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

,


Introduction
Consider the following nonconvex optimization problem min f x s.t.
g i x ≤ 0, i 1, . . ., m, x ∈ R n , 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 n f 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 Ω 0 is the feasible set of NP .
The classical l 1 exact penalty function 1 is 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 l 1 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 l 1 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 l 1 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.

Smooth Approximation to l 1 Exact Penalty Function
In this section we give a family of penalty functions, which decreasingly approximate the l 1 exact penalty function.At first we consider a class of smooth function φ : R → R with the following properties: 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 VI rφ t/r increases with respect to r > 0, for any t ∈ R; The following functions are often used in the smooth approximation of the l 1 exact penalty function and satisfy properties I -IV .
We now use φ • to construct the smooth penalty function 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, Therefore f β,r x smoothly approximates the l 1 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 .

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.
Step 1. Find x k such that 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: Lemma 3.1.Suppose that the assumption A 1 holds, then for any ε > 0, there exists k 0 ∈ N {1, 2, . ..}, such that for k ≥ k 0 , Proof.Suppose to the contrary that there exist an ε 0 > 0 and an infinite sequence K ⊆ N, such that for any k ∈ K, By the algorithm, we know that It follows from 3.4 that there exist a subsequence K 0 ⊆ K and an index i 0 ∈ I {1, . . ., m}, such that for any k ∈ K 0 , Thus, from the assumptions about f • , the properties about φ • , 3.5 and 3.6 , it follows that

3.7
This contradicts with A 1 .
Lemma 3.2.Suppose that the assumption A 1 holds, and x * is any accumulation point of {x k }, then x * ∈ Ω 0 , 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 By the arbitrariness of ε > 0, we have that x * ∈ Ω 0 .
Given x ∈ Ω 0 , we denote that I x {i ∈ I | g i x 0}.
Definition 3.3 see 12 .We say that x ∈ Ω 0 satisfies MFCQ, if there exists a h ∈ R n such that ∇g i x T h < 0, for any i ∈ 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

3.10
Definition 3.4.We say that the sequence {z k } k∈K satisfies GMFCQ, if there exist a subsequence K 0 ⊆ K and a vector h ∈ R n such that lim sup 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.
Proof.By 3.12 , we know that lim sup k∈K,k → ∞ g i z k ≥ 0 if and only if

3.13
Thus, I K I z * .By the assumption, there exists a h ∈ R n such that lim sup k∈K, k → ∞ ∇g i z k T h < 0, for any i ∈ I K .

3.14
We need two assumptions in the following: A 2 the sequence {∇f x k } and {∇g i x k }, i 1, . . ., m are both bounded; A 3 any subsequence of {x k } satisfies GMFCQ.
Theorem 3.6.Suppose that the assumptions A 1 , A 2 , and A 3 hold, then 1 there exists a k 0 such that for any k ≥ k 0 , 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 / ∈ Ω 0 .

3.16
By the algorithm, we know that

3.17
From the assumption A 3 and 3.16 , it follows that there exist K 0 ⊆ K and h ∈ R n such that lim sup k∈K 0 , k → ∞ ∇g i x k T h < 0, for any i ∈ I K 0 , 3.18 By 3.18 and the definition of I − K 0 , there exists a δ > 0, such that for all k ∈ K 0 ,

3.21
From the algorithm, we know that x k satisfies

3.22
Let k ∈ K 0 , from 3.22 we obtain that

3.23
We now analyze the three terms on the left side of 3.23 .
a By 3.17 and A 2 , 3.24 b By 3.21 , for any i ∈ I − K 0 , we have lim

3.25
From the properties of φ • and A 2 , we have that the second term satisfies lim 3.26 c From 3.19 , 3.20 , and the properties of φ • , it follows that where |I| denotes the number of the elements in I.
Now, by letting k → ∞, k ∈ K 0 , and taking the limit on both sides of 3.23 , we obtain from a -c that But by 3.19 and the properties of φ • , δ|I * K 0 |φ 0 > 0. This contradiction completes the proof of 1 .By 1 we know that there exists a k 0 , such that if k ≥ k 0 , then x k ∈ Ω 0 .Thus by the algorithm, when k ≥ k 0 , we have that

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 * ∈ Ω 0 .Thus by 3.22 , we obtain that In the second term of 3.31 , because i ∈ I \ I x * , so by 3.30 and the properties of φ • , we have In the third term of 3.31 , from the properties of φ • , the sequence {φ β k 0 g i x k /r k }, i ∈ I is nonnegative and bounded.Thus, there exists a subsequence 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 β k ,r k x k 0 is equivalent to

3.36
Therefore by 3.36 and the properties of φ • , we have for any x ∈ Ω 0 ,

3.37
From 3.37 , the arbitrariness of x ∈ Ω 0 and the nonnegativity of φ • , it follows that 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

3.39
On the other side, 3.37 implies that A 1 holds.Then from Lemma 3.2, we know x * ∈ Ω 0 .
Theorem 3.9.Suppose that NP is a convex programming problem, and the assumptions A 2 , A 3 hold, then Proof.Note that for NP which is convex, A 1 holds.By Theorem 3.6 there exists a k 0 , such that x k ∈ Ω 0 when k ≥ k 0 .Therefore from the algorithm, we have for any k ≥ k 0 , β k β k 0 .By 3.36 and the property VI of φ • , when k ≥ k 0 ,

3.40
Notice that x k ∈ Ω 0 k ≥ k 0 , by 3.37 and the properties of φ • , we have for k ≥ k 0 that inf 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 A 2 is satisfied.we can also check that {x k } satisfies GMFCQ.In fact, choose h −1, 1 T , then we have

3.44
On the other side, by the algorithm, we have x k ∈ Ω 0 and β k 1, for all k.By letting k → ∞, we get f β k ,r k x k ↓ 0 and f x k → 0. So by the algorithm we get a feasible solution sequence which is also optimal.
for any i ∈ I x * .3.33At last by letting k → ∞, k ∈ K 0 , and taking the limit on both sides of 3.31 , we obtain from 3.30 3.32 and 3.33 that Suppose that A 1 holds, {x k } is bounded, and any accumulation point x * of {x k } satisfies MFCQ, then 1 there exists a k 0 such that for any k ≥ k 0 k } is a KKT point of NP .