Homotopy method for a class of multiobjective optimization problems with equilibrium constraints

In this paper, we present a combined homotopy interior point method for solving multiobjective programs with equilibrium constraints. Under suitable conditions, we prove the existence and convergence of a smooth homotopy path from almost any interior point to a solution of the K-K-T system. Numerical results are presented to show the effectiveness of this algorithm.


1.
Introduction. Given functions f : R n+m → R p , g : R n+m → R s , G : R n+m → R l , and F : R n+m → R m . In this paper, we are devoted to the study of the multiobjective optimization problems with equilibrium constraints (MOPECs) in finite dimensional spaces defined as follows: where Z = {(x, y) ∈ R n+m : g(x, y) ≤ 0} is a nonempty closed convex set. For x ∈ X, S(x) is the solution set of a parametric variational inequality problem(PVI) where C(x) = {y ∈ R m : G(x, y) ≤ 0}, X = {x ∈ R n : (x, y) ∈ Z, f or some y ∈ R m }. Throughout this paper, we suppose that f , g and F are triply continuously differentiable and that G is sufficiently smooth. The feasible set of the MOPECs is denoted by E, which is nonempty in this paper.
In practice, many problems can be formulated as MOPECs (1), see reference [27]. It has extensive applications in practical areas such as traffic control, economic modeling, engineering design, and so on. When the number of the objects reduces to one, this kind of problem is the mathematical programs with equilibrium constraints (MPEC). It is well known that the constraints of MPEC destroy the convexity, connectedness and closeness of the feasible region [17]. Meanwhile, MOPECs has the complexity caused by multiple objects. Then, MOPECs is generally difficult to deal with.
During the past two decades, many researchers have paid attentions to the mathematical programs with equilibrium constraints. There were many literatures about basic theory, various effective algorithms and applications of the MPEC [4, 7-9, 12, 15, 20]. However, the papers about MOPECs were relatively few. They mainly studied the necessary optimization of the MOPECs and the constraints qualification of problem were abstract functions. In 2003, Ye and Zhu derived Fritz John type necessary optimality conditions and lead to Kuhn-Tucker type necessary optimality conditions under various constraint qualifications [27]. In 2007, Bao et al. used modern tools of variational analysis and generalized differentiation to gain the verified necessary optimality conditions for general problems [2]. In 2009, Boris S. Mordukhovich revealed a new Fredholm constraint qualification and established new qualified necessary optimality conditions for broad classes of the MOPECs in finite-dimensional and infinite-dimensional spaces [21].
In this paper, we solve the multiobjective optimization problems with equilibrium constraints by combined homotopy method. The homotopy method established by Kellogg et al. [11], Smale [23] and Chow et al. [3] is a powerful tool for solving nonlinear problems, fixed point problems and complementarity problems (see [14,16,26] and references therein). The advantage of this method is global convergence under certain weak assumptions and it was widely applied for solving other mathematic problems. In [24,29], the authors respectively solved multiobjective programming problem by homotopy method. Otherwise, the homotopy method was also used to deal with variational inequality problems [5,25,30]. In 2007, Li considered to solve MPEC by homotopy method in [13]. In this paper, we solve the multiobjective problem with general equilibrium constraints. Under certain assumptions, we get the equivalence formulation of the MOPECs (1). Then, under some weak conditions, the existence and convergence of a smooth homotopy path from almost any initial interior point to a solution of the KKT system of equivalence problem.
The rest of this paper is organized as follows. In Section 2, we reformulate MOPECs as a general multiobjective programming under suitable assumptions. In Section 3, we recall some definitions and properties. In Section 4, we construct a new combined homotopy mapping and prove the existence and the convergence of a smooth homotopy path from almost any interior initial point to a K-K-T point of MOPECs under some assumptions. In Section 5, we give two numerical results to show the effectiveness of this algorithm. In Section 6, the conclusions are given.

Preparation.
In what follows we consider the follower's parametric variational problem:

Assumption 1.
(A1) For each x ∈ X and i = 1, · · · , l, G i (x, ·) is a convex function in the second argument.
Lemma 2.1. Let F be a twice continuously differentiable, G i (i ∈ {i, · · · , l}) be triply continuously differentiable. Suppose that each G i (i ∈ {1, · · · , l}) is convex function in the second argument. Then y ∈ S(x) if and only if there exists a unique u ∈ R l + such that where U = diag(u), (3) is called the K-K-T system of problem (2) (see [10]).
We denote the multipliers u ∈ R l + satisfying (3) by M (x, y). Let us write an explicit formulation for the set : where I(x, y) = {i : G i (x, y) = 0}. It is easy to show that multiplier map M is closed by applying to a limit argument to the K-K-T system (3). where In order to introduce the following lemmas, we need to introduce the sequentially bounded constraint qualification (SBCQ) and the Mangasarian-Fromovitz constraint qualification (MFCQ) for the MOPECs (1), are (SBCQ): For any convergent sequence (x (k) , y (k) ) ⊂ E, there exists for each k multiplier vector u (k) ∈ M (x (k) , y (k) ) and u (k) is bounded; (MFCQ): The MFCQ is said to hold at a vector y ∈ C(x) if there exists a vector v ∈ R l such that v T ∇ y g i (x, y) < 0, f oralli ∈ I(x, y).

Proposition 1. [17]
Let F , g be twice continuously differentiable and each G i (i ∈ {1, · · · , l}) be triply continuously differentiable; let Z be closed. If the MFCQ holds at all pairs (x, y) ∈ E, then the SBCQ holds on E.
Let F , g be twice continuously differentiable and each G i (i ∈ {1, · · · , l}) be triply continuously differentiable; let Z be closed. Suppose that each G i (x, ·) is convex for all x ∈ X. Assume that the SBCQ holds on E for the setvalued map M defined above, then the problem (1) is equivalent to the following minimization problem in the variables (x, y, u) With this setup, MOPECs (1) is equivalent to the problem (5). For solving optimization problem (5), we construct the following homotopy equation: where θ = (x, y, u) T , e = (1, 1, · · · , 1) T ∈ R l , t ∈ (0, 1].
3. Some definitions and properties. For convenience, let For solving the MOPECs (7), except Assumptions A1-A3, we make the following assumptions hold, which will be used throughout this paper.
Remark 1. The Assumption A5 is called the normal cone condition.  (parametric form of the Sard theorem on a smooth manifold; see [28]). Let Q, N , P be smooth manifolds of dimensions q, m, p. Respectively, let ϕ : Q × N → P be a C r map, where r > max{0, m − p}. If 0 ∈ P is a regular value of ϕ, then for almost all α ∈ Q, 0 is a regular value of ϕ(α, ·).

Lemma 4.2. [29]
Suppose that H is defined as in (9) and Assumptions A3, A4 hold. Then for almost all initial points t), then the projection of the smooth curve Γ w (0) on the component λ is bounded.
Theorem 4.4. Suppose f, g, F are triply continuously differentiable and G is sufficiently smooth, Assumptions A1-A5 hold. Then, when t → 0, the solution of K-K-T system (8) exists and for almost all the initial point is nonempty and if (w * , 0) is the limit point of Γ w (0) , w * is the solution of the K-K-T system (8).
We know that the function H w (0) (w, 1) = 0 has a unique solution (w (0) , 1) in the set Ω(1) × {1}, then, case (i) is not possible. Obviously, by Theorem 4.3, case (ii) will not happen. Then, case (iii) is the unique possible case and w * is the solution of the K-K-T system (8).
Step 1. Compute an initial point.
If t > 1, let h k = h k