Homotopy Method for a General Multiobjective Programming Problem under Generalized Quasinormal Cone Condition

Abstract

A combined interior point homotopy continuation method is proposed for solving general multiobjective programming problem. We prove the existence and convergence of a smooth homotopy path from almost any interior initial interior point to a solution of the KKT system under some basic assumptions.

1. Introduction

In this paper, for any two vectors 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇 and 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑛)^𝑇 in 𝑅^𝑛, we use the following conventions: 𝑦𝑦<=𝑧𝑧,,iiffff𝑦𝑦_𝑖𝑖=𝑧_𝑖,𝑖=1,2,…,𝑛;𝑦≦𝑧,iff𝑦_𝑖<𝑧_𝑖,𝑖=1,2,…,𝑛;𝑦≤𝑧,iff𝑦_𝑖≤𝑧_𝑖,𝑖=1,2,…,𝑛;≤𝑧_𝑖,𝑦≠𝑧,𝑖=1,2,…,𝑛.(1.1)

We consider the following multiobjective programming problem:

(MOP) ms.itn.ℎ𝑔𝑓((𝑥𝑥())𝑥≦=),00,,(1.2) where 𝑓=(𝑓₁,𝑓₂,…,𝑓_𝑝)^𝑇∶𝑅^𝑛→𝑅^𝑝, 𝑔=(𝑔₁,𝑔₂,…,𝑔_𝑚)^𝑇∶𝑅^𝑛→𝑅^𝑚, and ℎ=(ℎ₁,ℎ₂,…,ℎ_𝑠)^𝑇∶𝑅^𝑛→𝑅^𝑠.

For 𝜆=(𝜆₁,𝜆₂,…,𝜆_𝑝)^𝑇∈𝑅^𝑝, let 𝑅^𝑛+ and 𝑅^𝑛++ denote the nonnegative and positive orthant of 𝑅^𝑛. Respectively, let Ω={𝑥∈𝑅^𝑛∣𝑔(𝑥)≦0,ℎ(𝑥)=0},Ω₀={𝑥∈𝑅^𝑛∣𝑔(𝑥)<0,ℎ(𝑥)=0},𝜕Ω=ΩΩ₀Λ₊,=𝜆∈𝑅^𝑝+∣^𝑝_𝑖=1𝜆_𝑖=1,Λ⁺⁺=𝜆∈𝑅^𝑝++∣^𝑝_𝑖=1𝐼={1,2,…,𝑚},𝐽={1,2,…,𝑠},𝜆_𝑖=1,(1.3) and let 𝐵(𝑥)=𝑖∈{1,2,…,𝑚}∣𝑔_𝑖(𝑥)=0(1.4) denote the active index set at a given point.

MOP has important application in many practical fields like production planning, structural designing, portfolio selection, and so forth. Research on it can be traced back to Pareto [1], Von Neumann and Morgenstern [2], and Koopmans [3] or even earlier. Especially, more and more attention has been paid to the homotopy method since Kellogg et al. [4], Smale [5], and Chow et al. [6] published the remarkable papers. The homotopy method now becomes an important tool for numerically solving complementary, variational inequalities, convex multiobjective programming, and nonlinear mathematical programming et al. [7–12] as a globally convergent method.

Among many methods, the weighed sum method is popular and efficient. It transforms the MOP to a single-objective programming [13]: ms.itn.𝑔𝜆(𝑥^𝑇)𝑓≦(𝑥0),_𝑝,_𝑖=1𝜆_𝑖=1,(1.5) where 𝜆 is the weight vector.

Recently, Song and Yao [14] generalize the combined homotopy interior point method to the general multi-objective programming problem under the so-called normal cone condition instead of the convexity condition about the feasible set. In that paper, they proved the existence of the homotopy path under the following assumptions:(A₁)Ω₀ is nonempty and bounded;(A₂) for all 𝑥∈Ω, the vectors {∇𝑔_𝑖(𝑥),𝑖∈𝐵(𝑥),∇ℎ_𝑗(𝑥),𝑗∈𝐽} are linearly independent;(A₃) for all 𝑥∈Ω, {𝑥+∑_{𝑖∈𝐵(𝑥)}𝑢_𝑖∇𝑔_𝑖(𝑥)+∑_𝑗∈𝐽𝑧_𝑗∇ℎ_𝑗(𝑥)∶𝑧_𝑗∈𝑅,𝑢_𝑖≥0,𝑗∈𝐽,𝑖∈𝐵(𝑥)}⋂Ω={𝑥}.

In [14], the combined homotopy method was given as follows: 𝐻𝜔,𝜔⁽⁰⁾,𝑡=⎡⎢⎢⎢⎢⎢⎢⎢⎣(1−𝑡)(∇𝑓(𝑥)𝜆+∇𝑔(𝑥)𝑢)+∇ℎ(𝑥)𝑧+𝑡𝑥−𝑥⁽⁰⁾𝑈×𝑔(𝑥)−ℎ𝑡𝑈(𝑥)₍₀₎(1−𝑡)1−_𝑝×𝑔𝑥⁽⁰⁾_𝑖∑₌₁𝜆_𝑖𝑒−𝑡𝜆−𝜆⁽⁰⁾⎤⎥⎥⎥⎥⎥⎥⎥⎦=0,(1.6) where 𝑥⁽⁰⁾∈Ω₀, 𝑢⁽⁰⁾>0, 𝜆⁽⁰⁾>0, and ∑^𝑝𝑖=1𝜆_𝑖(0)=1. However, the solution simply yields 𝜆=𝜆⁽⁰⁾ for all 𝑡∈(0,1]. That is, 𝜆 is fixed. In fact, from the last equation, we have 𝑝(1−𝑡)+(𝑝𝑡−𝑝−𝑡)∑^𝑝𝑖=1𝜆_𝑖+𝑡=0. According to this, we know that 𝜆≡𝜆⁰ for all of 𝑡∈[0,1].

That is, these methods are all solving the single-objective programming problem.

In [15], they present the concept of “positive linear independent” and weaken the assumptions than the ones in [14]. But in order to extend their results to a broader class of nonconvex multi-objective programming problems, we construct a new homotopy equation under generalized quasinorm cone condition in this paper and 𝜆 is not fixed in the calculation process.

The paper is organized as follows. In Section 2, we recall some notations and preliminaries results. In Section 3, we construct a new combined homotopy mapping and prove the existence and convergence of a smooth homotopy path from almost any interior initial point to the KKT points of MOP under some assumptions.  In Section 4, numerical results are given,which show that the method is feasible and effective.

2. Some Definitions and Properties

Definition 2.1. Let 𝑈⊂𝑅^𝑛 be an open set, and let 𝜑∶𝑈→𝑅^𝑃 be a smooth mapping. If Range[𝜕𝜑(𝑥)/𝜕𝑥]=𝑅^𝑝 for all 𝑥∈𝜑⁻¹(𝑦), then 𝑦∈𝑅^𝑝 is a regular value and 𝑥∈𝑅^𝑛 is a regular point.

Definition 2.2. Let 𝜂_𝑖∶𝑅^𝑛→𝑅^𝑛(𝑖=1,2,…,𝑚) and 𝛽_𝑗∶𝑅^𝑛→𝑅^𝑛(𝑗=1,2,…,𝑠). For any 𝑥∈Ω,{∇𝑔_𝑖(𝑥),𝜂_𝑖(𝑥)∶𝑖∈𝐵(𝑥)} is said to be positive linear independent with respect to 𝛽(𝑥), if 𝛽(𝑥)𝑧+_{𝑖∈𝐵(𝑥)}𝑦_𝑖∇𝑔_𝑖(𝑥)+𝑢_𝑖𝜂_𝑖(𝑥)=0,𝑧∈𝑅^𝑠,𝑦_𝑖≥0,𝑢_𝑖≥0(2.1) implies that 𝑧=0,𝑦_𝑖=0,𝑢_𝑖=0(𝑖∈𝐵(𝑥)),(2.2) where 𝛽(𝑥)=(𝛽₁(𝑥),…,𝛽_𝑠(𝑥)).

Lemma 2.3 (parametric form of the Sard theorem on a smooth manifold; see [16]). 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 2.4 (inverse image theorem; see [17]). If 0 is a regular value of the mapping 𝜑_𝛼(⋅)≜𝜑(𝛼,⋅), then 𝜑^𝛼−1(0) consists of some smooth manifolds.

Lemma 2.5 (classification theorem of one-dimensional manifold; see [17]). A one-dimensional smooth manifold is diffeomorphic to a unit circle or a unit interval.

The following four basic assumptions are commonly used in this paper:(C₁)Ω₀ is nonempty and bounded;(C₂) for any 𝑥∈Ω and 𝑡∈[0,1], there exists map 𝜂(𝑥) and 𝛽(𝑥), such that {∇𝑔_𝑖(𝑥),𝜂_𝑖(𝑥)∶𝑖∈𝐵(𝑥)} is positive linear independent with respect to ∇ℎ(𝑥)+𝑡(𝛽(𝑥)−∇ℎ(𝑥)); (C₃) for any 𝑥∈𝜕Ω, 𝑥+_{𝑖∈𝐵(𝑥)}𝑢_𝑖𝜂_𝑖(𝑥)+𝛽(𝑥)𝑧∶𝑧∈𝑅^𝑠,𝑢_𝑖≥0,𝑖∈𝐵(𝑥)Ω={𝑥}(2.3) (generalized quasinormal cone condition);(C₄) for any 𝑥∈Ω, ∇ℎ(𝑥)^𝑇𝛽(𝑥) is nonsingular.

Remark 2.6. If Ω satisfies the assumptions (A₁)–(A₃), then it necessarily satisfies the assumptions (C₁)–(C₄).
In fact, if we choose 𝜂(𝑥)=∇𝑔(𝑥) and 𝛽(𝑥)=∇ℎ(𝑥), then it is easy to get the result. Clearly, if Ω satisfies the assumptions (C₁)–(C₄), then it does not necessarily satisfies the assumptions (A₁)–(A₃).

3. Main Results

Let 𝑥∈Ω be a KKT point of MOP; our aim is to find (𝜆,𝑢,𝑧)∈𝑅^+𝑝+𝑚×𝑅^𝑠, such that ∇𝑓(𝑥)𝜆+1∇𝑈−𝑔𝑔((_𝑝𝑥𝑥))𝑢=+0∇,ℎ(𝑥)𝑧=0,((33..11ba))_𝑖=1𝜆_𝑖=0,(3.1c)where ∇𝑓(𝑥)=(∇𝑓₁(𝑥),…,∇𝑓_𝑝(𝑥))∈𝑅^𝑛×𝑝, ∇𝑔(𝑥)=(∇𝑔₁(𝑥),…,∇𝑔_𝑚(𝑥))∈𝑅^𝑛×𝑚, ∇ℎ(𝑥)=(∇ℎ₁(𝑥),…,∇ℎ_𝑚(𝑥))∈𝑅^𝑛×𝑠.

Meanwhile, the KKT system of MOP is (3.1a)–(3.1c).

For a convex multi-objective programming problem, the solution of the MOP can be obtained from the KKT system. And for a nonconvex multi-objective programming problem, it is significant that we can obtain a solution of the KKT system.

To solve the KKT system (3.1a)–(3.1c), we construct a homotopy equation as follows: 𝐻𝜔=,𝜔⎡⎢⎢⎢⎢⎢⎢⎢⎣(1⁽⁰⁾,−𝑡𝑡)∇𝑓(𝑥)𝜆+∇𝑔(𝑥)𝑢+𝑡𝜂(𝑥)𝑢₂𝑈×𝑔(𝑥)−+ℎ𝑡𝑈[(∇𝑥)ℎ₍₀₎(𝑥)+𝑡(𝛽(𝑥)−∇ℎ(𝑥))]𝑧+𝑡𝑥−𝑥⁽⁰⁾(1−𝑡)1−_𝑝×𝑔𝑥⁽⁰⁾_𝑖∑₌₁𝜆_𝑖𝑒−𝑡𝜆^5/2=0,−𝜆⁽⁰⁾^5/2⎤⎥⎥⎥⎥⎥⎥⎥⎦(3.2) where 𝜔⁽⁰⁾=(𝑥⁽⁰⁾,𝜆⁽⁰⁾,𝑢⁽⁰⁾,𝑧⁽⁰⁾)∈Ω₀×Λ⁺⁺×𝑅^𝑚++×{0}, 𝜔=(𝑥,𝜆,𝑢,𝑧)∈Ω×𝑅^{𝑝+𝑚+𝑠}, 𝑢²=(𝑢²¹,𝑢²²…,𝑢^2𝑚)^𝑇∈𝑅^𝑚, 𝜆^5/2=(𝜆^15/2,𝜆^25/2,…,𝜆^𝑝5/2)^𝑇∈𝑅^𝑝, 𝑈=diag(𝑢), 𝑒=(1,1,…,1)^𝑇∈𝑅^𝑝, and 𝑡∈[0,1].

As 𝑡=1, the homotopy equation (3.2) becomes 𝑈𝛽𝑔((𝑥𝑥))𝑧−+ℎ𝑈(𝑥𝑥)₍₀₎−=𝑥0,⁽⁰⁾=0,((33..33ba))𝜆𝑔𝑥⁽⁰⁾=0,(3.3c)_5/2=𝜆⁽⁰⁾^5/2.(3.3d)

By the assumption (C₃), we get 𝑧=0, 𝑥=𝑥⁽⁰⁾. Since 𝑔(𝑥⁽⁰⁾)<0 and 𝑥=𝑥⁽⁰⁾, (3.3c) implies that 𝑢=𝑢⁽⁰⁾. Equation (3.3d) shows that 𝜆=𝜆⁽⁰⁾. That is, 𝐻(𝜔,𝜔⁽⁰⁾,1)=0 with respect to 𝜔 has only one solution 𝜔=𝜔⁽⁰⁾=(𝑥⁽⁰⁾,𝜆⁽⁰⁾,𝑢⁽⁰⁾,0).

As 𝑡=0, 𝐻(𝜔,𝜔⁽⁰⁾,𝑡)=0 turns to the KKT system (3.1a)–(3.1c).

For a given 𝜔⁽⁰⁾, rewrite 𝐻(𝜔,𝜔⁽⁰⁾,𝑡) as 𝐻_𝜔(0)(𝜔,𝑡). The zero-point set of 𝐻_𝜔(0) is 𝐻^𝜔−1₍₀₎=(𝜔,𝑡)∈Ω×𝑅^𝑝+𝑚++×𝑅^𝑠×(0,1]∶𝐻𝜔,𝜔⁽⁰⁾,𝑡=0.(3.4)

Theorem 3.1. Suppose 𝑓,𝑔, and ℎ are three times continuous differentiable functions. In addition, let the assumptions (C₁)-(C₂) hold and 𝜂_𝑖,𝛽_𝑗 twice times continuously differentiable functions. Then for almost all initial points 𝜔⁽⁰⁾∈Ω₀×Λ⁺⁺×𝑅^𝑚++×{0}, 0 is a regular value of 𝐻_𝜔(0) and 𝐻^𝜔−1₍₀₎ consists of some smooth curves. Among them, a smooth curve, say Γ_𝜔(0), is starting from (𝜔⁽⁰⁾,1).

Proof. Denote the Jacobi matrix of 𝐻(𝜔,𝜔⁽⁰⁾,𝑡) by 𝐷𝐻(𝜔,𝜔⁽⁰⁾,𝑡). For any 𝜔⁽⁰⁾∈Ω₀×Λ⁺⁺×𝑅^𝑚++×{0} and 𝑡∈[0,1], we have 𝐷𝐻(𝜔,𝜔⁽⁰⁾,𝑡)=(𝜕𝐻/𝜕𝜔,𝜕𝐻/𝜕𝜔⁽⁰⁾,𝜕𝐻/𝜕𝑡). Now, we consider the submatrix of 𝐷𝐻(𝜔,𝜔⁽⁰⁾,𝑡).
For any (𝑥,𝑥⁽⁰⁾,𝜆⁽⁰⁾,𝑢⁽⁰⁾)∈𝑅^𝑛×Ω₀×Λ⁺⁺×𝑅^𝑚++, 𝜕𝐻𝜕𝑥,𝑥⁽⁰⁾,𝜆⁽⁰⁾,𝑢⁽⁰⁾=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑄−𝑡𝐼^𝑛∇ℎ(𝑥)_𝑇00𝑈∇𝑔(𝑥)_𝑇0000−𝑡𝑈⁽⁰⁾∇0𝑔𝑥⁽⁰⁾^𝑇50−𝑡diag𝑔𝑥⁽⁰⁾2𝑡𝜆⁽⁰⁾^3/2𝐼_𝑝0⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(3.5) where 𝑄=(1−𝑡)(∑^𝑝𝑖=1𝜆_𝑖∇²𝑓_𝑖(𝑥)+∑^𝑚𝑗=1𝑢_𝑗∇²𝑔_𝑗(𝑥)+𝑡∑^𝑚𝑗=1𝑢^2𝑗∇𝜂_𝑗(𝑥))+[∇²ℎ(𝑥)+𝑡(∇𝛽(𝑥)−∇²ℎ(𝑥))]𝑧+𝑡𝐼_𝑛.
We obtain that rank𝜕𝐻𝜕𝑥,𝑥⁽⁰⁾,𝜆⁽⁰⁾,𝑢⁽⁰⁾=𝑛+𝑝+𝑚+𝑠.(3.6)
That is, 0 is a regular value of 𝐻. By parametric form of the Sard theorem, for almost all 𝜔⁽⁰⁾∈Ω₀×Λ⁺⁺×𝑅^𝑚++×{0}, 0 is a regular value of 𝐻_𝜔(0). By inverse image theorem, 𝐻^𝜔−1₍₀₎(0) consists of some smooth curves. Since 𝐻(𝜔⁽⁰⁾,𝜔⁽⁰⁾,1)=0, there must be a smooth curve, denoted by Γ_𝜔(0), that starts from (𝜔⁽⁰⁾,1).

Theorem 3.2. Let assumptions (C₁)-(C₂) hold. For a given 𝜔⁽⁰⁾=(𝑥⁽⁰⁾,𝜆⁽⁰⁾,𝑢⁽⁰⁾,𝑧⁽⁰⁾)∈Ω₀×Λ⁺⁺×𝑅^𝑚++×{0}, if 0 is a regular value of 𝐻_𝜔(0), then the projection of the smooth curve Γ_𝜔(0) on the component 𝜆 is bounded.

Proof. Suppose that the conclusion does not hold. Since (0,1] is bounded, there exists a sequence {(𝜔^(𝑘),𝑡_𝑘)}⊂Γ_𝜔(0), such that 𝑡_𝑘→𝑡_∗,‖‖𝜆^(𝑘)‖‖→∞.(3.7) From the last equality of (3.2), we have ⎛⎜⎜⎜⎜⎜⎜⎝1−𝑡_𝑘1−𝑡_𝑘1⋯−𝑡_𝑘⎞⎟⎟⎟⎟⎟⎟⎠−⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝1−𝑡^𝑘𝜆^1(𝑘)+1−𝑡_𝑘∑_𝑖≠1𝜆^𝑖(𝑘)+𝑡_𝑘𝜆^1(𝑘)^5/21−𝑡_𝑘𝜆^2(𝑘)+1−𝑡_𝑘∑_𝑖≠2𝜆^𝑖(𝑘)+𝑡_𝑘𝜆^2(𝑘)^5/21−𝑡_𝑘⋯𝜆^𝑝(𝑘)+1−𝑡_𝑘∑_𝑖≠𝑝𝜆^𝑖(𝑘)+𝑡_𝑘𝜆^𝑝(𝑘)^5/2⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠−𝑡_𝑘⎛⎜⎜⎜⎜⎜⎜⎜⎝𝜆¹⁽⁰⁾^5/2𝜆²⁽⁰⁾^5/2𝜆⋯_𝑝(0)^5/2⎞⎟⎟⎟⎟⎟⎟⎟⎠=0.(3.8) If we assume ‖𝜆^(𝑘)‖→+∞(𝑘→∞), this hypothesis implies 𝑖∈{1,2,…,𝑝}∶lim_𝑘→∞𝜆^𝑖(𝑘)=∞≠Φ.(3.9)
Since 𝑡_𝑘→𝑡_∗, 𝜆^(𝑘)>0, it follows that the second part in the left-hand side of some equations in (3.8) tends to infinity as 𝑘→∞. But the other two parts are bounded. This is impossible. Thus, the component 𝜆 is bounded.

Theorem 3.3. Let 𝑓,𝑔, and ℎ be three times continuous differentiable functions. In addition, let the assumptions (C₁)–(C₄) hold and 𝜂_𝑖,𝛽_𝑗 twice times continuously differentiable functions. Then, for almost all of 𝜔⁽⁰⁾∈Ω₀×Λ⁺⁺×𝑅^𝑚++×{0}, 𝐻^𝜔−1₍₀₎(0) contains a smooth curve Γ_𝜔(0)⊂Ω×𝑅^𝑝+×𝑅^𝑚+×𝑅^𝑠×(0,1], which starts from (𝜔⁽⁰⁾,1). As 𝑡→0, the limit set 𝑇×{0}⊂Ω×Λ⁺×𝑅^𝑚+×𝑅^𝑠×{0} of Γ_𝜔(0) is nonempty and every point in 𝑇 is a solution of the KKT system (3.1a)–(3.1c).

Proof. From the homotopy equation (3.2), it is easy to see that Γ_𝜔(0)⊂Ω×𝑅^𝑝+×𝑅^𝑚+×𝑅^𝑠×(0,1]. By Theorem 3.1, for almost all 𝜔⁽⁰⁾∈Ω₀×Λ⁺⁺×𝑅^𝑚++×{0}, 0 is a regular value of 𝐻_𝜔(0) and 𝐻^𝜔−1₍₀₎ contains a smooth curve Γ_𝜔(0) starting from (𝜔⁽⁰⁾,1). By the classification theorem of one-dimensional smooth manifolds, Γ_𝜔(0) is diffeomorphic to a unit circle or the unit interval (0,1].
Noticing that ||||𝜕𝐻_𝜔(0)𝜔⁽⁰⁾,1𝜕𝜔||||=|||||||||||||||𝐼^𝑛∇ℎ𝑥₍₀₎00𝛽𝑥⁽⁰⁾𝑈^𝑇000₍₀₎∇𝑔0𝑥⁽⁰⁾^𝑇−50diag𝑔𝑥⁽⁰⁾02𝜆⁽⁰⁾^3/2=(−1)_𝑠𝐼_𝑝00|||||||||||||||||diag𝑔𝑥⁽⁰⁾|||||−52𝜆⁽⁰⁾^3/2𝐼_𝑝||||||∇ℎ𝑥₍₀₎^𝑇𝛽𝑥⁽⁰⁾|||.(3.10)
Because 𝑔(𝑥⁽⁰⁾)<0, 𝜆⁽⁰⁾∈Λ⁺⁺ and by the assumption (C₄), we know that [𝜕𝐻_𝜔(0)(𝜔⁽⁰⁾,1)/𝜕𝜔] is nonsingular. Therefore, the smooth curve Γ_𝜔(0) starts from (𝜔⁽⁰⁾,1) diffeomorphic to (0,1].
Let (𝜔^∗,𝑡_∗) be a limit point of Γ_𝜔(0); only three cases are possible:(a)(𝜔^∗,𝑡_∗)∈Ω×Λ⁺×𝑅^𝑚+×𝑅^𝑠×{0}; (b)(𝜔^∗,𝑡_∗)∈𝜕(Ω₀×𝑅^+𝑝+𝑚)×𝑅^𝑠×(0,1];(c)(𝜔^∗,𝑡_∗)∈Ω×𝑅^+𝑝+𝑚×𝑅^𝑠×{1}.
Because 𝐻(𝜔⁽⁰⁾,𝜔⁽⁰⁾,1)=0 has a unique solution (𝜔⁽⁰⁾,1), the case (c) will not happen.
In case (b), because Ω and (0,1] are bounded sets and by the assumption (C₂), for any 𝑥∈Ω and 𝑡∈[0,1], there exists map 𝜂(𝑥) and 𝛽(𝑥) such that, {∇𝑔_𝑖(𝑥),𝜂_𝑖(𝑥)∶𝑖∈𝐵(𝑥)} is positive linear independent with respect to ∇ℎ(𝑥)+𝑡(𝛽(𝑥)−∇ℎ(𝑥)). From the first equation of (3.2), we get that the component 𝑧 of Γ_𝜔(0) is bounded.
If case (b) holds, then there exists a sequence {(𝜔^(𝑘),𝑡_𝑘)}⊂Γ_𝜔(0), such that ‖‖𝜔^(𝑘),𝑡_𝑘‖‖→∞.(3.11)
Because Ω and (0,1] are bounded, there exists a subsequence (denoted also by {(𝜔^(𝑘),𝑡_𝑘)}⊂Γ_𝜔(0)) such that 𝑥^(𝑘)→𝑥^∗,𝜆^(𝑘)→𝜆^∗,‖‖𝑢^(𝑘)‖‖→∞,𝑧^(𝑘)→𝑧^∗,𝑡_𝑘→𝑡_∗,as𝑘→∞.(3.12)
By the third equation of (3.2), we have 𝑔𝑥^(𝑘)=𝑡_𝑘𝑈^(𝑘)⁻¹𝑈⁽⁰⁾𝑔𝑥⁽⁰⁾.(3.13) Hence, the active index set 𝐵(𝑥^∗) is nonempty.
From the first equation of (3.2), it follows that 1−+𝑡_𝑘∇ℎ∇𝑓𝑥_(𝑘)𝑥^(𝑘)𝜆^(𝑘)+∇𝑔𝑥^(𝑘)𝑢^(𝑘)+𝑡_𝑘𝜂𝑥^(𝑘)𝑢^(𝑘)²+𝑡_𝑘𝛽𝑥^(𝑘)−∇ℎ𝑥^(𝑘)𝑧^(𝑘)+𝑡_𝑘𝑥^(𝑘)−𝑥⁽⁰⁾=0.(3.14)
(i) When 𝑡_∗=1, rewrite (3.14) as _𝑗∈_{𝐵(𝑥∗)}1−𝑡_𝑘∇𝑔_𝑗𝑥^(𝑘)𝑢^𝑗(𝑘)+𝑡_𝑘𝜂_𝑗𝑥^(𝑘)𝑢^𝑗(𝑘)²+𝑡_𝑘=−1𝛽−𝑥𝑡^(𝑘)_𝑘𝑧^(𝑘)+1−𝑡_𝑘∇ℎ𝑥^(𝑘)𝑧^(𝑘)+𝑥^(𝑘)−𝑥⁽⁰⁾_{𝑗∉𝐵(𝑥∗)}∇𝑔_𝑗𝑥^(𝑘)𝑢^𝑗(𝑘)+𝑡_𝑘𝜂_𝑗𝑥^(𝑘)𝑢^𝑗(𝑘)²+∇𝑓𝑥^(𝑘)𝜆^(𝑘)−𝑥^(𝑘)−𝑥⁽⁰⁾.(3.15)
From the fact that 𝑢^𝑗(𝑘) is bounded for 𝑗∉𝐵(𝑥^∗) and by the assumptions (C₁) and (C₂), when 𝑘→∞, we observe that _𝑗∈_{𝐵(𝑥∗)}∇𝑔^𝑗𝑥^∗lim_𝑘→∞1−𝑡_𝑘𝑢^𝑗(𝑘)+𝜂_𝑗𝑥^∗lim_𝑘→∞1−𝑡_𝑘𝑡_𝑘𝑢^𝑗(𝑘)²=−𝛽𝑥^∗𝑧^∗+𝑥^∗−𝑥⁽⁰⁾.(3.16)
It is easy to see that the right-hand side of the equation is bounded. By the assumption (C₂), we have lim_𝑘→∞1−𝑡_𝑘𝑢^𝑗(𝑘)=0,lim_𝑘→∞1−𝑡_𝑘𝑡_𝑘𝑢^𝑗(𝑘)²=𝛼_𝑗,𝑗∈𝐵𝑥^∗,(3.17) where 𝛼_𝑗≥0.
Then, we have 𝑥⁽⁰⁾=𝑥^∗+𝛽𝑥^∗𝑧^∗+_{𝑗∈𝐵(𝑥∗)}𝛼_𝑗𝜂_𝑗𝑥^∗,(3.18) which contradicts the assumption (C₃).
(ii) When 𝑡_∗∈[0,1), rewrite (3.14) as _𝑗∈_{𝐵(𝑥∗)}1−𝑡_𝑘∇𝑔_𝑗𝑥^(𝑘)𝑢^𝑗(𝑘)+𝑡_𝑘𝜂_𝑗𝑥^(𝑘)𝑢^𝑗(𝑘)²=−1−𝑡_𝑘_{𝑗∉𝐵(𝑥∗)}∇𝑔^𝑗𝑥^(𝑘)𝑢^𝑗(𝑘)+𝑡_𝑘𝜂_𝑗𝑥^(𝑘)𝑢^𝑗(𝑘)²+∇𝑓𝑥^(𝑘)𝜆^(𝑘)−𝑡_𝑘𝑥^(𝑘)−𝑥⁽⁰⁾−∇ℎ𝑥^(𝑘)+𝑡_𝑘𝛽𝑥^(𝑘)−∇ℎ𝑥^(𝑘)𝑧^(𝑘).(3.19)
We know that, since Ω and 𝑢^𝑗(𝑘), 𝑗∉𝐵(𝑥^∗) are bounded as 𝑘→∞, the right-hand side of (3.19) is bounded. But by the assumption (C₂), if 𝑢^𝑗(𝑘)→∞(𝑗∈𝐵(𝑥^∗)), then the left-hand side of (3.19) is infinite; this is a contradiction.
As a conclusion, (a) is the only possible case, and 𝜔^∗ is a solution of the KKT system.
Let 𝑠 be the arc-length of Γ_𝜔(0). We can parameterize Γ_𝜔(0) with respect to 𝑠.

Theorem 3.4. The homotopy path Γ_𝜔(0) is determined by the following initial-value problem for the ordinary differential equation 𝐷𝜔𝐻(0)_𝜔(0)(=𝜔𝜔(𝑠)₍₀₎,𝑡(𝑠))⎡⎢⎢⎢⎢⎢⎢⎣𝜔𝜇⋅⋅⎤⎥⎥⎥⎥⎥⎥⎦=0,,𝑡(0)=1.(3.20) The component 𝜔^∗ of the solution point (𝜔(𝑠^∗),𝑡(𝑠^∗)),   for 𝑡(𝑠^∗)=0, is the solution of the KKT system.

4. Algorithm and Numerical Example

Algorithm 4.1. MOP’s Euler-Newton method.

Step 1. Give an initial point (𝜔⁽⁰⁾,1)∈Ω₀×𝑅^𝑝+𝑚++×{0}×{1}, an initial step length ℎ₀>0, and three small positive numbers 𝜀₁,𝜀₂,𝜀₃. Let 𝑘∶=0.

Step 2. Compute the direction 𝛾^(𝑘) of the predictor step.(a)Compute a unit tangent vector 𝜉^(𝑘)∈𝑅^{𝑛+𝑝+𝑚+𝑠+1} of Γ_𝜔0 at (𝜔⁽⁰⁾,𝑡_𝑘).(b)Determine the direction 𝛾^(𝑘) of the predictor step. If the sign of the determinant |||^𝐷𝐻_𝜔(0)^{(𝜔(𝑘),𝑡}_𝑘^)𝜉_𝑇(𝑘)||| is (−1)^{𝑝+𝑚+𝑠+𝑝𝑚+𝑝𝑠+𝑚𝑠+1}, take 𝛾^(𝑘)=𝜉^(𝑘). If the sign of the determinant |||^𝐷𝐻_𝜔(0)^{(𝜔(𝑘),𝑡}_𝑘^)𝜉_𝑇(𝑘)||| is (−1)^{𝑝+𝑚+𝑠+𝑝𝑚+𝑝𝑠+𝑚𝑠}, take 𝛾^(𝑘)=−𝜉^(𝑘).

Step 3. Compute a corrector point (𝜔^(𝑘+1),𝑡_𝑘+1): 𝜔_(𝑘+1)𝜔^∗^(𝑘),𝑡_∗_𝑘=𝜔^(𝑘),𝑡_𝑘+ℎ_𝑘𝛾^(𝑘),,𝑡_𝑘+1=𝜔^∗^(𝑘),𝑡_∗_𝑘−𝐷𝐻^𝜔(0)𝜔^∗^(𝑘),𝑡_∗_𝑘⁺𝐻_𝜔(0)𝜔^∗^(𝑘),𝑡_∗_𝑘,(4.1) where 𝐷𝐻_𝜔(0)(𝜔,𝑡)⁺=𝐷𝐻_𝜔(0)(𝜔,𝑡)^𝑇𝐷𝐻_𝜔⁽⁰⁾(𝜔,𝑡)𝐷𝐻_𝜔(0)(𝜔,𝑡)^𝑇⁻¹(4.2) is the Moore-Penrose inverse of 𝐷𝐻_𝜔(0)(𝜔,𝑡). If ‖𝐻_𝜔(0)(𝜔^(𝑘+1),𝑡_𝑘+1)‖≤𝜀₁, let ℎ_𝑘+1=min{ℎ₀,2ℎ_𝑘}, and go to Step 4. If ‖𝐻_𝜔(0)(𝜔^(𝑘+1),𝑡_𝑘+1)‖∈(𝜀₁,𝜀₂), let ℎ_𝑘+1=ℎ_𝑘, and go to Step 4. If ‖𝐻_𝜔(0)(𝜔^(𝑘+1),𝑡_𝑘+1)‖≥𝜀₂, let ℎ_𝑘+1=max{(1/2)ℎ₀,(1/2)ℎ_𝑘}, and go to Step 3.

Step 4. If 𝜔^(𝑘+1)∈Ω×𝑅^+𝑝+𝑚×𝑅^𝑠 and 𝑡_𝑘+1>𝜀₃, let 𝑘=𝑘+1 and go to Step 2.
If 𝜔^(𝑘+1)∈Ω×𝑅^+𝑝+𝑚×𝑅^𝑠 and 𝑡_𝑘+1<−𝜀₃, let ℎ_𝑘∶=ℎ_𝑘(𝑡_𝑘/(𝑡_𝑘−𝑡_𝑘+1)), go to Step 3, and recompute (𝜔^(𝑘+1),𝑡_𝑘+1) for the initial point (𝜔^(𝑘),𝑡_𝑘).
If 𝜔^(𝑘+1)∉Ω×𝑅^+𝑝+𝑚×𝑅^𝑠, let ℎ_𝑘∶=(ℎ_𝑘/2)(𝑡_𝑘/(𝑡_𝑘−𝑡_𝑘+1)), go to Step 3, and recompute (𝜔^(𝑘+1),𝑡_𝑘+1) for the initial point (𝜔^(𝑘),𝑡_𝑘).
If 𝜔^(𝑘+1)∈Ω×𝑅^+𝑝+𝑚×𝑅^𝑠, and |𝑡_𝑘+1|≤𝜀₃, then stop.

Example 4.2 (see [9]). Considermin𝑓=min𝑥²¹+𝑥²²+𝑥²³+𝑥²⁴+𝑥²⁵,3𝑥₁+2𝑥₂−13𝑥³s.t.𝑔₁+0.01𝑥₄−𝑥₅³;ℎ(𝑥)=𝑥²¹+𝑥²²+𝑥²³+𝑥²⁴−10≤0;₁ℎ(𝑥)=4𝑥₁−2𝑥₂+0.8𝑥₃+0.6𝑥₄+0.5𝑥²⁵=0;₂(𝑥)=𝑥₁+2𝑥₂−𝑥₃−0.5𝑥₄+𝑥₅−2=0.(4.3)
The results are shown in Table 1.

Table 1 

Results for Example 4.2.

Example 4.3. Considerms.itn.𝑔₁𝑓=min𝑥²¹+𝑥²²,𝑥₁;𝑔(𝑥)=−𝑥₂−6≤0;₂ℎ(𝑥)=𝑥₂−6≤0;₁(𝑥)=𝑥₁−𝑥²²−3=0.(4.4)
(1) Since ∇ℎ(𝑥)=(1,−2𝑥₂)^𝑇, it is easy to see that the assumption (A₃) in [14] and the assumption (C₃) in [15] are not satisfied at most points in feasible set. Hence, we introduce the functions 𝜂_𝑖(𝑥)(𝑖=1,2) and 𝛽(𝑥).
(2) Let 𝜂_𝑖(𝑥)=∇𝑔_𝑖(𝑥)(𝑖=1,2) and 𝛽(𝑥)=(10,0)^𝑇. It is easily verified that the feasible set satisfies the assumptions (C₁)–(C₄). The results are shown in Table 2.

Table 2 

Results for Example 4.3.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant no. 11171003), Key Project of Chinese Ministry of Education (Grant no. 211039), and the Jilin Province Natural Science Foundation (Grant no. 20101597).