A Noninterior Path following Algorithm for Solving a Class of Multiobjective Programming Problems

1 College of Mathematics, Luoyang Normal University, Luoyang 471022, China 2 Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012, China 3 Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China 4National Key Laboratory of Science and Technology onAdvanced Composites in Special Environments, Harbin Institute of Technology, Harbin 150080, China

To solve linear programming, in 1984, Karmarkar [5] proposed a projective scaling algorithm, which is the first efficient polynomial-time algorithm in practice and hence competitive with the widely used simplex algorithm, which has no polynomiality although it is also efficient for linear programming.It was noted that Karmarkar's projective scaling algorithm is equivalent to a projected Newton barrier algorithm [6].Based on Karmarkar's projective scaling algorithm, ones developed various central path following algorithms (or, in other terms, interior point methods and homotopy methods, see [7][8][9][10][11][12], etc.), which replaced projective scaling transformation of Karmarkar's algorithm with affine scaling transformation.This modification can relax the particular assumptions on the simplex structure by Karmarkar's algorithm.Later, the central path following algorithms were extended to solve convex nonlinear programming problems (see [13][14][15][16], etc.).It should be pointed out that all these central path following algorithms are globally convergent, but their global convergence results were obtained under the assumptions that the logarithmic barrier function is strictly convex and the solution set is nonempty and bounded.
Since Kellogg et al. [17] and Smale [18] proposed the notable homotopy method, this method has become a powerful solution tool with global convergence in finding solutions for various nonlinear problems, for example, zeros or fixed points of maps; see [19][20][21][22][23][24] and so forth.Furthermore, in [25], for convex nonlinear programming problems, by using the ideas of homotopy methods, Lin et al. proposed a new interior point method, which is called the combined homotopy interior point (CHIP) method, for solving convex programming problems.In that paper, the authors removed the convexity condition of the logarithmic barrier function and the nonemptiness of the solution set.In [26], by taking a piecewise technique, under the commonly used conditions in the literature, Yu et al. obtained the polynomiality of the CHIP method.Their results show that the efficiency of the CHIP method is also very well.The advantages mentioned above attract more and more researchers' attention and the CHIP method has been applied to various areas such as fixed point problems [27,28], variational inequalities [29,30], and bilevel programming problems [31].Furthermore, in 2008, for a class of nonconvex MOP problems, Song and Yao developed a new CHIP method [32].In that paper, the authors constructed a new combined homotopy and thus obtained the existence of an interior path from a known interior point to a KKT point of (1).
It is well-known that the choice of initial points plays an important role in the computational efficiency of the predictor-corrector algorithms (for a good introduction and a complete survey about the predictor-corrector algorithms, one can refer to the books [33,34], etc.) resulting from the CHIP algorithm.Here it should be pointed out that, for parametric programming (see [35][36][37][38][39][40][41][42], etc., for related works), the predictor-corrector algorithms also had successful applications (see [35,43], etc.).But in [32], initial points are generally confined in the interior of the feasible set, which is not easily localized for many cases; hence it is essential to enlarge the scope of choice of initial points.To this end, in this paper, we apply proper perturbations to the constraint functions and hence develop a noninterior path following algorithm.With the new approach, we are capable of choosing initial points more easily.This can improve the computational efficiency of the predictor-corrector algorithms greatly compared to before.
Another purpose of this paper is to solve MOP problems in a broader class of nonconvex sets than those in [32].To complete this work, we introduce  2 mappings   (,   ) ∈   ( = 1, . . ., ) and   (, V  ) ∈   ( = 1, . . ., ), which can make us extend the results in [32] to more general nonconvex sets.
In this paper, under the commonly used conditions in the literature, a bounded smooth homotopy path from a given initial point to a KKT point of (1) can be proven to exist.This forms the theoretical base of the noninterior path following algorithm.Numerically tracing the smooth path can lead to an implementable globally convergent algorithm for MOP problems.An explicit advantage of the noninterior path following algorithm is that the induced predictorcorrector algorithm has global convergence, compared with some locally convergent algorithms, for example, the notable Newton's algorithms [33,34].Although the usual continuation methods (see [44][45][46], etc.) are globally convergent, they require that the partial derivative of the mapping  in (9) with respect to  is nonsingular.This requirement is often not easily satisfied in practice (see [33,34], etc.).However, by the parameterized Sard theorem, the noninterior path following algorithm only requires that the mapping  in ( 9) is of full row rank.This is another advantage of the algorithm presented in this paper.In addition, compared with the results in [32], we can solve MOP problems on more general nonconvex sets, and we also enlarge the scope of choice of initial points to the exterior of the feasible set.
This paper is organized as follows.In Section 2, we apply proper perturbations to the constrained functions, based on which, we construct a new combined homotopy and hence develop a noninterior path following algorithm.In Section 3, we use the predictor-corrector algorithm resulting from the noninterior path following algorithm to compute some experimental examples to illustrate the results of this paper.Finally, we make some conclusions in Section 4.
In [32], Song and Yao developed a new CHIP method to solve the KKT point of (1) in a class of nonconvex sets; the main result of that paper is formulated as follows.
Remark 2. (1) In [32], the initial point  (0) is a strictly feasible point and has to satisfy the following constraints: ( (0) ) < 0 and ℎ( (0) ) = 0.However, it is not easy to choose such an initial point in practice when the constraint functions () and ℎ() are complex.For example, when the feasible set is ), it is difficult to choose the initial point  (0) satisfying all the constraints for this example.This difficulty may reduce the computational efficiency of the algorithm in [32].In this paper, we can choose the initial point  (0) arbitrarily in   , by selecting proper parameters  and  according to the signs of ( (0) ) and ℎ( (0) ).Since the initial point for the example mentioned above can be chosen arbitrarily in   , this modification can improve the computational efficiency of algorithms greatly compared to before.In addition, compared with some locally convergent methods, for example, the notable Newton's methods, the method proposed in this paper is a globally convergent method, whose initial points can be chosen more easily.
(2) In [32], the authors required that the feasible set must satisfy the so-called normal cone condition, which is a generalization of the convexity condition (Figure 1).If the feasible set is a convex set, then it satisfies the normal cone condition.On the other hand, if the feasible set satisfies the normal cone condition, then the outer normal cone of the feasible set at a boundary point  can not meet the interior of the feasible set but meets the feasible set only at .In this paper, we extend the results in [32] to more general nonconvex sets.If the feasible set satisfies the normal cone condition, let   (,   ) = ∇  ()  ,  = 1, . . ., ,   (, V  ) = ∇ℎ  ()V  ,  = 1, . . ., ; then it necessarily satisfies assumptions ( 1 )-( 4 ).Conversely, the conclusion does not hold.This point can be illustrated by Examples 1-4 in Section 3.
In the following, we recall some basic definitions and results from differential topology, which will be used in our main result of this paper.
The inverse image theorem (see [47]) tells us that if 0 is a regular value of the map , then  −1 (0) consists of some smooth curves.The regularity of  can be obtained by the following lemma.
Lemma 4 (transversality theorem, see [21]).Let , , and  be smooth manifolds with dimensions , , and p, respectively.Let  ⊂  be a submanifold of codimension  (i.e., p = + dimension of ).Consider a smooth map Φ :  ×  → .If Φ is transversal to , then, for almost all  ∈ , Φ  (⋅) = Φ(, ⋅) :  →  is transversal to .Recall that a smooth map ℎ :  →  is transversal to  if where ℎ is the Jacobi matrix of ℎ and    and    denote the tangent spaces of  and  at , respectively.
In this paper,  = {0}, so the transversality theorem is corresponding to the parameterized Sard theorem on smooth manifolds.
The -component of  * is a KKT point of (1).
It follows from Theorem 3.2 in [32] that the projection of the smooth curve Γ  (0) onto the -plane is bounded.
By the above discussion, we obtain that case (a) is the only possible case, and thus the -component of  * is a KKT point of (1).
Theorem 7. The homotopy path Γ  (0) is determined by the following initial value problem to the ordinary differential equation: where  is the arclength of the curve Γ  (0) .

Numerical Results
By using the homotopy (9) and the predictor-corrector algorithm, several numerical examples are given to illustrate the work in this paper.To illustrate that our result is an extension of the work in [32], we choose some examples whose feasible sets do not satisfy the normal cone condition but satisfy assumptions ( 1 )-( 4 ).In addition, the initial points are chosen not certainly to be in the interior of the feasible sets.In each example, we set  1 = 1⋅−3,  2 = 1⋅−6, and ℎ 0 = 0.02.The behaviors of homotopy paths are shown in Figures 2, 3, 4, and 5. Computational results are given in Table 1, where  (0) denotes the initial point, IT the number of iterations,  the value of ‖  (0) ( () ,   )‖ when the algorithm stops, and  * the KKT point.

Figure 1 :
Figure 1: The nonconvex set satisfying the normal cone condition.