On the Convergence of a Smooth Penalty Algorithm without Computing Global Solutions

Abstract

We consider a smooth penalty algorithm to solve nonconvex optimization problem based on a family of smooth functions that approximate the usual exact penalty function. At each iteration in the algorithm we only need to find a stationary point of the smooth penalty function, so the difficulty of computing the global solution can be avoided. Under a generalized Mangasarian-Fromovitz constraint qualification condition (GMFCQ) that is weaker and more comprehensive than the traditional MFCQ, we prove that the sequence generated by this algorithm will enter the feasible solution set of the primal problem after finite times of iteration, and if the sequence of iteration points has an accumulation point, then it must be a Karush-Kuhn-Tucker (KKT) point. Furthermore, we obtain better convergence for convex optimization problem.

1. Introduction

Consider the following nonconvex optimization problem msi.nt.𝑓𝑔(𝑥_𝑖)𝑥∈(𝑥𝑅)≤_𝑛0,𝑖=1,…,𝑚,,(NP) where 𝑓,𝑔_𝑖∶𝑅^𝑛→𝑅,𝑖=1,…,𝑚, are all continuously differentiable functions. Without loss of generality, we suppose throughout this paper that inf_𝑥∈𝑅^𝑛𝑓(𝑥)≥0, because otherwise we can substitute 𝑓(𝑥) by exp(𝑓(𝑥)). Let Ω_𝜀={𝑥∈𝑅^𝑛∣𝑔_𝑖(𝑥)≤𝜀,𝑖=1,…,𝑚} be the relax feasible set for 𝜀>0. Then Ω₀ is the feasible set of (NP).

The classical 𝑙₁ exact penalty function [1] is 𝑓_𝛽(𝑥)=𝑓(𝑥)+𝛽^𝑚_𝑖=1𝑔_𝑖(𝑥)⁺,(1.1) where 𝛽>0 is a penalty parameter, and 𝑔_𝑖(𝑥)⁺=max0,𝑔_𝑖(𝑥),𝑖=1,…,𝑚.(1.2)

The obvious advantage of the traditional exact penalty functions such as the 𝑙₁ exact penalty function is that when the penalty parameter is sufficiently large, their global optimal solutions exist and are optimal solutions of (NP). But they also have obvious disadvantage, that is, their nonsmoothness, which prevent the use of many efficient unconstrained optimization algorithms (such as Gradient-type or Newton-type algorithm). Therefore the study on the smooth approximation of exact penalty functions has attracted broad interests in scholars [2–8]. In recent years based on the smooth approximation of the exact penalty function, several smooth penalty methods are given to solve (NP). For example, [9] gives a smooth penalty method based on approximating the 𝑙₁ exact penalty function. Under the assumptions that the optimal solution satisfies MFCQ and the iterate sequence is bounded, it is proved that the iterative sequence will enter the feasible set and every accumulation point is the optimal solution of (NP). In [10, 11], smooth penalty methods are considered based on approximating low-order exact penalty functions. Reference [10] proves the similar results as [9] under very strict conditions (some of them are uneasy to check). The conditions for convergence of the smooth penalty algorithm in [11] are weaker than that in [10], but in [11] it is only proved that the accumulation point of the iterate sequence is a Fritz-John (FJ) point of (NP).

In the algorithms given by [9–11], at each iteration a global optimal solution of the smooth penalty problem is needed. As we all know, it is very difficult to find a global optimal point of a nonconvex function. To avoid this difficulty, in this paper we give a smooth penalty algorithm based on the smooth approximation of the 𝑙₁ exact penalty function. The feature of this algorithm lies in that only a stationary point of the penalty function is needed to compute at each iteration. To prove the convergence of this algorithm, we first establish a generalized Mangasarian-Fromovitz constraint qualification condition (GMFCQ) weaker and more comprehensive than the traditional MFCQ. Under this condition, we prove that the iterative sequence of the algorithm will enter the feasible set of (NP). Moreover, we prove that if the iterative sequence has accumulation points, then each of them is a KKT point of (NP). Finally, we apply this algorithm to solve convex optimization and get better convergence results.

The rest of this paper is organized as follows. In the next section, we give a family of smooth penalty functions. In Section 3 based on the smooth penalty functions given in Section 2, we propose an algorithm for (NP) and analyze its convergence under the GMFCQ condition. We give an example that satisfies GMFCQ at last in this section.

2. Smooth Approximation to 𝑙₁ Exact Penalty Function

In this section we give a family of penalty functions, which decreasingly approximate the 𝑙₁ exact penalty function. At first we consider a class of smooth function 𝜙∶𝑅→𝑅₊ with the following properties:(I)𝜙(⋅) is a continuously differentiable convex function with 𝜙^′(0)>0;(II)lim_{𝑡→−∞}𝜙(𝑡)=𝑎, where 𝑎 is a nonnegative constant;(III)𝜙(𝑡)≥𝑡, for any 𝑡>0;(IV)lim_𝑡→+∞(𝜙(𝑡))/𝑡)=1.From (I)–(IV), it follows that 𝜙 satisfies(V)0≤𝜙^(𝑡)≤1, for any 𝑡∈𝑅, and lim_{𝑡→−∞}𝜙^(𝑡)=0,lim_𝑡→+∞𝜙^(𝑡)=1;(VI)𝑟𝜙(𝑡/𝑟) increases with respect to 𝑟>0, for any 𝑡∈𝑅;(VII)𝑟𝜙(𝑡/𝑟)↓𝑡⁺(𝑟↓0), for any 𝑡∈𝑅.

The following functions are often used in the smooth approximation of the 𝑙₁ exact penalty function and satisfy properties (I)–(IV).(1)𝜙(𝑡)=log(1+𝑒^𝑡).(2)𝜙(𝑡)=(𝑡+√𝑡²+4)/2. (3)𝜙(𝑡)=^{𝑒𝑡,𝑡≤0;𝑡+1,𝑡>0.}

We now use 𝜙(⋅) to construct the smooth penalty function 𝑓_𝛽,𝑟(𝑥)=𝑓(𝑥)+𝑟^𝑚_𝑖=1𝜙𝛽𝑔^𝑖(𝑥)𝑟,(2.1) where 𝛽≥1 is a penalty parameter.

By (VII), we easily know when 𝑟→0⁺,𝑓_𝛽,𝑟(𝑥) decreasingly converges to 𝑓_𝛽(𝑥), that is, 𝑓_𝛽,𝑟(𝑥)=𝑓(𝑥)+𝑟^𝑚_𝑖=1𝜙𝛽𝑔^𝑖(𝑥)𝑟↓𝑓(𝑥)+𝛽^𝑚_𝑖=1𝑔_𝑖(𝑥)⁺.(2.2) Therefore 𝑓_𝛽,𝑟(𝑥) smoothly approximates the 𝑙₁ exact penalty function, where 𝑟 decreases to improve the precision of the approximation. It is worth noting that the smooth function 𝜙(⋅) and penalty function 𝑓_𝛽,𝑟(⋅) given in this paper make substantive improvement of the corresponding functions given in [9]. This gives 𝑓_𝛽,𝑟(⋅) better convergence properties (refer to (2.2) and Theorem 3.9).

3. The Algorithm and Its Convergence

We propose a penalty algorithm for (NP) in this section based on computing the stationary point of 𝑓_𝛽,𝑟(⋅). We assume that for any 𝛽≥1 and 0<𝑟≤1,𝑓_𝛽,𝑟(⋅) always has stationary point.

Algorithm
Step 0. Given 𝑥⁰∈𝑅^𝑛,𝛽₁=1,𝑟₁=1,0<𝜂₁<1, and 𝜂₂>1. Let 𝑘=1. Step 1. Find 𝑥^𝑘 such that ∇𝑓_{𝛽𝑘,𝑟𝑘}𝑥^𝑘=0.(3.1)Step 2. Put 𝑟_𝑘+1=𝜂₁𝑟_𝑘, 𝛽_𝑘+1=𝛽^𝑘𝜂₂if𝑥^𝑘∈Ω₀,𝛽_𝑘otherwise.(3.2)Step 3. Let 𝑘=𝑘+1 and return to Step 1.

Let {𝑥^𝑘} be the iterative sequence generated by the algorithm. We shall use the following assumption:(𝐴₁) the penalty function value sequence {𝑓_{𝛽𝑘,𝑟𝑘}(𝑥^𝑘)} is bounded.

Lemma 3.1. Suppose that the assumption (𝐴₁) holds, then for any 𝜀>0, there exists 𝑘₀∈𝑁={1,2,…}, such that for 𝑘≥𝑘₀, 𝑥^𝑘∈Ω_𝜀.(3.3)

Proof. Suppose to the contrary that there exist an 𝜀₀>0 and an infinite sequence 𝐾⊆𝑁, such that for any 𝑘∈𝐾, 𝑥^𝑘∉Ω_𝜀0.(3.4) By the algorithm, we know that 𝛽_𝑘⟶+∞(𝑘⟶∞).(3.5) It follows from (3.4) that there exist a subsequence 𝐾₀⊆𝐾 and an index 𝑖₀∈𝐼={1,…,𝑚}, such that for any 𝑘∈𝐾₀, 𝑔_𝑖0𝑥^𝑘>𝜀₀.(3.6) Thus, from the assumptions about 𝑓(⋅), the properties about 𝜙(⋅), (3.5) and (3.6), it follows that 𝑓_{𝛽𝑘,𝑟𝑘}𝑥^𝑘=𝑓𝑥^𝑘+𝑟_𝑘𝑚_𝑖=1𝜙𝛽^𝑘𝑔_𝑖𝑥^𝑘≥𝑓𝑥_𝑘𝑟_𝑘+𝑟_𝑘𝜙𝛽^𝑘𝜀₀≥𝜙𝛽_𝑘𝑟_𝑘𝜀₀/𝑟_𝑘𝛽_𝑘𝜀₀/𝑟_𝑘𝛽^𝑘𝜀₀⟶+∞𝑘⟶∞,𝑘∈𝐾₀.(3.7) This contradicts with (𝐴₁).

Lemma 3.2. Suppose that the assumption (𝐴₁) holds, and 𝑥^∗ is any accumulation point of {𝑥^𝑘}, then 𝑥^∗∈Ω₀, that is, 𝑥^∗ is a feasible solution of (NP).

Proof. By Lemma 3.1, we obtain that for any 𝜀>0 and every sufficiently large 𝑘,𝑥^𝑘∈Ω_𝜀. Let 𝑥^∗ be an accumulation point of {𝑥^𝑘}, then there exists a subsequence {𝑥^𝑘}_𝑘∈𝐾 such that 𝑥^𝑘→𝑥^∗(𝑘∈𝐾,𝑘→∞). Therefore 𝑥^∗∈Ω_𝜀.(3.8) By the arbitrariness of 𝜀>0, we have that 𝑥^∗∈Ω₀.

Given 𝑥∈Ω₀, we denote that 𝐼(𝑥)={𝑖∈𝐼∣𝑔_𝑖(𝑥)=0}.

Definition 3.3 (see [12]). We say that 𝑥∈Ω₀ satisfies MFCQ, if there exists a ℎ∈𝑅^𝑛 such that ∇𝑔_𝑖𝑥^𝑇ℎ<0,forany𝑖∈𝐼𝑥.(3.9)

In the following we propose a kind of generalized Mangasarian-Fromovitz constraint qualification (GMFCQ).

Let 𝐾⊆𝑁 be a subsequence, and for sequence {𝑧^𝑘}_𝑘∈𝐾 in 𝑅^𝑛 denote two index sets as 𝐼𝐼⁺(𝐾)=𝑖∈𝐼limsup_{𝑘∈𝐾,𝑘→∞}𝑔_𝑖𝑧^𝑘≥0₋,(𝐾)=𝑖∈𝐼limsup_{𝑘∈𝐾,𝑘→∞}𝑔_𝑖𝑧^𝑘<0.(3.10)

Definition 3.4. We say that the sequence {𝑧^𝑘}_𝑘∈𝐾 satisfies GMFCQ, if there exist a subsequence 𝐾₀⊆𝐾 and a vector ℎ∈𝑅^𝑛 such that limsup_{𝑘∈𝐾0,𝑘→∞}∇𝑔_𝑖𝑧^𝑘^𝑇ℎ<0,forany𝑖∈𝐼⁺𝐾₀.(3.11)

Under some circumstances, the sequence {𝑥^𝑘} may satisfy that ‖𝑥^𝑘‖→+∞(𝑘→∞), which can be seen for the example in the last part of this section. At this time MFCQ can not be applied, but GMFCQ can. The following proposition shows that Definition 3.4 is a substantive generalization of Definition 3.3.

Proposition 3.5. Suppose that {𝑧^𝑘}_𝑘∈𝐾 satisfies lim_{𝑘∈𝐾,𝑘→∞}𝑧_𝑘=𝑧^∗∈Ω₀.(3.12) If 𝑧^∗ satisfies MFCQ, then {𝑧^𝑘}_𝑘∈𝐾 satisfies GMFCQ.

Proof. By (3.12), we know that limsup_{𝑘∈𝐾,𝑘→∞}𝑔_𝑖(𝑧^𝑘)≥0 if and only if lim_{𝑘∈𝐾,𝑘→∞}𝑔_𝑖𝑧^𝑘=𝑔_𝑖𝑧^∗=0.(3.13) Thus, 𝐼⁺(𝐾)=𝐼(𝑧^∗). By the assumption, there exists a ℎ∈𝑅^𝑛 such that limsup_{𝑘∈𝐾,𝑘→∞}∇𝑔_𝑖𝑧^𝑘^𝑇ℎ<0,forany𝑖∈𝐼⁺(𝐾).(3.14)

We need two assumptions in the following:(𝐴₂) the sequence {∇𝑓(𝑥^𝑘)} and {∇𝑔_𝑖(𝑥^𝑘)},𝑖=1,…,𝑚 are both bounded;(𝐴₃) any subsequence of {𝑥^𝑘} satisfies GMFCQ.

Theorem 3.6. Suppose that the assumptions (𝐴₁), (𝐴₂), and (𝐴₃) hold, then(1)there exists a 𝑘₀ such that for any 𝑘≥𝑘₀, 𝑥^𝑘∈Ω₀;(3.15)(2)any accumulation point of {𝑥^𝑘} is a KKT point of (NP).

Proof. If (1) does not hold, that is, there exists a subsequence 𝐾⊆𝑁 such that for any 𝑘∈𝐾, it holds that 𝑥^𝑘∉Ω₀.(3.16) By the algorithm, we know that lim_𝑘→∞𝛽_𝑘=+∞.(3.17) From the assumption (𝐴₃) and (3.16), it follows that there exist 𝐾₀⊆𝐾 and ℎ∈𝑅^𝑛 such that 𝐼limsup_{𝑘∈𝐾0,𝑘→∞}∇𝑔_𝑖𝑥^𝑘^𝑇ℎ<0,forany𝑖∈𝐼⁺𝐾₀,(3.18)_∗𝐾₀=𝑖∈𝐼∣𝑔_𝑖𝑥^𝑘>0,forany𝑘∈𝐾₀≠∅,𝐼^∗𝐾₀⊆𝐼⁺𝐾₀.(3.19) By (3.18) and the definition of 𝐼⁻(𝐾₀), there exists a 𝛿>0, such that for all 𝑘∈𝐾₀, ∇𝑔𝑔_𝑖𝑥^𝑘^𝑇ℎ≤−𝛿,forany𝑖∈𝐼⁺𝐾₀,(3.20)_𝑖𝑥^𝑘≤−𝛿,forany𝑖∈𝐼⁻𝐾₀.(3.21) From the algorithm, we know that 𝑥^𝑘 satisfies ∇𝑓𝑥^𝑘+^𝑚_𝑖=1𝛽_𝑘𝜙^′𝛽^𝑘𝑔_𝑖𝑥^𝑘𝑟_𝑘∇𝑔^𝑖𝑥^𝑘=0.(3.22) Let 𝑘∈𝐾₀, from (3.22) we obtain that ∇𝑓𝑥^𝑘^𝑇ℎ𝛽_𝑘+_{𝑖∈𝐼−𝐾0}𝜙^𝛽^𝑘𝑔_𝑖𝑥^𝑘𝑟_𝑘∇𝑔^𝑖𝑥^𝑘^𝑇ℎ+_{𝑖∈𝐼+𝐾0}𝜙^𝛽^𝑘𝑔_𝑖𝑥^𝑘𝑟_𝑘∇𝑔^𝑖𝑥^𝑘^𝑇ℎ=0.(3.23) We now analyze the three terms on the left side of (3.23).(a)By (3.17) and (𝐴₂), lim_{𝑘∈𝐾0,𝑘→∞}∇𝑓𝑥^𝑘^𝑇ℎ𝛽_𝑘=0.(3.24)(b)By (3.21), for any 𝑖∈𝐼⁻(𝐾₀), we have lim_{𝑘∈𝐾0,𝑘→∞}𝛽_𝑘𝑔_𝑖𝑥^𝑘𝑟_𝑘=−∞.(3.25) From the properties of 𝜙(⋅) and (𝐴₂), we have that the second term satisfies lim_{𝑘∈𝐾0,𝑘→∞𝑖∈𝐼}_{−𝐾0}𝜙^𝛽^𝑘𝑔_𝑖𝑥^𝑘𝑟_𝑘∇𝑔^𝑖𝑥^𝑘^𝑇ℎ=0.(3.26)(c)From (3.19), (3.20), and the properties of 𝜙(⋅), it follows that _𝑖∈𝐼_+𝐾0𝜙^𝛽^𝑘𝑔_𝑖𝑥^𝑘𝑟_𝑘∇𝑔^𝑖𝑥^𝑘^𝑇ℎ≤−𝛿_{𝑖∈𝐼+𝐾0}𝜙^𝛽^𝑘𝑔_𝑖𝑥^𝑘≤−𝛿_{𝑖∈𝐼∗𝐾0}𝑟_𝑘𝜙^𝛽^𝑘𝑔_𝑖𝑥^𝑘≤−𝛿||𝐼_∗𝑟_𝑘𝐾₀||𝜙^(0),(3.27) where |𝐼| denotes the number of the elements in 𝐼.Now, by letting 𝑘→∞,𝑘∈𝐾₀, and taking the limit on both sides of (3.23), we obtain from (a)–(c) that 𝛿||𝐼^∗𝐾₀||𝜙^(0)≤0.(3.28) But by (3.19) and the properties of 𝜙(⋅),𝛿|𝐼^∗(𝐾₀)|𝜙^(0)>0. This contradiction completes the proof of (1).
By (1) we know that there exists a 𝑘₀, such that if 𝑘≥𝑘₀, then 𝑥^𝑘∈Ω₀. Thus by the algorithm, when 𝑘≥𝑘₀, we have that 𝛽_𝑘=𝛽_𝑘0.(3.29) Suppose that 𝑥^∗ is an accumulation point of {𝑥^𝑘}, then there exists a subsequence {𝑥^𝑘}_𝑘∈𝐾, such that lim_{𝑘∈𝐾,𝑘→∞}𝑥^𝑘=𝑥^∗.(3.30) By Lemma 3.2, 𝑥^∗ is a feasible point of (NP), that is, 𝑥^∗∈Ω₀. Thus by (3.22), we obtain that ∇𝑓𝑥^𝑘+_{𝑖∈𝐼⧵𝐼(𝑥∗)}𝛽_𝑘0𝜙^𝛽^𝑘0𝑔_𝑖𝑥^𝑘𝑟_𝑘∇𝑔^𝑖𝑥^𝑘+_{𝑖∈𝐼(𝑥∗)}𝛽_𝑘0𝜙^𝛽^𝑘0𝑔_𝑖𝑥^𝑘𝑟_𝑘∇𝑔^𝑖𝑥^𝑘=0.(3.31) In the second term of (3.31), because 𝑖∈𝐼⧵𝐼(𝑥^∗), so by (3.30) and the properties of 𝜙^(⋅), we have lim_{𝑘∈𝐾,𝑘→∞}𝜙^𝛽^𝑘0𝑔_𝑖𝑥^𝑘𝑟_𝑘=0.(3.32) In the third term of (3.31), from the properties of 𝜙^(⋅), the sequence {𝜙^(𝛽_𝑘0𝑔_𝑖(𝑥^𝑘)/𝑟_𝑘)},𝑖∈𝐼 is nonnegative and bounded. Thus, there exists a subsequence 𝐾₀⊆𝐾 such that lim_{𝑘∈𝐾0,𝑘→∞}𝛽_𝑘0𝜙^𝛽^𝑘0𝑔_𝑖𝑥^𝑘𝑟_𝑘=𝜆^𝑖≥0,forany𝑖∈𝐼𝑥^∗.(3.33) At last by letting 𝑘→∞,𝑘∈𝐾₀, and taking the limit on both sides of (3.31), we obtain from (3.30)(3.32) and (3.33) that ∇𝑓𝑥^∗+_{𝑖∈𝐼(𝑥∗)}𝜆_𝑖∇𝑔_𝑖𝑥^∗=0.(3.34)

By Lemma 3.2, Proposition 3.5, and Theorem 3.6, we obtain the following conclusion.

Corollary 3.7. Suppose that (𝐴₁) holds, {𝑥^𝑘} is bounded, and any accumulation point 𝑥^∗ of {𝑥^𝑘} satisfies MFCQ, then(1)there exists a 𝑘₀ such that for any 𝑘≥𝑘₀𝑥^𝑘∈Ω₀;(3.35)(2)any accumulation point of {𝑥^𝑘} is a KKT point of (NP).

When (NP) is a convex programming problem, that is, the functions 𝑓 and 𝑔_𝑖,𝑖∈𝐼 of (NP) are all convex functions, the algorithm has better convergence results.

Theorem 3.8. Suppose (NP) is a convex programming problem, then every accumulation point of {𝑥^𝑘} is an optimal solution of (NP).

Proof. Since 𝑓(⋅),𝑔_𝑖(⋅),𝑖∈𝐼 are convex, and 𝜙(⋅) is increasing, then for any 𝛽>0 and 𝑟>0,𝑓_𝛽,𝑟(⋅) is convex. Thus ∇𝑓_{𝛽𝑘,𝑟𝑘}(𝑥^𝑘)=0 is equivalent to 𝑥^𝑘∈argmin_{𝑥∈𝑅𝑛}𝑓_{𝛽𝑘,𝑟𝑘}(𝑥).(3.36) Therefore by (3.36) and the properties of 𝜙(⋅), we have for any 𝑥∈Ω₀, 𝑓_{𝛽𝑘,𝑟𝑘}𝑥^𝑘=𝑓𝑥^𝑘+𝑟_𝑘𝑚_𝑖=1𝜙𝛽^𝑘𝑔_𝑖𝑥^𝑘≤𝑓𝑟_𝑘𝑥+𝑟_𝑘𝑚_𝑖=1𝜙𝛽^𝑘𝑔_𝑖𝑥≤𝑓𝑟_𝑘𝑥+𝑟_𝑘𝑚𝜙(0).(3.37) From (3.37), the arbitrariness of 𝑥∈Ω₀ and the nonnegativity of 𝜙(⋅), it follows that 𝑓𝑥^𝑘≤inf_𝑥∈Ω0𝑓(𝑥)+𝑟_𝑘𝑚𝜙(0).(3.38) Suppose that 𝑥^∗ is an accumulation point of {𝑥^𝑘}, there exists a subsequence 𝐾⊆𝑁 such that lim_{𝑘∈𝐾,𝑘→∞}𝑥^𝑘=𝑥^∗. Thus, by (3.38), we have 𝑓𝑥^∗≤inf_𝑥∈Ω0𝑓(𝑥).(3.39) On the other side, (3.37) implies that (𝐴₁) holds. Then from Lemma 3.2, we know 𝑥^∗∈Ω₀.

Theorem 3.9. Suppose that (NP) is a convex programming problem, and the assumptions (𝐴₂), (𝐴₃) hold, then(1)there exists a 𝑘₀, for any 𝑘≥𝑘₀,{𝑓_{𝛽𝑘,𝑟𝑘}(𝑥^𝑘)} decreases to inf_𝑥∈Ω0𝑓(𝑥).(2)lim_𝑘→∞𝑓(𝑥^𝑘)=inf_𝑥∈Ω0𝑓(𝑥).

Proof. Note that for (NP) which is convex, (𝐴₁) holds. By Theorem 3.6 there exists a 𝑘₀, such that 𝑥^𝑘∈Ω₀ when 𝑘≥𝑘₀. Therefore from the algorithm, we have for any 𝑘≥𝑘₀,𝛽_𝑘=𝛽_𝑘0. By (3.36) and the property (VI) of 𝜙(⋅), when 𝑘≥𝑘₀, 𝑓_{𝛽𝑘0,𝑟𝑘+1}𝑥^𝑘+1=≤𝑓𝑓𝑥_{𝛽𝑘0,𝑟𝑘+1𝑘}𝑥^𝑘+𝑟_𝑚𝑘+1_𝑖=1𝜙𝛽^𝑘0𝑔_𝑖𝑥^𝑘≤𝑓𝑥_𝑘𝑟_𝑘+1+𝑟^𝑘𝑚_𝑖=1𝜙𝛽^𝑘0𝑔_𝑖𝑥^𝑘≤𝑓_{𝛽𝑘0,𝑟𝑘}𝑟_𝑘𝑥^𝑘.(3.40) Notice that 𝑥^𝑘∈Ω₀(𝑘≥𝑘₀), by (3.37) and the properties of 𝜙(⋅), we have for 𝑘≥𝑘₀ that inf_𝑥∈Ω0𝑓(𝑥)≤≤𝑓𝑓𝑥^𝑘_{𝛽𝑘0,𝑟𝑘}≤inf_𝑥∈Ω0𝑥^𝑘𝑓(𝑥)+𝑟_𝑘𝑚𝜙(0).(3.41) Combining (3.40) with (3.41), we obtain the conclusion.

Example 3.10. Consider that min_𝑥∈Ω0𝑓(𝑥)=(1/4)(𝑥₁−𝑥₂)²,Ω₀={𝑥∈𝑅²∣𝑔(𝑥)=𝑥₁−𝑥₂≤0}.
This is a convex case. Denote its optimal solution by Ω^∗0={𝑥^∗∈Ω₀∣𝑥^∗1−𝑥^∗2=0} and let 𝜙(𝑡)=(𝑡+√𝑡²+4)/2. We consider 𝑓_𝛽,𝑟(⋅), that is, 𝑓_𝛽,𝑟(𝑥)=14𝑥¹−𝑥₂²+𝑟2⎛⎜⎜⎝𝛽²𝑟²𝑥₁−𝑥₂²+4+𝛽𝑟𝑥¹−𝑥₂⎞⎟⎟⎠.(3.42) Because 𝑓_𝛽,𝑟(⋅) is convex, thus ∇𝑓_𝛽,𝑟(𝑥^∗)=0 if and only if 𝑥^∗∈argmin_𝑥∈𝑅²𝑓_𝛽,𝑟(𝑥). By the algorithm, we get stationary points as 𝑥^𝑘=⎛⎜⎜⎝𝑘+𝑘𝛼_𝑘⎞⎟⎟⎠,𝑘=0,1,…,(3.43) where 𝛼_𝑘>0 and lim_𝑘→∞𝛼_𝑘=0. Here {𝑥^𝑘} has no accumulation point, that is, lim_𝑘→∞‖𝑥^𝑘‖=+∞. Thus in the analysis of convergence, MFCQ may not be appropriate to be applied as a constraint qualification condition for this example. But for any 𝑘∈𝑁, we have ∇𝑓(𝑥^𝑘)=(−(1/2)𝛼_𝑘,(1/2)𝛼_𝑘)^𝑇, ∇𝑔(𝑥^𝑘)=(1,−1)^𝑇, which implies that assumption (𝐴₂) is satisfied. we can also check that {𝑥^𝑘} satisfies GMFCQ. In fact, choose ℎ=(−1,1)^𝑇, then we have lim_𝑘→∞lim_𝑘→∞𝑔𝑥^𝑘=lim_𝑘→∞𝛼_𝑘=0,∇𝑔𝑥^𝑘^𝑇ℎ=(1,−1)^𝑇(−1,1)=−2<0.(3.44) On the other side, by the algorithm, we have 𝑥^𝑘∈Ω₀ and 𝛽_𝑘=1, for all 𝑘. By letting 𝑘→∞, we get 𝑓_{𝛽𝑘,𝑟𝑘}(𝑥^𝑘)↓0 and 𝑓(𝑥^𝑘)→0. So by the algorithm we get a feasible solution sequence which is also optimal.

Acknowledgments

This paper was supported by the National Natural Science Foundations of China (10971118, 10701047, and 10901096). The authors would like to give their thanks to the editor and anonymous referees for their valuable suggestions and comments.