A Two-Parametric Class of Merit Functions for the Second-Order Cone Complementarity Problem

We propose a two-parametric class of merit functions for the second-order cone complementarity problem (SOCCP) based on the one-parametric class of complementarity functions. By the new class of merit functions, the SOCCP can be reformulated as an unconstrainedminimization problem.The new class ofmerit functions is shown to possess some favorable properties. In particular, it provides a global error bound if F andG have the joint uniformCartesianP-property. And it has bounded level sets under aweaker condition than the most available conditions. Some preliminary numerical results for solving the SOCCPs show the effectiveness of the merit function method via the new class of merit functions.

Recently great attention has been paid to the SOCCP, since it has a variety of engineering and management applications, such as filter design, antenna array weight design, truss design, and grasping force optimization in robotics [1,2].Furthermore, the SOCCP contains a wide class of problems, such as nonlinear complementarity problems (NCP) and second-order cone programming (SOCP) [3,4].For example, the SOCCP with  1 =  2 = ⋅ ⋅ ⋅ =   = 1 and () =  for any  ∈   is the NCP, and the KKT conditions for the SOCP reduce to the SOCCP.
(3) Such a  is called a merit function for the SOCCP.Thus the SOCCP is equivalent to the following unconstrained smooth (global) minimization problem: min ∈   ( () ,  ()) .
A popular choice of  is the Fischer-Burmeister (FB) merit function  FB (, ) := 1 2      FB (, )     2 , ( where  FB :   ×   →   is the vector-valued FB function defined by with  2 =  ∘  denoting the Jordan product between  and itself and √ being a vector such that (√) 2 = .The function  FB is shown to be a merit function for the SOCCP in [14].
In this paper, we consider the two-parametric class of merit functions defined by   1 , 2 (, ) :=  1  0 (, ) +   2 (, ) , (7) where  0 :   ×   →  + and   :   ×   →  + are given, respectively, by (, ) := 1 2       (, ) with (⋅) + denoting the metric projection on the second-order cone ,  1 > 0 and  2 ∈ (0, 4).Here   :   →   is the one-parametric class of SOC complementarity functions [16] defined by where  ∈ (0, 4) is an arbitrary but fixed parameter.When  = 2,   reduces to the vector-valued FB function given by (6), and as  → 0, it becomes a multiple of the vector-valued residual function Thus, the one-parametric class of vector-valued functions (10) covers two popular second-order cone complementarity functions.Hence the two-parametric class of merit functions defined as ( 7)-( 10) includes a broad class of merit functions.We will show that the SOCCP can be reformulated as the following unconstrained smooth (global) minimization problem: If  1 = 1 and  2 = 2, the function  in (12) induced by the new class of merit functions   1 , 2 reduces to [17] fLT () :=  0 ( () ,  ()) + 1 2      FB ( () ,  ()) with  0 given as (8).It has been shown that fLT provides a global error bound if  and  are jointly strongly monotone, and it has bounded level sets if  and  are jointly monotone and a strictly feasible solution exists [17].In contrast, the merit function  FB lacks these properties.Motivated by these works, we aim to study the twoparametric class of merit functions for the SOCCP defined as ( 7)- (10) and its favorable properties in this paper.We also prove that the class of merit functions provides a global error bound if  and  have the joint uniform Cartesian property, which will play an important role in analyzing the convergence rate of some iterative methods for solving the SOCCP.And it has bounded level sets under a rather weak condition, which ensures that the sequence generated by a descent method has at least one accumulation point.
The organization of this paper is as follows.In Section 2, we review some preliminaries including the Euclidean Jordan algebra associated with SOC and some results about the one-parametric class of SOC complementarity functions.In Section 3, based on the one-parametric class of SOC complementarity functions, we propose a two-parametric class of merit functions for the second-order cone complementarity problem (SOCCP), which is shown to possess some favorable properties.In Section 4, we show that the class of merit functions provides a global error bound if  and  have the joint uniform Cartesian -property, and it has bounded level sets under a rather weak condition.Some preliminary numerical results are reported in Section 5.And we close this paper with some conclusions in Section 6.
In what follows, we denote the nonnegative orthant of  by  + .We use the symbol ‖ ⋅ ‖ to denote the Euclidean norm defined by ‖‖ := √    for a vector  or the corresponding induced matrix norm.For simplicity, we often use  = ( 1 ;  2 ) for the column vector  = ( 1 ,   2 )  .For the SOC   , int   , and bd   mean the topological interior and the boundary of   , respectively.

Preliminaries
In this section, we recall some preliminaries, which include Euclidean Jordan algebra [3,18] associated with the SOC  and some results used in the subsequent analysis.
Without loss of generality, we may assume that  = 1 and  =   in Sections 2 and 3.
First, we recall the Euclidean Jordan algebra associated with the SOC and some useful definitions.The Euclidean Jordan algebra for the SOC   is the algebra defined by with  = (1, 0, . . ., 0) ∈   being its unit element.Given an element  = ( 1 ;  2 ) ∈  ×  −1 , we define where  represents the ( − 1) × ( − 1) identity matrix.It is easy to verify that  ∘  = () for any  ∈   .Moreover, () is symmetric positive definite (and hence invertible) if and only if  ∈ int   .Now we give the spectral factorization of vectors in   associated with the SOC   .Let  = ( 1 ;  2 ) ∈  ×  −1 .Then  can be decomposed as where  1 ,  2 and  (1) ,  (2) are the spectral values and the associated spectral vectors of  given by for  = 1, 2, with any  ∈  −1 such that ‖‖ = 1.It is obvious that ‖ (1) ‖ = ‖ (2) ‖ = 1/ √ 2. By the spectral factorization, a scalar function can be extended to a function for the SOC.
The following results, describing the special properties of the function   given as (10), will play an important role in the subsequent analysis.

A Two-Parametric Class of Merit Functions
In this section, we study the two-parametric class of merit functions   1 , 2 given by ( 7)- (10).As we will see,   1 , 2 has some favorable properties.The most important property is that the SOCCP can be reformulated as the global minimization of the function () given as (12).Moreover, the function  provides a global error bound and bounded level sets under weak conditions, which will be shown in the next section.
Proof.By following the proof of Lemma 4.1 [16] and using Proposition 6, we can show that the desired results hold.

Error Bound and Bounded Level Sets
By Proposition 8, we see that the SOCCP is equivalent to the global minimization of the function ().In this section, we show that the function  provides a global error bound for the solution of the SOCCP and has bounded level sets, under rather weak conditions.
In this section, we consider the general case that  ⊂   is the Cartesian product of SOCs; that is,  =   1 ×   2 × ⋅ ⋅ ⋅ ×    with ,  1 , . . .,   ≥ 1,  =  1 +  2 + ⋅ ⋅ ⋅ +   .Thus, we obtain and therefore the results in Sections 2 and 3 can be easily extended to the general case.
First, we discuss under what condition the function  provides a global error bound for the solution of the SOCCP.To this end, we need the concepts of Cartesian -properties introduced in [21] for a nonlinear transformation, which are natural extensions of the -properties on Cartesian products in   established by Facchinei and Pang [22].Recently, the Cartesian -properties are extended to the context of general Euclidean Jordan algebra associated with symmetric cones [20].Definition 9.The mappings  = ( 1 , . . .,   ) and  = ( 1 , . . .,   ) are said to have (i) the joint uniform Cartesian -property if there exists a constant  > 0 such that, for every ,  ∈   , there is an index  ∈ {1, 2, . . ., } such that ⟨  () −   () ,   () −   ()⟩ ≥       −      2 ; (54) (ii) the joint Cartesian -property if, for every ,  ∈   with  ̸ = , there is an index  ∈ {1, 2, . . ., } such that Now we show that the function  provides a global error bound for the solution of the SOCCP if  and  have the joint uniform Cartesian -property.Proposition 10.Let  be given by ( 7)-( 10) and (12).Suppose that  and  have the joint uniform Cartesian -property and the SOCCP has a solution  * .Then there exists a constant  > 0 such that, for any  ∈   , ) () and hence (  ) → +∞.This contradicts the fact that {  } ⊆   (γ).
It should be noted that Condition 11 is rather weak to guarantee the boundedness of level sets of .As far as we know, the weakest condition available to ensure the boundedness of level sets is the following condition given by [20].
Condition 13 (see [20]).For any sequence {  } ⊆     [20].Hence they all implies Condition 11, since the SCCP includes the SOCCP.Therefore, Condition 11 is a weaker condition than the most available conditions to guarantee the boundedness of level sets.and Gap denotes the (average) value of |⟨(), ()⟩| when the algorithm terminates.We solve the linear SOCCPs of different dimensions with size  from 50 to 1000 and  = 1.The random problems of each size are generated 10 times, and the test results with different parameters  1 > 0 and  2 ∈ (0, 4) are listed in Tables 1, 2, and 3. From the results of these tables, we give several observations.(i) All the random problems have been solved in very short CPU time.
(ii) The problem size slightly affects the number of iterations.
(iii) For the same dimension of linear SOCCPs, choices of different parameters  1 > 0 and  2 ∈ (0, 4) generally do not affect the number of iterations and the CPU time.

Conclusions
In this paper, we have studied a two-parametric class of merit functions for the second-order cone complementarity problem based on the one-parametric class of complementarity functions.The new proposed class of merit functions includes a broad class of merit functions, since the one-parametric class of complementarity functions is closely related to the famous natural residual function and Fischer-Burmeister function.The new class of merit functions has been shown to possess some favorable properties.In particular, it provides a global error bound if  and  have the joint uniform Cartesian -property.And it has bounded level sets under a weaker condition than the most available conditions [20,23].

Table 3 :
Numerical results for SOCCPs with ( 1 ,  2 ) = (10, 3.5).Some preliminary numerical results for solving the SOCCPs show the effectiveness of the merit function method via the new class of merit functions.