New Convergence Properties of the Primal Augmented Lagrangian Method

Abstract

New convergence properties of the proximal augmented Lagrangian method is established for continuous nonconvex optimization problem with both equality and inequality constrains. In particular, the multiplier sequences are not required to be bounded. Different convergence results are discussed dependent on whether the iterative sequence {𝑥^𝑘} generated by algorithm is convergent or divergent. Furthermore, under certain convexity assumption, we show that every accumulation point of {𝑥^𝑘} is either a degenerate point or a KKT point of the primal problem. Numerical experiments are presented finally.

1. Introduction

In this paper, we consider the following nonlinear programming problem:min𝑓(𝑥)s.t.𝑔_𝑖(𝑥)ℎ≤0,𝑖=1,…,𝑚;_𝑗(𝑥)=0,𝑗=1,…,𝑙;𝑥∈Ω, (𝑃) where 𝑓,𝑔_𝑖∶ℝ^𝑛→ℝ for each 𝑖=1,…,𝑚 and ℎ_𝑗∶ℝ^𝑛→ℝ for each 𝑗=1,…,𝑙 are all continuously differentiable functions, Ω is a nonempty and closed set in ℝ^𝑛. Denoted by 𝑋 the feasible region and by 𝑋^∗ the solution set.

Augmented Lagrangian algorithms are very popular tools for solving nonlinear programming problems. At each outer iteration of these methods, a simpler optimization problem is solved, for which efficient algorithms can be used, especially when the problems are large. The most famous augmented Lagrangian algorithm based on the Powell-Hestenes-Rockafellar [1–3] formula has been successfully used for defining practical nonlinear programming algorithms [4–7]. At each iteration, a minimization problem with simple constraints is approximately solved whereas Lagrange multipliers and penalty parameters are updated in the master routine. The advantage of the Augmented Lagrangian approach over other methods is that the subproblems can be solved using algorithms that can deal with a very large number of variables without making use of factorization of matrices of any kind.

An indispensable assumption in the most existing global convergence analysis for augmented Lagrangian methods is that the multiplier sequence generated by the algorithms is bounded. This restrictive assumption confines applications of augmented Lagrangian methods in many practical situation. The important work on this direction includes [8], where global convergence of modified augmented Lagrangian methods for nonconvex optimization with equality constraints was established; and Andreani et al. [4] and Birgin et al. [9] investigated the augmented Lagrangian methods using safeguarding strategies for nonconvex constrained problems. Recently, for inequality-constrained global optimization, Luo et al. [10] established the convergence properties of the primal-dual method based on four types of augmented Lagrangian functions without the boundedness assumption of the multiplier sequence. More information can be found in [5, 11, 12].

In this paper, for the optimization problem (𝑃) with both equality and inequality constraints, we further study the convergence property of the proximal Lagrangian method without requiring the boundedness of multiplier sequences. The main contribution of this paper lies in the following three aspects. First, more general constraints are considered, without restricting only inequality constraints as in [10, 13] and requiring boundedness of 𝑋 as in [9]. Second, an essential assumption on the global convergence properties given in [4–7, 9, 10] is that the iterative sequence {𝑥^𝑘} must be convergent in advance; here, we further discuss the case when {𝑥^𝑘} is divergent and develop a necessary and sufficient condition for {𝑓(𝑥^𝑘)} converging to the optimal value of primal problem. Third, the definition of degeneration in [9, 10] is extended from inequality constraint to both inequality and equality constraints.

This paper is organized as follows. In Section 2, we propose the multiplier algorithm and study its global convergence properties. Preliminary numerical results are reported in Section 3. The conclusion is drawn in Section 4.

2. Multiplier Algorithms

The primal augmented Lagrangian function for (𝑃) is𝐿(𝑥)+𝑐𝑥,𝜆,𝜇,𝑐∶=𝑓2^𝑙_𝑗=1ℎ_𝑗(𝑥)+𝜇_𝑗𝑐²+^𝑚_𝑖=1max0,𝑔_𝑖(𝑥)+𝜆_𝑖𝑐², (2.1) where (𝑥,𝜆,𝜇,𝑐)∈ℝ^𝑛×ℝ^𝑚×ℝ^𝑙×ℝ₊₊, and ℝ₊₊ denotes the all positive real scalars, that is, ℝ₊₊={𝑎∈ℝ∣𝑎>0}.

Given (𝑥,𝜆,𝜇,𝑐), the augmented Lagrangian relaxation problem associated with the augmented Lagrangian 𝐿 is defined bymin𝐿𝑥,𝜆,𝜇,𝑐s.t.𝑥∈Ω. (𝐿_{𝜆,𝜇,𝑐}) Given 𝜀≥0, then the 𝜀-optimal solution set of (𝐿_{𝜆,𝜇,𝑐}), denoted by 𝑆^∗(𝜆,𝜇,𝑐,𝜀), is defined as𝑥∈Ω∣𝐿𝑥,𝜆,𝜇,𝑐≤inf_𝑥∈Ω𝐿𝑥,𝜆,𝜇,𝑐+𝜀. (2.2)

If Ω is closed and bounded, then the global optimal solution of (𝐿_{𝜆,𝜇,𝑐}) exists. However, if Ω is unbounded, then (𝐿_{𝜆,𝜇,𝑐}) maybe unsolvable. To overcome this difficultly, we assume throughout this paper that 𝑓 is bounded on Ω from below, that is,𝑓_∗∶=inf_𝑥∈Ω𝑓(𝑥)>−∞. (2.3) This assumption is rather mild in optimization programming, because otherwise the objective function 𝑓 can be replaced by 𝑒^𝑓(𝑥). It ensures that the 𝜀-optimal solution set with 𝜀>0 always exists, since 𝐿(𝑥,𝜆,𝜇,𝑐) is bounded from below by (2.1) and (2.3).

Recall that a vector 𝑥^∗ is said to be a KKT point of (𝑃) if there exist 𝜆^∗_𝑖≥0 for each 𝑖=1,…,𝑚 and 𝜇^∗_𝑗 for each 𝑗=1,…,𝑙 such that0∈∇𝑓(𝑥)+∑^𝑚_𝑖=1𝜆_𝑖∇𝑔_𝑖(𝑥)+∑^𝑙_𝑗=1𝜇_𝑗∇ℎ_𝑗(𝑥)+𝒩_Ω𝑥^∗,𝜆^∗_𝑖𝑔_𝑖𝑥^∗=0,∀𝑖=1,…,𝑚, (2.4) where 𝒩_Ω(𝑥^∗) denotes the normal cone of Ω at 𝑥^∗. The collection set of all 𝜆^∗ and 𝜇^∗ satisfying (2.4) is denoted by Λ(𝑥^∗).

The multiplier algorithm based on the primal augmented Lagrangian 𝐿 is proposed below. One of its main features is that the Lagrangian multipliers associated with equality and inequality constraints are not restricted to be bounded, which makes the algorithm applicable for many problems in practice.

Algorithm 2.1 (Multiplier algorithm based on 𝐿). Step 1. Select an initial point 𝑥⁰∈ℝ^𝑛, 𝜆⁰≥0, 𝜇₀∈ℝ, 𝑐₀>0, and 𝜀₀≥0. Set 𝑘∶=0.Step 2. Compute 𝜆_𝑖^𝑘+1=max0,𝑐_𝑘𝑔_𝑖𝑥^𝑘+𝜆^𝑘_𝑖,∀𝑖=1,…,𝑚, (2.5)𝜇_𝑗^𝑘+1=𝜇^𝑘_𝑗+𝑐_𝑘ℎ_𝑗𝑥^𝑘,∀𝑗=1,…,𝑙, (2.6)𝜀_𝑘+1=𝜀_𝑘𝑘+1, (2.7)𝑐_𝑘+1≥(𝑘+1)max1,^𝑚_𝑖=1𝜆_𝑖^𝑘+12,^𝑙_𝑗=1𝜇_𝑗^𝑘+12. (2.8)Step 3. Find 𝑥^𝑘+1∈𝑆^∗(𝜆^k+1,𝜇^𝑘+1,𝑐_𝑘+1,𝜀_𝑘+1);Step 4. If 𝑥^𝑘+1∈𝑋 and (𝜆^𝑘+1,𝜇^𝑘+1)∈Λ(𝑥^𝑘+1), then STOP; otherwise, let 𝑘∶=𝑘+1 and go back to Step 2.

The iterative formula for 𝜀_𝑘+1 given in (2.7) is just used to guarantee its convergence to zero. In fact, in the practical numerical experiment, we can choose 𝜀_𝑘+1=𝜀_𝑘/𝑐_𝑘 to improve the convergence of the algorithm. The following lemma gives the relationship between the penalty parameter 𝑐_𝑘 and the multipliers 𝜆^𝑘 and 𝜇^𝑘.

Lemma 2.2. Let (𝜆^𝑘,𝜇^𝑘,𝑐_𝑘) be given as in Algorithm 2.1, then the following terms 𝜆^𝑘𝑐_𝑘,𝜇^𝑘𝑐_𝑘,𝜆^𝑘2𝑐_𝑘,𝜇^𝑘2𝑐_𝑘 (2.9) all approach to zero as 𝑘→∞.

Proof. This follows immediately from (2.8).

For establishing the convergence property of Algorithm 2.1, we first consider the perturbation analysis of (𝑃). Given 𝛼≥0, define the perturbation of feasible region as𝑋(𝛼)=𝑥∈Ω∣𝑔_𝑖(𝑥)≤𝛼,ℎ_𝑗(𝑥)≤𝛼,𝑖=1,…,𝑚,𝑗=1,…,𝑙, (2.10) and the perturbation of level set as𝐿(𝛼)=𝑥∈Ω∣𝑓(𝑥)≤𝑣(0)+𝛼. (2.11) It is clear that 𝑋(0) coincides with the feasible set of (𝑃). The corresponding perturbation function is given as𝑣(𝛼)=inf𝑓(𝑥)∣𝑥∈𝑋(𝛼). (2.12)

The following result shows that the perturbation value function is upper semicontinuous at zero.

Lemma 2.3. The perturbation function 𝑣 is upper semicontinuous at zero from right.

Proof. Since 𝑋(0)⊂𝑋(𝛼) for any 𝛼≥0, then 𝑣(𝛼)≤𝑣(0) by definition (2.12). This implies that limsup_𝛼→0+𝑣(𝛼)≤𝑣(0).

Lemma 2.4. Let (𝜆^𝑘,𝜇^𝑘,𝑐_𝑘,𝜀_𝑘) be given as in Algorithm 2.1. For any 𝜀>0, one has 𝑆^∗𝜆^𝑘,𝜇^𝑘,𝑐_𝑘,𝜀_𝑘⊆𝑥∈Ω∣𝐿𝑥,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘≤𝑣(0)+𝜀, (2.13) whenever 𝑘 is sufficiently large.

Proof. For any given 𝜀, it follows from (2.7) and Lemma 2.4 that when 𝑘 is large enough, we have 12𝑐_𝑘^𝑚_𝑖=1𝜆^𝑘_𝑖𝑐_𝑘²+12𝑐_𝑘^𝑙_𝑖=1𝜇^𝑘_𝑖𝑐_𝑘²+𝜀_𝑘≤𝜀. (2.14) Therefore, for 𝑥∈𝑆^∗(𝜆^𝑘,𝜇^𝑘,𝑐_𝑘,𝜀_𝑘), 𝐿𝑥,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘≤inf𝐿𝑥,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘∣𝑥∈Ω+𝜀_𝑘≤inf𝐿𝑥,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘∣𝑥∈𝑋(0)+𝜀_𝑘𝑓(𝑥)(0+1≤inf∣𝑥∈𝑋)2𝑐_𝑘^𝑚_𝑖=1𝜆^𝑘_𝑖𝑐_𝑘²+12𝑐_𝑘^𝑙_𝑖=1𝜇^𝑘_𝑖𝑐_𝑘²+𝜀_𝑘≤𝑣(0)+𝜀. (2.15)

Lemma 2.5. Let (𝜆^𝑘,𝜇^𝑘,𝑐_𝑘) be given as in Algorithm 2.1. For any 𝜀>0, one has 𝑥∈Ω∣𝐿𝑥,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘≤𝑣(0)+𝜀⊆𝑋(𝜀). (2.16) whenever 𝑘 is sufficiently large.

Proof. We prove this result by the way of contradiction. Suppose that we can find an 𝜀₀>0 and an infinite subsequence 𝐾⊆{1,2,…} such that 𝑧^𝑘∈𝑥∈Ω∣𝐿𝑥,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘≤𝑣(0)+𝜀,∀𝑘∈𝐾, (2.17) but 𝑧^𝑘∉𝑋𝜀₀,∀𝑘∈𝐾. (2.18) It follows from (2.17) that 𝑣(0)+𝜀≥𝐿𝑧^𝑘,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘=𝑓𝑧^𝑘+𝑐_𝑘2^𝑙_𝑗=1ℎ_𝑗𝑧^𝑘+𝜇^𝑘_𝑗𝑐_𝑘²+^𝑚_𝑖=1max0,𝑔_𝑖𝑧^𝑘+𝜆^𝑘_𝑖𝑐_𝑘². (2.19)
Since 𝑧^𝑘∉𝑋(𝜀₀), it needs to consider the following two cases.
Case 1. There exist an index 𝑗₀ and an infinite subsequence 𝐾₀⊆𝐾 such that |ℎ_𝑗0(𝑧^𝑘)|>𝜀₀. It then follows from (2.19) that 𝑣(0)+𝜀≥𝑓_∗+𝑐_𝑘2^𝑙_𝑗=1ℎ_𝑗𝑧^𝑘+𝜇^𝑘_𝑗𝑐_𝑘²+^𝑚_𝑖=1max0,𝑔_𝑖𝑧^𝑘+𝜆^𝑘_𝑖𝑐_𝑘²≥𝑓_∗+𝑐_𝑘2ℎ_𝑗0𝑧^𝑘+𝜇^𝑘_𝑗0𝑐_𝑘². (2.20)
Using Lemma 2.2 and the fact that |ℎ_𝑗0(𝑧^𝑘)|≥𝜀₀ gives us ℎ_𝑗0𝑧^𝑘+𝜇^𝑘_𝑗0𝑐_𝑘²≥12𝜀₀, (2.21) whenever 𝑘 is sufficiently large. This, together with (2.20), yields 𝑣(0)=+∞ by taking 𝑘∈𝐾₀ approaching to ∞, which leads to a contradiction.Case 2. There exist an index 𝑖₀ and an infinite subsequence 𝐾₀⊆𝐾 such that 𝑔_𝑖0(𝑧^𝑘)>𝜀₀. It follows from (2.19) that 𝑣(0)+𝜀≥𝑓_∗+𝑐_𝑘2^𝑙_𝑗=1ℎ_𝑗𝑧^𝑘+𝜇^𝑘_𝑗𝑐_𝑘²+^𝑚_𝑖=1max0,𝑔_𝑖𝑧^𝑘+𝜆^𝑘_𝑖𝑐_𝑘²≥𝑓_∗+𝑐_𝑘2_𝑖≠𝑖0max0,𝑔_𝑖𝑧^𝑘+𝜆^𝑘_𝑖𝑐_𝑘²+max0,𝑔_𝑖0𝑧^𝑘+𝜆^𝑘_𝑖0𝑐_𝑘²≥𝑓_∗+𝑐_𝑘2max0,𝑔_𝑖0𝑧^𝑘+𝜆^𝑘_𝑖0𝑐_𝑘²≥𝑓_∗+𝜀₀𝑐_𝑘4, (2.22) where the last step is due to Lemma 2.2, since 𝑔_𝑖0(𝑧^𝑘)>𝜀₀ and 𝜆^𝑘_𝑖0/𝑐_𝑘→0. Taking limits in the above inequality yields 𝑣(0)=+∞, which is a contradiction. This completes the proof.

Lemma 2.6. Let (𝜆^𝑘,𝜇^𝑘,𝑐_𝑘) be given as in Algorithm 2.1. For any 𝜀>0, one has 𝑥∈Ω∣𝐿𝑥,𝜆^k,𝜇^𝑘,𝑐_𝑘≤𝑣(0)+𝜀⊆𝐿(𝜀),∀𝑘=1,2,…. (2.23)

Proof. For an arbitrarily 𝑥∈{𝑥∈Ω∣𝐿(𝑥,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘)≤𝑣(0)+𝜀}, we have 𝑓(𝑥)=𝐿𝑥,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘−𝑐_𝑘2^𝑙_𝑗=1ℎ_𝑗(𝑥)+𝜇^𝑘_𝑗𝑐_𝑘²+^𝑚_𝑖=1max0,𝑔_𝑖(𝑥)+𝜆^𝑘_𝑖𝑐_𝑘²≤𝑣(0)+𝜀. (2.24) The proof is complete.

With these preparation, the global convergence property of Algorithm 2.1 can be given, which shows that if the algorithm terminates in finite steps, then we obtain a KKT point of (𝑃); otherwise, every limit point of {𝑥^𝑘} would be the optimal solution of (𝑃).

Theorem 2.7. Let {𝑥^𝑘} be the iterative sequence generated by Algorithm 2.1. Then if {𝑥^𝑘} is terminated in finite steps, then one gets a KKT point of (𝑃); otherwise, every limit point of {𝑥^𝑘} belongs to 𝑋^∗.

Proof. According to the construction of Algorithm 2.1, the first part is clear. It remains to prove the second part. Let 𝜀>0 be given. It follows from Lemmas 2.4–2.6 that when 𝑘 is large enough, we have 𝑆^∗𝜆^𝑘,𝜇^𝑘,𝑐_𝑘,𝜀_𝑘⊆𝑥∈Ω∣𝐿𝑥,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘(0)(𝜀)(𝜀).≤𝑣+𝜀⊆𝑋∩𝐿 (2.25) Thus, 𝑥^𝑘∈𝑋(𝜀)∩𝐿(𝜀). (2.26) Note that 𝑋(𝜀) and 𝐿(𝜀) are closed, due to the continuity of 𝑓, 𝑔_𝑖 for all 𝑖=1,…,𝑚 and ℎ_𝑗 for all 𝑗=1,…,𝑙 and the closeness of Ω. Taking the limit in (2.26) yields 𝑥^∗∈𝑋(𝜀)∩𝐿(𝜀), which further shows that 𝑥^∗∈𝑋(0)∩𝐿(0), since 𝜀>0 is arbitrary, that is, 𝑥^∗∈𝑋^∗. The proof is complete.

The foregoing result is applicable to the case when {𝑥^𝑘} at least has an accumulation point. However, a natural question arises: how does the algorithm perform as {𝑥^𝑘} is divergent? The following theorem gives an answer.

Theorem 2.8. Let {𝑥^𝑘} be an iterative sequence generated by Algorithm 2.1. Then, lim_𝑘→∞𝑓𝑥^𝑘=𝑣(0) (2.27) if and only if 𝑣(𝛼) is lower semicontinuous at 𝛼=0 from right.

Proof. We first show the sufficiency. According to the proof of Theorem 2.7 (recall (2.26)), we know that 𝑣(𝜀)≤𝑓𝑥^𝑘≤𝑣(0)+𝜀, (2.28) whenever 𝑘 is sufficiently large. Since 𝑣(𝛼) is lower semicontinuous at 𝛼=0 from right, taking the lower limitation in (2.28) yields 𝑣(0)≤liminf_𝜀→0+𝑣(𝜀)≤liminf_𝑘→∞𝑓𝑥^𝑘≤limsup_𝑘→∞𝑓𝑥^𝑘≤𝑣(0), (2.29) that is, lim_𝑘→∞𝑓𝑥^𝑘=𝑣(0). (2.30) We now show the necessity. Suppose on the contrary that 𝑣 is not lower semicontinuous at zero from right, then there exist 𝛿₀>0 and 𝜀_𝑗→0⁺(as𝑗→∞) such that 𝑣𝜀_𝑗≤𝑣(0)−𝛿₀,∀𝑗=1,2,…,𝑚. (2.31) For any given 𝑘, since 𝜀_𝑗→0 we can choose a subsequence 𝑗_𝑘 satisfying 𝜀_𝑗𝑘𝑐_𝑘⟶0as𝑘⟶∞. (2.32) In addition, let 𝑧^𝑘∈𝑋(𝜀_𝑗𝑘) with 𝑓(𝑧^𝑘)≤𝑣(𝜀_𝑗𝑘)+𝛿₀/2, which further implies 𝑓(𝑧^𝑘)≤𝑣(0)−𝛿₀/2 by (2.31). Therefore, 𝑓𝑥^𝑘=𝐿𝑥^𝑘,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘−𝑐_𝑘2^𝑙_𝑗=1ℎ_𝑗𝑥^𝑘+𝜇^𝑘_𝑗𝑐_𝑘²+^𝑚_𝑖=1max0,𝑔_𝑖𝑥^𝑘+𝜆^𝑘_𝑖𝑐_𝑘²≤inf_𝑥∈Ω𝐿𝑥,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘+𝜀_𝑘≤𝑓𝑧^𝑘+𝑐_𝑘2^𝑙_𝑗=1ℎ_𝑗𝑧^𝑘+𝜇^𝑘_𝑗𝑐_𝑘²+^𝑚_𝑖=1max0,𝑔_𝑖z^𝑘+𝜆^𝑘_𝑖𝑐_𝑘²+𝜀_𝑘(0)−𝛿≤𝑣⁰2+𝑐_𝑘2^𝑙_𝑗=1𝜀_𝑗𝑘+𝜇^𝑘_𝑗𝑐_𝑘²+^𝑚_𝑖=1max0,𝜀_𝑗𝑘+𝜆^𝑘_𝑖𝑐_𝑘²+𝜀_𝑘, (2.33) where the last step is due to the fact |ℎ_𝑗(𝑧^𝑘)|≤𝜀_𝑗𝑘 and 𝑔_𝑖(𝑧^𝑘)≤𝜀_𝑗𝑘 since 𝑧^𝑘∈𝑋(𝜀_𝑗𝑘). Taking limits in both sides of (2.33) and using (2.7), (2.27), and Lemma 2.2, we get 𝑣(0)=lim_𝑘→∞𝑓𝑥^𝑘(0)−𝛿≤𝑣⁰2, (2.34) which leads to a contradiction. The proof is complete.

Note that in many practical cases, the set Ω typically stands for a more simple constraint, for example, a box or a bounded polytope [7]. Hence, we conclude this paper by considering the case of Ω is a bounded, closed, and convex subset of ℝ^𝑛. In this case, the global optimal solution of the augmented Lagrangian relaxation problem always exists. Hence, we choose 𝜀₀=0 in Step 1 of Algorithm 2.1, which in turn implies that 𝜀_𝑘=0 for all 𝑘 by (2.7). First, however, we need to extend the definition of degenerate from inequality constraint as in [10] to both inequality and equality constraints.

Definition 2.9. A point 𝑥^∗∈𝑋 is said to be degenerate if there exists 𝜆^∗∈ℝ^𝑚₊ and 𝜇^∗∈ℝ^𝑙 such that _{(𝑥𝑖∈𝐼∗)}𝜆^∗_𝑖+^𝑙_𝑗=1𝜇^∗_𝑗>0,𝒫_Ω𝑥^∗−^𝑙_𝑗=1𝜇^∗_𝑗∇ℎ_𝑗𝑥^∗−_{(𝑥𝑖∈𝐼∗)}𝜆^∗_𝑖∇𝑔_𝑖𝑥^∗=𝑥^∗, (2.35) where 𝒫_Ω(𝑥) denotes the projection of 𝑥 onto Ω and 𝐼(𝑥^∗)={𝑖∣𝑔_𝑖(𝑥^∗)=0,𝑖=1,…,𝑚}.

Theorem 2.10. Suppose that Ω is a bounded, closed, and convex set of ℝ^𝑛. Let 𝜀₀=0 and {𝑥^𝑘} be the iterative sequence generated by Algorithm 2.1. Then, every accumulation point of {𝑥^𝑘}, say 𝑥^∗, is either a degenerate or a KKT point of (𝑃).

Proof. Noting that 𝜀_𝑘=0 for all 𝑘 by 𝜀₀=0 and (2.7), then {𝑥^𝑘} is a global optimal solution of 𝐿(𝑥,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘) by Step 3 in Algorithm 2.1. Applying the well-known optimality condition of optimization problem to the augmented Lagrangian relaxation problem (𝐿_{𝜆,𝜇,𝑐}) yields −∇_𝑥𝐿𝑥,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘∈𝒩_Ω𝑥^𝑘, (2.36) where 𝒩_Ω(𝑥^𝑘) is the normal cone of Ω at 𝑥^𝑘. This together with (2.5) and (2.6) means that 𝒫_Ω𝑥^𝑘−∇𝑓𝑥^𝑘−^𝑙_𝑗=1𝜇_𝑗^𝑘+1∇ℎ_𝑗𝑥^𝑘−^𝑚_𝑖=1𝜆_𝑖^𝑘+1∇𝑔_𝑖𝑥^𝑘=𝑥^𝑘, (2.37) where we have used the basic property of normal cone of convex set. Let 𝒦 be an infinite subsequence in {1,2,…} such that {𝑥^𝑘}_𝒦→𝑥^∗∈Ω. Consider now the following two cases.Case 1. Either {𝜆^𝑘+1}_𝒦 or {𝜇^𝑘+1}_𝒦 is unbounded. In this case, we must have 𝑇^𝑘∶=^𝑚_𝑖=1𝜆_𝑖^𝑘+1+^𝑙_𝑗=1𝜇_𝑗^𝑘+1⟶∞,𝑘⟶∞,𝑘∈𝒦. (2.38) Since 0≤𝜆_𝑖^𝑘+1/𝑇^𝑘≤1 and 0≤𝜇_𝑖^𝑘+1/𝑇^𝑘≤1 are bounded, we can assume by passing a subsequence if necessary that 𝜆_𝑖^𝑘+1𝑇^𝑘⟶𝜆^∗_𝑖,𝜇_𝑗^𝑘+1𝑇^𝑘⟶𝜇^∗_𝑗,𝑖=1,…,𝑚,𝑗=1,…,𝑙. (2.39) Clearly, 𝜆^∗_𝑖 and 𝜇^∗_𝑗 are not all zeros. On the other hand, since 𝒩_Ω(𝑥^∗) is cone, then it follows from (2.36) that −1𝑇^𝑘∇_𝑥𝐿𝑥,𝜆^𝑘,𝜇^𝑘,𝑐_𝑘∈𝒩_Ω𝑥^𝑘, (2.40) from which and using the basic property of normal cone of convex set, we further have 𝒫_Ω𝑥^𝑘−1𝑇^𝑘∇𝑓𝑥^𝑘+^𝑙_𝑗=1𝜇_𝑗^𝑘+1∇ℎ_𝑗𝑥^𝑘+^𝑚_𝑖=1𝜆_𝑖^𝑘+1∇𝑔_𝑖𝑥^𝑘=𝑥^𝑘. (2.41) Since 𝑥^𝑘→𝑥^∗ and 𝑇^𝑘→∞ as 𝑘∈𝒦→∞, we obtain from (2.39) and (2.41) 𝒫_Ω𝑥^∗−^𝑙_𝑗=1𝜇^∗_𝑗∇ℎ_𝑗𝑥^∗−^𝑚_𝑖=1𝜆^∗_𝑖∇𝑔_𝑖𝑥^∗−𝑥^∗=0, (2.42) where we have used the continuity of the projection operator.
If 𝑖∉𝐼(𝑥^∗), then 𝑔_𝑖(𝑥^∗)<0. Since 𝑐_𝑘→∞, we have 𝑐_𝑘𝑔_𝑖(𝑥^𝑘)→−∞ as 𝑘∈𝒦→∞. Using (2.5) and Lemma 2.2, we obtain lim_{𝑘→∞,𝑘∈𝒦}𝜆_𝑖^𝑘+1=lim_{𝑘→∞,𝑘∈𝒦}max0,𝑔_𝑖𝑥^𝑘+𝜆^𝑘_𝑖𝑐_𝑘𝑐_𝑘=0,∀𝑖∉𝐼𝑥^∗, (2.43) which, together with (2.39), implies that 𝜆^∗_𝑖=0 for all 𝑖∉𝐼(𝑥^∗). Therefore, we obtain from (2.42) that 𝒫_Ω𝑥^∗−^𝑙_𝑗=1𝜇^∗_𝑗∇ℎ_𝑗𝑥^∗−_{(𝑥𝑖∈𝐼∗)}𝜆^∗_𝑖∇𝑔_𝑖𝑥^∗−𝑥^∗=0. (2.44) So 𝑥^∗ is degenerate.Case 2. Both {𝜆^𝑘+1}_𝒦 and {𝜇^𝑘+1}_𝒦 are bounded. In this case, we can assume without loss of generality that lim_{𝑘→∞,𝑘∈𝒦}𝜆^𝑘_𝑖=𝜆^∗_𝑖≥0,lim_{𝑘→∞,𝑘∈𝒦}𝜇^𝑘_𝑖=𝜇^∗_𝑖. (2.45) Taking limits in (2.37) gives rise to 𝒫_Ω𝑥^∗−∇𝑓𝑥^∗−^𝑙_𝑗=1𝜇^∗_𝑗∇ℎ_𝑗𝑥^∗−^𝑚_𝑖=1𝜆^∗_𝑖∇𝑔_𝑖𝑥^∗=𝑥^∗, (2.46) which is equivalent to −∇𝑓𝑥^∗−^𝑙_𝑗=1𝜇^∗_𝑗∇ℎ_𝑗𝑥^∗−^𝑚_𝑖=1𝜆^∗_𝑖∇𝑔_𝑖𝑥^∗∈𝒩_Ω𝑥^∗. (2.47)
We claim that 𝑥^∗ is a feasible point. In fact, if 𝑔_𝑖(𝑥^∗)>0 for some 𝑖, then 𝑐_𝑘𝑔_𝑖(𝑥^𝑘)→∞ as 𝑘∈𝒦₁→∞. From (2.5), we must have 𝜆_𝑖^𝑘+1→∞, contradicting the boundedness of {𝜆_𝑖^𝑘+1}_𝑘∈𝒦. Note that (2.6) can be rewritten as 𝜇_𝑗^𝑘+1𝑐_𝑘=𝜇^𝑘_𝑗𝑐_𝑘+ℎ_𝑗𝑥^𝑘. (2.48) Taking limits in both sides and using the boundedness of {𝜇_𝑖^𝑘+1}_𝑘∈𝒦, we obtain that ℎ_𝑗(𝑥^∗)=0 for all 𝑗=1,2,…,𝑙. Thus, 𝑥^∗ is a feasible solution of (𝑃) as claimed.
If 𝑖∉𝐼(𝑥^∗), that is, 𝑔_𝑖(𝑥^∗)<0, then following almost the same argument as in Case 1, we can show that 𝜆^∗_𝑖=0 (cf. (2.43)). Therefore, 𝑔_𝑖𝑥^∗≤0,𝜆^∗_𝑖𝑔_𝑖𝑥^∗=0,𝑖=1,…,𝑚;ℎ_𝑗𝑥^∗=0,𝑗=1,2,…,𝑙. (2.49) This together with (2.47) implies that 𝑥^∗ is a KKT point of (𝑃) and 𝜆^∗, 𝜇^∗ are the corresponding Lagrangian multipliers.

3. Numerical Reports

To give some insight into the behavior of our proposed algorithm presented in this paper, we solve the following nonlinar programming problems. The test was done at a PC of Pentium 4 with 2.8 GHz CPU and 1.99 GB memory, and the computer codes were written in MATLAB 7.0. Numerical results are reported in Tables 1–4, where 𝑘 is the number of iterations, 𝑐_𝑘 is the penalty parameter, 𝑥^𝑘 is iterative point found by the algorithm, and 𝑓(𝑥^𝑘) is the objective value.

Table 1 

Result of Example 3.1.

Table 2 

Result of Example 3.2.

Table 3 

Result of Example 3.3.

Table 4 

Result of Example 3.4.

Example 3.1 (see [14]). It holds that min0.5𝑥₁+𝑥₂²+50𝑥₂−𝑥₁²+𝑥²₃+𝑥₃−sin𝑥₁+𝑥₂s.t.𝑥₁−1²+𝑥₂−1²+𝑥₃−1²−1.5≤0. (3.1)

Example 3.2 (see [14]). Consider min0.5𝑥₁+𝑥₂²+50𝑥₂−𝑥₁²+sin²𝑥₁+𝑥₂s.t.𝑥₁−1²+𝑥₂−1²+𝑥₃−1²−1.5≤0. (3.2)

Example 3.3. It holds that min1000−𝑥²₁−2𝑥²₂−𝑥²₃−𝑥₁𝑥₂−𝑥₁𝑥₃s.t.𝑥²₁+2𝑥²₂+𝑥²₃−25=0. (3.3)

Example 3.4. Consider min1000−𝑥²₁−2𝑥²₂−𝑥²₃−𝑥₁𝑥₂−𝑥₁𝑥₃s.t.8𝑥₁+14𝑥₂+7𝑥₃−56=0−𝑥_𝑖≤0,𝑖=1,2,3. (3.4)

4. Conclusions

Augmented Lagrangian methods are useful tools for solving many practical nonconvex optimization problems. In this paper, new convergence property of proximal augmented Lagrangian algorithm is established without requiring the boundedness of multiplier sequences. It is proved that if the algorithm terminates in finite steps, then we obtain a KKT point of the primal problem; otherwise, the iterative sequence {𝑥^𝑘} generalized by algorithm converges to optimal solution. Even if {𝑥^𝑘} is divergent, we also present a necessary and sufficient condition for the convergence of {𝑓(𝑥^𝑘)} to the optimal value. Moreover, under suitable assumptions, we show that every accumulation point of the iterative sequence generated by the algorithm is either a degenerate or is a KKT point of the primal problem. As our future work, one of the interesting and important topics is whether these nice properties could be extended to more general cone programming, for example, nonlinear semidefinite programming or second-order cone programming.

Acknowledgments

The authors would like to thank the referees for their valuable comments, which greatly improved the presentation of the paper. Research of the first author was partly supported by the National Natural Science Foundation of China (11101248, 11026047) and Shandong Province Natural Science Foundation (ZR2010AQ026). Research of the second author was partly supported by the National Natural Science Foundation of China (11171247). Research of the forth author was partly supported by the National Natural Science Foundation of China (10971118).