cpu res

500 10 5.000 0.251 8.864e-09 500 10 5.500 0.289 4.905e-07

1000 10 5.000 0.632 2.165e-08 1000 10 5.500 0.616 7.184e-08

1500 9 5.000 1.224 1.537e-07 1500 9 6.000 1.466 4.654e-09

2000 10 5.000 2.145 1.599e-07 2000 10 6.500 2.866 3.151e-08 2500 9 5.000 3.519 3.897e-08 2500 10 6.000 4.477 4.320e-08 3000 10 5.000 5.161 9.769e-08 3000 10 6.500 7.348 1.743e-07 3500 7 5.000 7.415 2.226e-07 3500 10 8.000 11.957 5.674e-07 4000 9 5.000 9.974 5.795e-08 4000 10 7.000 14.875 2.166e-08 4500 8 5.000 13.075 2.374e-07 4500 10 7.000 19.204 2.433e-08

Table 2: Comparisons of non-monotone Algorithm 4.1 and monotone Algorithm in [25] for Example 5.1

three functions, we are interested in using the number of iterations as a performance measure
*for Algorithm 4.1 with ϕ*1*(µ, α), ϕ*2*(µ, α), and ϕ*3*(µ, α), respectively. The performance plot*
*based on iteration number is presented in Figure 5. From this figure, we also see that ϕ*3*(µ, α)*
*working with Algorithm 4.1 has the best numerical performance, followed by ϕ*4*(µ, α). In*
other words, in view of “iteration numbers”, there has

*ϕ*3*(µ, α) > ϕ*1*(µ, α) > ϕ*2*(µ, α)*
*where “>” means “better performance”.*

We are also interested in using the computing time as performance measure for Algorithm
*4.1 with diﬀerent ϕ**i**(µ, α), i = 1, 2, 3. The performance plot based on computing time is*
*presented in Figure 6. From this figure, we can also see the function ϕ*3*(µ, t) has best*
performance. In other words, in view of “computing time”, there has

*ϕ*_{3}*(µ, α) > ϕ*_{1}*(µ, α) > ϕ*_{2}*(µ, α)*
*where “>” means “better performance”.*

In summary, for the Example 5.1, no matter the number of iterations or the computing
*time is taken into account, the function ϕ*3*(µ, α) is the best choice for the Algorithm 4.1.*

**Example 5.2. Consider the system (1.5) with inequalities only, where x***∈ IR*^{5}, *K*^{5} =
*K*^{3}*× K*^{2} and

*f (x) :=*

*24(2x*_{1}*− x*2)^{3}*+ exp(x*_{1}*+ x*_{3})*− 4x*4*+ x*_{5}

*−12(2x*1*− x*2)^{3}*+ 3(3x*2*+ 5x*3*)/*√

*1 + (3x*2*+ 5x*3)^{2}*− 6x*4*− 7x*5

*−exp(x*1*− x*3*) + 5(3x*2*+ 5x*3*)/*√

*1 + (3x*2*+ 5x*3)^{2}*− 3x*4*+ 5x*5

*4x*1*+ 6x*2*+ 3x*3*− 1*

*−x*1*+ 7x*2*− 5x*3+ 2

*⪯*_{K}^{5} *0.*

This problem is taken from [17].

Figure 5: Performance profile of iteration numbers for Example 5.1.

Figure 6: Performance profile of computing time for Example 5.1.

Example 5.2 is tested 20 times for 20 random starting points. Similar to the case of Example 5.1, besides using Algorithm 4.1 to test Example 5.2, we have also tested it using the monotone smoothing-type algorithm in [25]. From Table 3, we see that there is no big diﬀerence regarding performance between these two algorithms for Example 5.2.

Moreover, Figure 7 shows the performance profile of iteration number in Algorithm
4.1 for Example 5.2 on 100 test problems with random starting points. The three solvers
*correspond to Algorithm 4.1 with ϕ*_{1}*(µ, α), ϕ*_{2}*(µ, α), and ϕ*_{3}*(µ, α), respectively. From this*
*figure, we see that ϕ*_{3}*(µ, α) working with Algorithm 4.1 has the best numerical performance.*

*followed by ϕ*_{2}*(µ, t). In summary, from the viewpoint of “iteration numbers”, we conclude*
that

*ϕ*3*(µ, α) > ϕ*2*(µ, α) > ϕ*1*(µ, α),*
*where “>” means “better performance”.*

Non-monotone Monotone

suc iter cpu res suc iter cpu res

20 13.500 0.002 5.835e-08 20 8.750 0.005 1.2510e-07

Table 3: Comparisons of non-monotone Algorithm 4.1 and monotone Algorithm in [25] for Example 5.2

Figure 7: Performance profile of iteration number for Example 5.2.

**Example 5.3. Consider the system of equalities and inequalities (1.5), where**
*f (x) :=*(

*f**I**(x)*^{T}*, f**E**(x)** ^{T}*)

*T*

*x∈ IR*^{6}*,*
with

*f**I**(x)* =

*−x*^{4}1

*3x*^{3}_{2}*+ 2x*2*− x*3*− 5x*^{2}3

*−4x*^{2}2*− 7x*3*+ 10x*^{3}_{3}

*−x*^{3}4*− x*5

*x*_{5}*+ x*_{6}

*⪯*_{K}^{5}=*K*^{3}*×K*^{2} *0,*

*f**E**(x)* = *2x*1*+ 5x*^{2}_{2}*− 3x*^{2}3*+ 2x*4*− x*5*x*6*− 7.*

**Example 5.4. Consider the system of equalities and inequalities (1.5), where**
*f (x) :=*(

*f*_{I}*(x)*^{T}*, f*_{E}*(x)** ^{T}*)

*T*

*x∈ IR*^{6}*,*
with

*f**I**(x)* =

*−e*^{5x}^{1}*+ x*2

*x*2*+ x*^{3}_{3}

*−3e*^{x}^{4}
*5x*5*− x*6

* ⪯*^{K}^{4}^{=}^{K}^{2}^{×K}^{2} *0,*

*f**E**(x)* =

[ *3x*1*+ e*^{x}^{2}^{+x}^{3}*− 2x*4*− 7x*5*+ x*6*− 3*
*2x*^{2}_{1}*+ x*2*+ 3x*3*− (x*4*− x*5)^{2}*+ 2x*6*− 13*

]

*= 0.*

**Example 5.5. Consider the system of equalities and inequalities (1.5), where**

*f (x) :=*(

*f*_{I}*(x)*^{T}*, f*_{E}*(x)** ^{T}*)

*T*

*x∈ IR*^{7}*,*
with

*f**I**(x)* =

*3x*^{3}_{1}
*x*2*− x*3

*−2(x*4*− 1)*^{2}
*sin(x*5*+ x*6)
*2x*6*+ x*7

*⪯*_{K}^{5}=*K*^{2}*×K*^{3} *0,*

*f*_{E}*(x)* =

[ *x*_{1}*+ x*_{2}*+ 2x*_{3}*x*_{4}*+ sin x*_{5}*+ cos x*_{6}*+ 2x*_{7}
*x*^{3}_{1}*+ x*2+√

*x*^{2}_{3}*+ 3 + 2x*4*+ x*5*+ x*6*+ 6x*7

]

*= 0.*

Exam fun suc *c* *σ* iter cpu res

5.2 *ϕ*1 20 5 *0.02* *13.500* *0.002* 5.835e-08

5.2 *ϕ*_{2} 20 5 *0.02* *8.450* *0.001* 5.134e-07

5.2 *ϕ*_{3} 20 5 *0.02* *8.600* *0.002* 2.260e-07

5.3 *ϕ*1 20 1 *0.02* *21.083* *0.009* 8.165e-07

5.3 *ϕ*2 17 1 *0.02* *14.647* *0.001* 2.899e-07??

5.3 *ϕ*3 17 1 *0.02* *18.529* *0.002* 7.167e-07

5.4 *ϕ*_{1} 20 *0.5* *0.002* *46.750* *0.033* 1.648e-07

5.4 *ϕ*_{2} 2 *0.5* *0.002* *420.000* *0.499* 9.964e-07

5.4 *ϕ*_{3} 0 *0.5* *0.002* Fail Fail Fail

5.5 *ϕ*1 20 *0.1* *0.002* *14.250* *0.009* 6.251e-07

5.5 *ϕ*2 20 *0.1* *0.002* *13.250* *0.001* 6.532e-07

5.5 *ϕ*3 20 *0.1* *0.002* *12.650* *0.001* 6.016e-07

Table 4: Average performance of Algorithm4.1 for Examples 5.2-5.5

Table 4 shows the numerical results including three smoothing functions (fun) used to
solve the problems, the number (suc) that Algorithm 4.1 successfully solves every generated
*problem, the parameters c and σ, the average iteration numbers (iter), the average CPU*
time (cpu) in seconds and the average residual norm*∥H(z)∥ (res) for Examples 5.2-5.5 with*
random initializations, respectively. Performance profiles are provided as below.

Figure 8 and Figure 9 are the performance profiles in terms of iteration number for
Example 5.3 and Example 5.5. From the Figure 8, we see that although the best probability
*of the function ϕ*3 is lower, but the ratio that can be solved in a large number of problems
is higher than that of the other two. In this case, the diﬀerence between the three functions
*is not obvious. From the Figure 9, we can also see the function ϕ*3has best performance.

In summary, below are our numerical observations and conclusions.

1. The Algorithm 4.1 is eﬀective. In particular, the numerical results show that our proposed method is better than the algorithm with monotone line search studied in [25]

when solving the system of inequalities under the order induced by second-order cone.

Figure 8: Performance profile of iteration number for Example 5.3.

Figure 9: Performance profile of iteration number for Example 5.5.

*2. For Examples 5.1 and 5.2, the function ϕ*3 outperforms much better than the others.

For the rest problems, the diﬀerence of their numerical performance is very marginal.

3. For future topics, it is interesting to discover more eﬃcient smoothing functions and to apply the type of SOC-functions to other optimization problems involved second-order cones.

**References**

*[1] F. Alizadeh, D. Goldfarb, Second-order cone programming, Math. Program. 95 (2003)*
3–51.