A Comparison of Normal Cone Conditions for Homotopy Methods for Solving Inequality Constrained Nonlinear Programming Problems

Homotopy methods are powerful tools for solving nonlinear programming. Their global convergence can be generally established under conditions of the nonemptiness and boundness of the interior of the feasible set, the Positive Linear Independent Constraint Qualification (PLICQ), which is equivalent to the Mangasarian-Fromovitz Constraint Qualification (MFCQ), and the normal cone condition. This paper provides a comparison of the existing normal cone conditions used in homotopy methods for solving inequality constrained nonlinear programming.


Introduction
In this paper, we consider the following inequality constrained nonlinear programming (ICNLP) problem: min f x ð Þ, where x ∈ R n is the variable, f : R n ⟶ R and g i : R n ⟶ R, i = 1, ⋯, m, are three times continuously differentiable.
Over the past few decades, the theory, algorithms, and applications of nonlinear programming have been rapidly developed, and many numerical methods have been proposed, such as augmented Lagrangian methods, sequential quadratic programming methods, reduced gradient methods, interior point methods, and homotopy methods. Homotopy methods are powerful numerical methods for solving many nonlinear problems. The primary advantage of homotopy methods is that their global convergence can be established under fairly weak assumptions, and the starting points can be chosen rather freely. A comprehensive introduction of the homotopy methods can be found in, e.g., the books [1,2]. The first homotopy method for solving nonlinear programming was proposed for general convex programming in [3]. Among these available homotopy methods for solving nonlinear programming, most of them are designed for solving nonconvex ICNLP problems.
In 1995, the combined homotopy interior point (CHIP) method was proposed for solving nonconvex ICNLP problems in [4,5]. Hereafter, many modified CHIP methods have been proposed for nonconvex ICNLP problems. The global convergence of these homotopy methods can be generally established under three conditions on the original problem: the nonemptiness and boundedness of the interior of the feasible set; the Positive Linear Independent Constraint Qualification (PLICQ), which is equivalent to the Mangasarian-Fromovitz Constraint Qualification (MFCQ) (see [6]); and one type of normal cone conditions, which guarantees the boundedness of the homotopy path near the starting hyperplane. It is well known that the first two conditions are generally used in numerical methods for solving nonlinear programming. The normal cone conditions are generalization of the convexity of the feasible set and extend these homotopy methods from convex programming to nonconvex programming. In addition, a probability-one homotopy method was also proposed for solving nonconvex ICNLP problems in [7]; its global convergence was established under the nonemptiness and boundedness of a parametrized feasible set, Arrow-Hurwicz-Uzawa constraint qualification, and that the homotopy path does not go to infinity near the starting hyperplane. In recent years, these homotopy methods have been extended to fixed point problems, variational inequalities, semidefinite programming, multiobjective programming, constrained sequential minimax problems, and so on.
In this paper, we present the typical normal cone conditions for homotopy methods for solving ICNLP problems, along with the corresponding homotopy maps and global convergence. We give a comparison of four normal cone conditions, including the normal cone condition, the quasinormal cone condition, the pseudocone condition, and the weak normal cone condition. Their relations are discussed in detail for the first time. Some typical nonconvex sets are presented. The comparison can help us to identify features of these normal cone conditions and the corresponding homotopy methods and may motivate us to give some improved homotopy methods for specialized nonconvex programming.
To obtain our results, we conclude this section with some notations. Throughout this paper, Ω = fx ∈ R n | g i ðxÞ ≤ 0, i = 1, ⋯, mg represents the feasible set of the ICNLP problem (1); Ω 0 = fx ∈ R n jg i ðxÞ < 0, i = 1, ⋯, mg denotes the interior of Ω; ∂Ω = ΩΩ 0 means the boundary of Ω. IðxÞ = fi ∈ f1, ⋯, mg | g i ðxÞ = 0g represents the active index set of inequality constraints at x. R m + and R m ++ denote the nonnegative and positive quadrant of R m , respectively. I indicates the identity matrix. k·k denotes the Euclidean norm. For a function F : R n ⟶ R m , F −1 ðCÞ = fx ∈ R n | FðxÞ ∈ Cg is the inverse of the set C ∈ R m ; the n × m matrix ∇FðxÞ, whose ði, jÞth element is ∂F j ðxÞ/∂x i , is the transpose of the Jacobian of F. Assumption 1. Ω 0 is nonempty and bounded.

Normal Cone Conditions and Homotopy Methods for ICNLP Problems
There exist four typical normal cone conditions in homotopy methods for solving ICNLP problems in literatures; related results will be introduced in this section. For the nonconvex ICNLP problem (1), the first homotopy method, called the combined homotopy interior point (CHIP) method, was proposed in [4,5]. The combined homotopy map is constructed as where ðx, y, tÞ ∈ Ω × R m + × ½0, 1, ðx 0 , y 0 Þ ∈ Ω 0 × R m ++ , Y and Y 0 are diagonal matrices with the ith diagonal elements y i and y 0 i for i = 1, ⋯, m, respectively. Under Assumption 1, the assumption that f∇g i ðxÞ | i ∈ IðxÞg has a full column rank for any x ∈ ∂Ω, which can be replaced by Assumption 2, and the assumption that Ω satisfies the normal cone condition (see Definition 3), for almost all ðx 0 , y 0 Þ ∈ Ω 0 × R m ++ , the zero point set of (3) defines a smooth homotopy path Γ ðx 0 ,y 0 Þ ⊂ Ω 0 × R m ++ × ð0, 1, which starts from ðx 0 , y 0 , 1Þ and approaches to the hyperplane t = 0. For any limit point ðx * , y * , 0Þ of Γ ðx 0 ,y 0 Þ , x * is a KKT point of the problem (1), y * is the corresponding Lagrange multiplier.
According to Definition 3, the NCC means that, for any x ∈ ∂Ω, the set fx 0 − x | x 0 ∈ Ω 0 g does not intersect with the cone f∑ i∈IðxÞ y i ∇g i ðxÞ | y i ≥ 0, i ∈ IðxÞg. Moreover, if g i : R n ⟶ R, i = 1, ⋯, m, are convex, then Ω satisfies the normal cone condition; however, the reciprocal implication is not true, a typical counterexample is Figure 1). In [8], a modified CHIP method was presented. The homotopy map is defined as where ðx, y, tÞ ∈ Ω × R m and ηðxÞ is a positive independent map with respect to ∇gðxÞ (see Definition 4). Under Assumption 1, the assumption that f∇g i ðxÞ | i ∈ IðxÞg are linear independent for any x ∈ ∂Ω, which can be also replaced by Assumption 2, and the assumption that Ω satisfies the quasinormal cone condition related to the positive independent map ηðxÞ (see Definition 5), for almost all ðx 0 , y 0 Þ ∈ Ω 0 × R m ++ , the global convergence of a smooth homotopy path Γ ðx 0 ,y 0 Þ ⊂ Ω 0 × R m ++ × ð0, 1 starting from ðx 0 , y 0 , 1Þ and approaching to the hyperplane t = 0 can be established; then, a KKT point x * with the corresponding Lagrange multiplier y * can be obtained.

Advances in Mathematical Physics
Definition 4 (positive independent map, see [8]). If there exist smooth maps η i ðxÞ: then, ηðxÞ = ðη 1 ðxÞ, ⋯, η m ðxÞÞ is said to be a positive independent map with respect to ∇gðxÞ.
Definition 5 (quasinormal cone condition (QNCC) [8]). If there exists a smooth positive independent map ηðxÞ = ðη 1 ðxÞ, ⋯, η m ðxÞÞ with respect to ∇gðxÞ such that, for any x ∈ ∂Ω, then, Ω is said to satisfy the QNCC related to ηðxÞ. According to Definition 5, the QNCC means that, for any In [9], another modified CHIP method was proposed with the homotopy map where ðx, y, tÞ ∈ Ω × R m ÞÞ is a consistent hair map (see Definition 7). Similarly to the homotopy methods in [1,6], under Assumption 1, the assumption that f∇g i ðxÞ | i ∈ IðxÞg has a full column rank for any x ∈ ∂Ω, and the assumption that Ω satisfies the pseudocone condition with respect to the consistent hair map ηðx, zÞ (see Definition 8), the global convergence can be established.
Definition 9 (weak normal cone condition (WNCC) [10]). If there exists an open subset b Ω ⊂ Ω 0 such that, for any x ∈ ∂Ω, then, Ω is said to satisfy the WNCC with respect to b Ω. According to Definition 9, the WNCC means that there exists an open subset b Ω ⊂ Ω 0 such that for any x ∈ ∂Ω, the set fx 0 − x | x 0 ∈ b Ωg does not intersect with the cone f∑ i∈IðxÞ y i ∇g i ðxÞ | y i ≥ 0, i ∈ IðxÞg.

A Comparison of the Four Typical Normal Cone Conditions
In this section, for the first time, we study the relations of the four typical normal cone conditions introduced in Section 2, and some typical nonconvex sets are introduced.

Proposition 10.
If the PLICQ holds, the NCC implies the QNCC and PCC.
Proof. Suppose that the PLICQ holds, then ηðxÞ = ð∇g 1 ðxÞ, ⋯,∇g m ðxÞÞ is a positive independent map with respect to ∇gðxÞ by Definition 4, and ηðx, zÞ = ðz 1 ∇g 1 ðxÞ, ⋯, z m ∇g m ðxÞÞ is a consistent hair map of Ω by Definition 7. Then, Ω satisfies the QNCC with respect to ηðxÞ by Definition 5, and the PCC with respect to ηðx, zÞ by Definition 8.

The Global Convergence of the Homotopy Methods for Solving ICNLP
As shown in Section 2, the global convergence of these homotopy methods can be established under three conditions. In this section, we present some comments for these conditions. The boundedness of Ω 0 ensures that the variable x in the homotopy path keeps bounded. For Example 4.1, Ω 0 = fx ∈ R 2 | x 2 1 + x 2 < 0g is unbounded, we have x ⟶ ð−∞, − ∞Þ with y ⟶ 0 and t → 0 for the starting point x 0 = ð0,−1Þ with y 0 = 1 in the homotopy path defined by the CHIP method. Since the real-world ICNLP problems generally have the optimal solutions at finity, the variable x in the homotopy path always keeps bounded even if Ω 0 is unbounded. For Example 4.2, Ω 0 = fx ∈ R 2 | x 2 1 − x 2 < 0g is unbounded, but we have x ⟶ ð−0:5,0:25Þ with y ⟶ 1 and t ⟶ 0 for the starting point x 0 = ð0, 1Þ with y 0 = 1 in the homotopy path defined by the CHIP method. min Example 4.2. min The PLICQ (MFCQ) is the most widely used constraint qualification for the ICNLP problems, it ensures that the variable y in the homotopy path keeps bounded. For Example 4.3, the PLICQ does not hold at ð0,−1Þ ∈ ∂Ω, we have x ⟶ ð0,−1Þ that is not a KKT point, y ⟶ ð0,+∞,+ ∞Þ and t ⟶ 0 for the starting point x 0 = ð−2, 0Þ with y 0 = ð1, 1, 1Þ, and x ⟶ ð2:8242,−1Þ, y ⟶ ð0:1768,0, 0:6464Þ The normal cone conditions and the PLICQ ensure that the variable y in the homotopy path keeps bounded when the homotopy path tends to the hyperplane fðx, y, 1Þ | ðx, yÞ ∈ Ω × R m + g. Although our numerical experiences show that these homotopy methods always globally converge even if the normal cone conditions do not hold, the normal cone conditions are important to establish the global convergence of these homotopy methods in theory. For Example 4.4, the normal cone conditions do not hold for any x ∈ fx ∈ R 2 | x 2 1 + x 2 2 = 1g ⊂ ∂Ω, but we have x ⟶ ð2,−2Þ, y ⟶ ð0:25,0Þ and t ⟶ 0 for all tested starting points x 0 ∈ Ω 0 with y 0 = ð1, 1Þ in the homotopy path defined by the CHIP method.

Conclusion
In this paper, we provide a comparison of four typical normal cone conditions used in homotopy methods for solving inequality constrained nonlinear programming. The NCC holds for convex sets and a class of nonconvex sets. The WNCC, QNCC, and PCC are more weak than the NCC. However, the NCC is more convenient to use than the WNCC, QNCC and PCC. For the feasible set satisfying the QNCC or PCC, some auxiliary maps should be constructed in the homotopy maps. For the feasible set satisfying the WNCC, the starting point for the homotopy methods should be chosen from a special open subset of the feasible set. On the other hand, except for few special nonconvex sets, there does not exist a general way of checking a set to satisfy the NCC or construct the open subset, the positive independent map and the consistent hair map for the WNCC, QNCC, and PCC, respectively.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.