Abstract

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-Continuity Property, 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.

1. Introduction

Consider the following mathematical program with complementarity constraints (MPCC):where , , , and , are all continuously differentiable functions. MPCC (1) stems from many fields, such as shape design, economic equilibrium, and multilevel game (see [1, 2]). In the past two decades, it has attracted great interest of research from applied mathematicians and engineers. One of the main challenges in studying such a problem is that the constraints in (1) may fail to satisfy some standard constraint qualification at a feasible point. Thus, many fundamental theoretical results and powerful algorithms for an ordinary smooth optimization problem cannot be directly employed to solve (1). Actually, some specific approaches have been proposed for solving MPCC (1), such as the sequential quadratic programming approach in [3–8], the interior point methods in [9, 10], the penalty approach in [11–13], the lifting method in [14], the relaxation approach in [15–22], and the smoothing methods in [23–30].

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 function. Numerical experiments have shown that this smoothing method outperforms the existent ones, particularly in comparison with the state-of-the-art methods derived from the classical 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, [23–30]), the complementarity constraints , , 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 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. denotes the feasible region of Problem (1). For a function and a given vector , stands for the active index set of at , i.e., for all . For a given vector , denotes the support set of .

2. 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:where , , 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 saywhere 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.

Definition 2. A point is called an Approximate Karush-Kuhn-Tucker (AKKT) point of Problem (2) if there are sequences , and , where , such that

The well-known Karush-Kuhn-Tucker theorem declares that a local minimizer of Problem (2) is a stationary point if some constraint qualification holds. Linearly independent constraint qualification (LICQ) and Mangasarian-Fromovitz constraint qualification (MFCQ) are the most popular constraint qualifications.

Definition 3. A feasible point of (2) is said to satisfy the MFCQ if the gradients are linearly independent and there exists a vector such that

In [33], the above MFCQ is described by a concept of positive linearly dependent.

Definition 4 (see [33]). A finite set of vectors is said to be positive linearly dependent if there exists such that for all , , andConversely, if (6) holds if and only if , then the group of vectors is called to be positive linearly independent.

The following result has been proved in [33].

Lemma 5 (see [33]). A point satisfies the MFCQ if and only if the gradientsare positive-linearly independent.

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 such that

is outer semicontinuous (Definition 5.4 [35] )at ; that is,

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 such thatwhere , either , or ,  .

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 vectorsare linearly independent and there exists a vector such that

The following result holds which is similar to Lemma 5.

Lemma 10 (see [19]). A feasible point of Problem (1) satisfies MPCC-MFCQ if and only if the gradientsare positive-linearly independent.

Andreani [21] extended the definition of CCP from nonlinear programming to MPCC.

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 such thatis outer semicontinuous (Definition 5.4 [35] ) at ; that is,

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.

Lemma 12 (see [21]). MPCC-CCP holds at if and only if MPCC-AKKT point is an M-stationary point.

In the end of this section, we come to propose a new smoothing method of Problem (1).

We first note thatcan be written asClearly, (23) is equivalent toSincewe obtain an equivalent form of (22):More generally, we setwhere 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.

Lemma 13 (see [36]). Let be defined by (28). Suppose that , , are continuously differentiable. Then,
(a) is increasing with respect to , and .
(b) For any and , it holds that

On the basis of Lemma 13, we approximate by the following logarithm-exponential function:Then, it is natural that for the following complementarity constraintswe can approximate in (31) by the following system of inequalities:where is given byConsequently, the original MPCC (1) is approximated by the following new smooth optimization problem:We denote the feasible region of Problem (34) by . The Lagrange function of Problem (34) is as follows:where .

Since Problem (34) is a standard smooth optimization problem, many powerful optimization techniques can be directly applied to solve it (see [37–40]).

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 .

3. 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).

Lemma 15. Let be defined by (33). Then, (1) for all , is continuously differentiable and is decreasing with respect to .
(2) The gradient of is calculated bywhere(3) .
(4) Let be feasible for Problem (1). If , then , as and . If , then, , as and .

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 , we write and asandrespectively. Thus, it is easy to see that , as and . In a similar way, we can prove that , (, ) in the case that .

In virtue of Lemmas 10 and 15, we now prove that Problem (34) satisfies some constraint qualification under mild conditions.

Theorem 16. Let be a feasible point of Problem (1) such that MPCC-MFCQ is satisfied at . Then, there exist a neighborhood of and a sufficiently small such that Problem (34) satisfies the standard MFCQ at any point for any .

Proof. Since are all continuous, there exist a neighborhood and a positive constant such that for any and any point , we haveFor convenience of discussion, we denoteWe will first prove that the following equalitiesare true.
Actually, if , then andTherefore, . It says that .
In a similar fashion, we can prove that .
Noting that MPCC-MFCQ is satisfied at for Problem (1), we conclude from Lemma 10 that the following gradientsare positive-linearly independent.
Owing to the factsit follows from the result (4) in Lemma 15 that , ) and , ) as and .
Similar to the proof of Proposition 2.2 in [41], it is concluded that there are a neighborhood and a sufficiently small such that for all with , the vectorsare positive-linearly independent.
We now claim that if , then the standard MFCQ holds for Problem (34), where and .
Take . In view of Lemma 5, we should show thatif and only if all the multiplier vectors, , , , and are null ones. To see this, we rewrite (47) asFrom (46) and (48), it follows thatTaking into account for all , it yieldsSince ( and (, we know   (. Thus, (47) holds. The proof is completed.