A New Smoothing Method for Mathematical Programs with Complementarity Constraints Based on Logarithm-Exponential Function

We present a new smoothing method based on a logarithm-exponential function for mathematical program with complementarity constraints (MPCC). Different from the existing smoothing methods available in the literature, we construct an approximate smooth problem of MPCC by partly smoothing the complementarity constraints. With this new method, it is proved that the Mangasarian-Fromovitz constraint qualification holds for the approximate smooth problem. Convergence of the approximate solution sequence, generated by solving a series of smooth perturbed subproblems, is investigated. Under the weaker constraint qualification MPCC-Cone-ContinuityProperty, it is proved that any accumulation point of the approximate solution sequence is a M-stationary point of the original MPCC. Preliminary numerical results indicate that the developed algorithm based on the partly smoothing method is efficient, particularly in comparison with the other similar ones.

Among them, the smoothing method is one of the most popular approaches, which uses a smoothing function to approximate the complementarity constraints in (1).As a result, the original MPCC is reformulated into a standard smooth optimization model.Then, an approximate solution of MPCC is obtained by solving a series of smooth perturbed subproblems.Therefore, it is often necessary to prove theoretically that the sequence of approximate solutions converges to a stationary point (or an optimal solution) of the original MPCC.
Very recently, in [31], a partially smoothing Jacobian method is proposed for solving nonlinear complementarity problems with  0 function.Numerical experiments have shown that this smoothing method outperforms the existent ones, particularly in comparison with the state-of-the-art methodsderivedfromtheclassical Fischer Burmeister smoothing function and aggregation function.
Inspired by the idea of partly smoothing in [31], we intend to construct an approximate problem of (1) by partly smoothing the complementarity constraints in (1) such that the degree of approximation is improved between MPCC (1) and the constructed smooth problem.Specifically, in the existing results (see, for example, [23][24][25][26][27][28][29][30]), the complementarity constraints () ≥ 0, () ≥ 0, ()  () = 0 are often wholly approximated by a system of smooth equations with a perturbation parameter.In contrast, we construct an approximate problem of (1) only by replacing ()  () = 0 with a system of inequalities such that (1) is transformed into a standard smooth optimization problem.Consequently, under a weaker constraint qualification, called the MPCC-Cone-Continuity Property (MPCC-CCP), we can prove theoretically that any accumulation point of the approximate solution sequence is M-stationary to the original MPCC.Numerical experiments will be employed to show the efficiency of the proposed smoothing method, particularly in comparison with the other similar methods available in the literature.
The rest of the paper is organized as follows.In next section, we first review some concepts of nonlinear programming and MPCC; then we present a new smoothing method of Problem (1).Section 3 is devoted to establishment of convergence theory.In Section 4, numerical performance of the new method is reported.Final remarks are made in the last section.
Throughout this paper,   represents the -th component of a vector  and similar notations are used for vector-valued functions.ϝ denotes the feasible region of Problem (1).For a function  :   →   and a given vector  ∈   ,   () stands for the active index set of  at , i.e., { :   () = 0} for all  ∈   ().For a given vector , supp() ≜ { :   ̸ = 0} denotes the support set of .

Preliminaries and New Smoothing Approach
In this section, some basic concepts will be first stated, which are necessary to the development of a new smoothing method.Then, we will propose a new smoothing method of Problem (1).
A typical mathematical model of nonlinear programming (NLP) problems is expressed as follows: min  () where  :   → ,  :   →   , ℎ :   →   are all continuously differentiable functions.
Denote  the feasible region of Problem (2).Stationary points in  play a fundamental role in finding a minimizer of (2).Definition 1 (see [32]).A point  * ∈  is called a stationary point of Problem (2) if there exist  ∈   + and  ∈   such that ( * , , ) ∈  ++ is a KKT point of (2); that is to say where  and  are called the vectors of multipliers.Since it is usually not possible to solve the NLPs exactly, mostly of the standard NLPs method stops when the KKT conditions are satisfied approximately.Thus, approximate KKT point is very necessary.
The following result has been proved in [33].
Recently Andreani et al. [34] introduced a new CQ called Cone-Continuity Property (CCP) intimately related to the AKKT condition.
Definition 6 (see [34]).A feasible point  * of ( 2) is said to satisfy the CCP if the set-valued mapping  Î () such that is outer semicontinuous (Definition 5.4 [35] )at  * ; that is, lim sup It has been shown that CCP is strictly stronger than ACQ and weaker than CRSC in [34].
Next, we will extend the above concepts and results from NLP to MPCC.For an arbitrary feasible point  of (1), we first define the following index sets.
Similar to Definitions 1 and 2, we give definitions of different stationary points for the MPCC.
Definition 7 (see [19]).Let  * be a feasible point of Problem (1).Then, (a)  * is said to be W-stationary if there exist multiplier vectors  ∈   and ,  ∈   such that (b)  * is said to be M-stationary, if it is W-stationary and Definition 8 (see [21]).Let  * be a feasible point of Problem (1);  * is called a MPCC-AKKT point if there are sequences where supp( A definition of MFCQ for MPCC is presented similar to Definition 3. Definition 9 (see [19]).A feasible point  * of ( 1) is said to satisfy MPCC-MFCQ if and only if the vectors are linearly independent and there exists a vector  ∈   such that The following result holds which is similar to Lemma 5.
Andreani [21] extended the definition of CCP from nonlinear programming to MPCC.

Mathematical Problems in Engineering
Definition 11 (see [21]).A feasible point  * of ( 1) is said to satisfy MPCC-Cone-Continuity Property (MPCC-CCP) if and only if the set-valued mapping  Î   () such that is outer semicontinuous (Definition 5.4 [35] ) at  * ; that is, lim sup In [21], it has been shown that MPCC-CCP is strictly weaker than MPCC-RCPLD and MPCC-CCP implies the MPCC-Abadie CQ under certain assumption.Furthermore, MPCC-CCP is also independent of MPCC-quasinormality and MPCC-pseudonormality.The following lemma shows the relationship between MPCC-CCP and MPCC-AKKT.
In the end of this section, we come to propose a new smoothing method of Problem (1).
We first note that  ≥ 0,  ≥ 0,  = 0 (22) can be written as Clearly, ( 23) is equivalent to Since we obtain an equivalent form of ( 22): More generally, we set where   :  →  is continuously differentiable.Then,  can be approximated by a logarithm-exponential function [36]: The following result presents some nice properties of the logarithm-exponential function.
(b) For any  ∈   and  > 0, it holds that On the basis of Lemma 13, we approximate max{, } by the following logarithm-exponential function: Then, it is natural that for the following complementarity constraints: we can approximate ()  () = 0 in (31) by the following system of inequalities: where Φ  :   →   is given by . . .
Since Problem ( 34) is a standard smooth optimization problem, many powerful optimization techniques can be directly applied to solve it (see [37][38][39][40]).Remark 14.Unlike the existing smoothing methods, it is noted that (34) is obtained only by partly smoothing the complementarity constraints (31).
As an approximate problem of the MPCC (1) with a perturbation parameter , a critical issue should be addressed that concerns what is the relation between the optimal solutions of ( 34) and (1).Therefore, our next focus in this paper is to prove theoretically that the solution of the perturbed problem (34) tends to an optimal solution of (1) as  ↓ 0.

Convergence Analysis and Development of Algorithm
In this section, we will investigate the limiting behavior of stationary points of the perturbed subproblems.
We first study the constraint qualification of (34).
(2) The gradient of  , is calculated by ( (3) ) Let  * be feasible for Problem (1).If  ∈  +0 ( * ), then  Proof.From the definition of  , , it is easy to see that the first result holds.By direct calculation, we obtain the second result.The third is directly from the second one.It remains to prove the last result.
As  ∈  +0 ( * ), we write and respectively.Thus, it is easy to see that → 1 as  →  * and  ↓ 0. In a similar way, we can prove that In virtue of Lemmas 10 and 15, we now prove that Problem (34) satisfies some constraint qualification under mild conditions.
The following result establishes the relations between the optimal solutions of the original problem and that of the perturbed subproblem under the constraint qualification of MPCC-CCP.Theorem 17.Let {  } be a positive sequence which is convergent to zero as  → ∞.Suppose that {  } is a sequence, generated by the stationary points of the smooth problem (34) with perturbation parameter  =   .If  * is an accumulation point of the sequence {  } and MPCC-CCP holds at  * and { :   (  ) > 0,   (  ) > 0} ∩ (  ) = ⌀, then  * is an M-stationary point of the original MPCC (1).
Proof.From Lemma 12, it follows that we only need to show that  * is an MPCC-AKKT point.From Definition 8, it will be sufficient to show that there is subsequence {  } which is an MPCC-AKKT point subsequence.Since {  } is a stationary point sequence generated by the smooth problem (34) with perturbation parameter  =   , there exist Lagrangian multiplier vectors   ,   , and   such that and supp (  ) ⊆   (  ) ,   ≥ 0, Clearly, (51) can be rewritten as Since supp (  ) ∩ supp (  ) = 0, we can write and Then, (51) is equivalent to First, we will show that    = 0,  ∈  +0 ( * ) for enough large k.Since   (  ) ≥ 0,   (  ) ≥ 0, we decompose  +0 ( * ) into four subsets.
On the basis of Theorems 16 and 17, we now develop an implementable algorithm to solve the original MPCC (1) before the end of this section.
Step 2. Let   be the current parameter.Solve the following problem: min  () ) . (61) The optimal solution is referred to as  * . Step

Numerical Results
In this section, we investigate the numerical behavior of Algorithm 18.We compare Algorithm 18 with a similar algorithm developed by Facchinei et al. in [23] as they are used to solve the same test problems.All the test problems are from [23,42].The solution tolerance   is set to be 10 −3 for the problems 8(f)-8(j).For the other test problems, the solution tolerance   takes 10 −6 .As done in [23], the initial perturbation parameter is set to be 1, and  = 0.1 for reduction of the perturbation parameter.The corresponding computer procedures in MATLAB run in the following computer environment: 2.20GHz CPU, 1.75GB memory based operation system of Windows 7.
Numerical efficiency of the two algorithms is reported in Tables 1 and 2. For each algorithm, the optimal value of the objective function, the optimal solution, the number of iterations, and the achieved termination condition are recorded for evaluating the numerical performance.The used notations in Tables 1 and 2 are listed as follows.
Prob: the test problems; f  /f  : the optimal function value obtained by the algorithm in [23]/the optimal function value by Algorithm 18; x *  /x *  : the optimal solution obtained by the algorithm in [23]/the optimal solution obtained by Algorithm 18; k  /k  : the number of iterations as the algorithm in [23] stops/the number of iterations as Algorithm 18 stops; maxkio  /maxkio  : the maximal degree of constraint violation as the algorithm in [23] stops at the optimal solutions  * /the maximal degree of constraint violation as Algorithm 18 stops at the optimal solutions  * .
As done in [23], only the -part of the optimal solution  * is shown in Table 1.In Table 2, for the optimal solutions  * found by the algorithm in [23] and Algorithm 18, only the first component of  * is shown in x *  /x *  .From Tables 1 and 2, it is clear that (1) Algorithm 18 can obtain the same optimal function value for the other test problems as the algorithms in [23] except for the problem: Scholtes 5.For this test problem, our algorithm gets smaller (better) value of the objective function than the algorithm in [23].(2) By the two algorithms, almost the same optimal solutions have been obtained for all the test problems.(3) As an impressive performance, Algorithm 18 costs smaller number of iterations with higher accuracy in finding out the optimal solution.Actually, for the 41 ones out of all the 44 test problems, the number of iterations is smaller than that of the algorithm in [23].( 4) With regard to the termination conditions, for the 33 ones out of the 44 test problems, Algorithm 18 has less degree of constraint violation at the optimal solution than that of the algorithm in [23].
The numerical results in Tables 1 and 2 demonstrate that Algorithm 18 outperforms the algorithms in [23], and the proposed partly smoothing method in this paper is promising in solving MPCC.

Final Remarks
Different from the existing smoothing methods available in the literature, we have proposed a partly smoothing method based on the logarithm-exponential function for the mathematical programs with complementarity constraints.It has been proved that Mangasarian-Fromovitz constraint qualification holds for the constructed approximate smooth problem in our method.Under the weaker constraint qualification

Table 1 :
Comparison between different algorithms.

Table 2 :
Comparison between different algorithms (continued).CCP, it has been proved that any accumulation point of the approximate solution sequence is an M-stationary point of the original MPCC.Preliminary numerical results have demonstrated that the proposed smoothing method is more efficient than the other similar ones.