An Augmented Lagrangian Algorithm for Solving Semiinfinite Programming

Abstract

We present a smooth augmented Lagrangian algorithm for semiinfinite programming (SIP). For this algorithm, we establish a perturbation theorem under mild conditions. As a corollary of the perturbation theorem, we obtain the global convergence result, that is, any accumulation point of the sequence generated by the algorithm is the solution of SIP. We get this global convergence result without any boundedness condition or coercive condition. Another corollary of the perturbation theorem shows that the perturbation function at zero point is lower semi-continuous if and only if the algorithm forces the sequence of objective function convergence to the optimal value of SIP. Finally, numerical results are given.

1. Introduction

We consider the semi-infinite programming (SIP): inf𝑓(𝑥)∣𝑥∈𝑋, (1.1) where 𝑋={𝑥∈𝑅^𝑛∣𝑔(𝑥,𝑠)≤0,forall𝑠∈Ω}, the functions 𝑓∶𝑅^𝑛→𝑅 and 𝑔∶𝑅^𝑛×𝑅^𝑚→𝑅 are continuously differentiable. Ω⊂𝑅^𝑚 is a nonempty bounded and closed domain. In this paper, we assume that inf_{𝑥∈𝑅𝑛}𝑓(𝑥)>−∞. (1.2) This assumption is very mild, because the objective function 𝑓(𝑥) can be replaced by 𝑒^𝑓(𝑥) if the assumption is not satisfied.

Semi-infinite programming has wide applications such as engineering technology, optimal control, characteristic value calculation, and statistical design. Many methods have been proposed to solve semi-infinite programming (see [1–4]). As we know, the main difficulty for solving SIP is that it has infinite constraints. If transforming the infinite constraints into an integral function, SIP (1.1) is equivalent to a nonlinear programming with finite constraints.

For any given 𝑥∈𝑅^𝑛 and 𝑠∈Ω, let 𝑔(𝑥,𝑠)₊=max𝑔(𝑥,𝑠),0. (1.3) Define 𝜑∶𝑅^𝑛→𝑅 by 𝜑(𝑥)=_Ω𝑔(𝑥,𝑠)₊𝑑𝜇(𝑠), (1.4) where 𝜇≥0 is a given probability measure on Ω, that is, ∫_Ω𝑑𝜇(𝑠)=𝜇(Ω)=1. Thus SIP (1.1) can be reformulated as the following nonlinear programming (NP) with one equality constraint: inf_{𝑥∈𝑅𝑛}𝑓(𝑥)∣𝜑(𝑥)=0. (1.5) Then nonlinear programming (1.5) has the same optimal solution and optimal value with SIP (1.1).

For nonlinear programming with finite equality constraints, Hestenes [5] and Powell [6] independently proposed an augmented Lagrangian function by incorporating a quadratic penalty term in the conventional Lagrangian function. This augmented Lagrangian function avoids the shortcoming that the conventional Lagrangian function is only suitable for convex function. So the augmented Lagrangian function can be applied to nonconvex optimization problem. Later, the augmented Lagrangian function was extended to inequality constrained optimization problems and thoroughly investigated by Rockafellar [7]. Recently, Yang and Teo [8] and Rückmann and Shapiro [9] introduced the augmented Lagrangian function for SIP (1.1). In [9], necessary and sufficient conditions for the existence of corresponding augmented Lagrange multipliers were presented. [8] proposed a nonlinear Lagrangian method and established that the sequence of optimal values of nonlinear penalty problems converges to that of SIP (1.1), under the assumption that the level set of objective function is bounded. In this paper, using the equivalent relation of semi-infinite programming (1.1) and nonlinear programming (1.5), without any boundness condition, we present an augmented Lagrangian algorithm for SIP (1.1).

We notice that although the constraints of NP (1.5) are finite, but the constraint function is nonsmooth. Therefore, existing gradient-based optimization methods cannot be used to solve NP (1.5) directly. To overcome this inconvenience, we have to smooth the constraint function. For SIP (1.1), [10–13] presented semismooth Newton methods and smoothing Newton methods. They proved that each accumulation point is a generalized stationary point of SIP (1.1). However, at each iteration of these methods, a Hessian matrix needs to be computed. When the size of the problem is large, computing a Hessian matrix is very expensive. Based on exact 𝑙₁ penalty function that is approximated by a family of smoothing functions, a smoothed-penalty algorithm for solving NP (1.5) was proposed by [14]. They proved that if the constrained set is bounded or the objective function is coercive, the algorithm generates a sequence whose accumulation points are solutions of SIP (1.1).

In this paper, for SIP (1.1), we present a smooth augmented Lagrangian algorithm by smoothing the classical augmented Lagrangian function [7]. In this algorithm, we need not have to get an exact global optimal solution of unconstraint subproblem at each iteration. It is sufficient to search an inexact solution. It is not difficult to obtain an inexact solution, whenever the evaluation of the integral function is not very expensive. For this algorithm, we establish a perturbation theorem under mild conditions. As a corollary of the perturbation theorem, we obtain the global convergence result, that is, any accumulation point of the sequence generated by the algorithm is the solution of SIP (1.1). We get this global convergence result without any boundedness condition or coercive condition. It is noteworthy that the boundedness of the multiplier sequence is a sufficient condition in many literatures about Lagrangian method (see [15–17]). However, in our algorithm, the multiplier sequence can be unbounded. Another corollary of the perturbation theorem shows that the perturbation function at zero point is lower semi-continuous if and only if the algorithm forces the sequence of objective function convergence to the optimal value of SIP (1.1).

The paper is organized as follows. In the next section, we present a smooth augmented Lagrangian algorithm. In Section 3, we establish the perturbation theorem of the algorithm. By this theorem, we obtain a global convergence property and a sufficient and necessary condition in which the algorithm forces the sequence of objective functions convergence to the optimal value of SIP (1.1). Finally, we give some numerical results in Section 4.

2. Smooth Augmented Lagrangian Algorithm

Before we introduce the algorithm, some definitions and symbols need to be given. For 𝜀≥0, we define the relaxed feasible set of SIP (1.1) as follows: 𝑅_𝜀=𝑥∈𝑅^𝑛∣_Ω𝑔(𝑥,𝑠)₊𝑑𝜇(𝑠)≤𝜀. (2.1) Then 𝑅₀ is the feasible set of SIP (1.1). Let 𝑅^∗₀ be the set of optimal solutions of SIP (1.1). We assume that 𝑅^∗₀≠∅ in this paper.

The perturbation function is defined as follows: 𝜃_𝑓(𝜀)=inf_{𝑥∈𝑅𝜀}𝑓(𝑥). (2.2) Thus the optimal value of SIP (1.1) is 𝜃_𝑓(0)=inf_𝑥∈𝑅0𝑓(𝑥). (2.3) It is easy to show that 𝜃_𝑓(⋅) is upper semi-continuous at the point 𝜀=0.

For problem (1.5), the corresponding classical augmented Lagrangian function [7] is 𝐿(𝑥)+𝜌𝑥,𝜆,𝜌=𝑓2𝜑(𝑥)+𝜆𝜌²−𝜆²,2𝜌 (2.4) where 𝜆 is the Lagrangian multiplier and 𝜌 is the penalty parameter. On base of it, we introduce a class of smooth augmented Lagrangian function: 𝑓_𝑟(𝑥)+𝜌𝑥,𝜆,𝜌=𝑓2𝑟_Ω𝜙𝑔(𝑥,𝑠)𝑟(𝑠)+𝜆𝑑𝜇𝜌²−𝜆².2𝜌 (2.5) Here 𝑟 is the approximate parameter.

In the following, we suppose that the continuously differentiable function 𝜙∶𝑅→𝑅 satisfies(a)𝜙(⋅) is nonnegative and monotone increasing;(b) for any 𝑡>0, 𝜙(𝑡)≥𝑡;(c)lim_𝑡→+∞(𝜙(𝑡)/𝑡)=1.

It is easy to check that there are many continuously differentiable functions satisfying conditions (a), (b), and (c). For example, 𝜙₁(𝑡)=log1+𝑒^𝑡.𝜙₂(𝑡)=2𝑒^𝑡()𝜙,𝑡<0,𝑡+log1+𝑡+2,𝑡≥0.₃(𝑡)=𝑒^𝑡𝜙,𝑡<0,𝑡+1,𝑡≥0.₄(𝑡)=12𝑡+√𝑡²+4. (2.6) Using conditions (a) and (c), for any 𝑡∈𝑅, we have lim_𝑟→0+𝑡𝑟𝜙𝑟=max{0,𝑡}. (2.7) From the above equation, under conditions (a)–(c), the smooth function 𝑓_𝑟(𝑥,𝜆,𝜌) approximates to the classical augmented Lagrangian function 𝐿(𝑥,𝜆,𝜌) as 𝑟 approaches to zero, that is, lim_𝑟→0+𝑓_𝑟(𝑥)+𝜌𝑥,𝜆,𝜌=𝐿𝑥,𝜆,𝜌=𝑓2𝜑(𝑥)+𝜆𝜌²−𝜆².2𝜌 (2.8)

Based on the smooth augmented Lagrangian function 𝑓_𝜌(𝑥,𝛽,𝑟), we present the following smooth augmented Lagrangian algorithm.

Algorithm 2.1. Set 𝑥₀∈𝑅^𝑛,  𝑟₀>0,  𝜀₀>0,  𝜌₀>0,  𝜆₀∈(0,min{1/√𝑟₀,1/√𝜀₀}],  𝑘∶=0.
Step  1. Compute 𝑥_𝑘∈argmin_{𝑥∈𝑅𝑛}𝑓_𝑟𝑘𝑥,𝜆_𝑘,𝜌_𝑘. (2.9) Otherwise, seek on inexact global optimal solution 𝑥_𝑘 satisfying 𝑓_𝑟𝑘𝑥_𝑘,𝜆_𝑘,𝜌_𝑘≤inf_{𝑥∈𝑅𝑛}𝑓_𝑟𝑘𝑥,𝜆_𝑘,𝜌_𝑘+𝜀_𝑘. (2.10)Step  2. Set 𝑟_𝑘+1=(1/2)𝑟_𝑘, 𝜀_𝑘+1=(1/2)𝜀_𝑘, 𝜌_𝑘+1=𝜌_𝑘,𝑥_𝑘∈𝑅_𝜀𝑘;2𝜌_𝑘,𝑥_𝑘∉𝑅_𝜀𝑘,𝜆_𝑘+1∈10,min√𝑟_𝑘+1,1√𝜀_𝑘+1. (2.11)Step  3. Set 𝑘∶=𝑘+1, go back step 1.

Since 𝑓(𝑥) is bounded below and 𝜙(⋅) is nonnegative, an inexact solution satisfying (2.10) always exists. Thus Algorithm 2.1 is feasible.

3. Convergence Properties

In this section, by using a perturbation theorem of Algorithm 2.1, we will obtain a global convergence property, a sufficient and necessary condition that Algorithm 2.1 forces the sequence of objective functions convergence to the optimal value of SIP (1.1). To prove the perturbation theorem, we first give the following two lemmas.

Let Ω₊(𝑥)={𝑠∈Ω∣𝑔(𝑥,𝑠)>0}, Ω₀(𝑥)={𝑠∈Ω∣𝑔(𝑥,𝑠)=0}, Ω₋(𝑥)={𝑠∈Ω∣𝑔(𝑥,𝑠)<0}.

Lemma 3.1. Suppose that the point sequence {𝑥_𝑘} is generated by Algorithm 2.1. Then for any 𝜀>0, there exists a positive integer 𝑘₀ such that 𝑥_𝑘∈𝑅_𝜀, for all 𝑘>𝑘₀.

Proof.    
Case  1. When 𝑘→+∞, 𝜌_𝑘 tends to a finite number. From Algorithm 2.1, there exists a positive integer 𝑁₁ such that 𝑥_𝑘∈𝑅_𝜀𝑘 for all 𝑘>𝑁₁. Notice that 𝜀_𝑘→0, so for any 𝜀>0, there exists a positive integer 𝑁₂ such that 𝜀_𝑘<𝜀 for all 𝑘>𝑁₂. Therefore, when 𝑘>max{𝑁₁,𝑁₂}, we have 𝑥_𝑘∈𝑅_𝜀𝑘⊆𝑅_𝜀.
Case  2. When 𝑘→+∞, 𝜌_𝑘→+∞. We suppose that the conclusion does not hold. Then for 𝜀>0, there exists an infinite subsequence 𝐾⊆𝑁={1,2,3,…} such that 𝑥_𝑘∉𝑅_𝜀 for all 𝑘∈𝐾, that is, _Ω𝑔𝑥_𝑘,𝑠₊𝑑𝜇(𝑠)>𝜀. (3.1) Since 𝜀_𝑘→0, for the above 𝜀>0, there exists a positive integer 𝑁₃ such that 𝜀_𝑘<𝜀 for all 𝑘>𝑁₃. Then using (2.10) in Algorithm 2.1, we have 𝑓_𝑟𝑘𝑥_𝑘,𝜆_𝑘,𝜌_𝑘≤inf_{𝑥∈𝑅𝑛}𝑓_𝑟𝑘𝑥,𝜆_𝑘,𝜌_𝑘+𝜀. (3.2) Therefore by (3.1), (3.2), and 𝜙(𝑡) satisfying (a)-(b), for any 𝑘∈𝐾, 𝑘>𝑁₃, we derive that inf_{𝑥∈𝑅𝑛}𝑓_𝑟𝑘𝑥,𝜆_𝑘,𝜌_𝑘+𝜀≥𝑓_𝑟𝑘𝑥_𝑘,𝜆_𝑘,𝜌_𝑘=𝑓𝑥_𝑘+𝜌_𝑘2𝑟_𝑘Ω𝜙𝑔𝑥_𝑘,𝑠𝑟_𝑘(𝑠)+𝜆𝑑𝜇^𝑘𝜌_𝑘²−𝜆²_𝑘2𝜌_𝑘≥𝑓𝑥_𝑘+𝜌_𝑘2𝑟_{𝑘Ω+(𝑥𝑘)}𝜙𝑔𝑥_𝑘,𝑠𝑟_𝑘(𝑠)+𝜆𝑑𝜇^𝑘𝜌_𝑘²−𝜆²_𝑘2𝜌_𝑘≥𝑓𝑥_𝑘+𝜌_𝑘2_{Ω+(𝑥𝑘)}𝑔𝑥_𝑘(𝑠)+𝜆,𝑠𝑑𝜇^𝑘𝜌_𝑘²−𝜆²_𝑘2𝜌_𝑘≥𝑓𝑥_𝑘+𝜌_𝑘2𝜆𝜀+^𝑘𝜌_𝑘²−𝜆²_𝑘2𝜌_𝑘≥𝑓𝑥_𝑘+𝜌_𝑘2𝜀². (3.3) Note that {𝑓(𝑥_𝑘)} is bounded below and 𝜌_𝑘→+∞(𝑘→+∞), then we can obtain that 𝑓𝑥_𝑘+𝜌_𝑘2𝜀²⟶+∞,(𝑘∈𝐾,𝑘⟶+∞), (3.4) that is, inf_𝑥∈𝑅^𝑛𝑓_𝑟𝑘(𝑥,𝜆_𝑘,𝜌_𝑘)→+∞(𝑘∈𝐾,𝑘→+∞). However, on the other hand, since 𝑅₀≠∅, we can choose 𝑥∈𝑅₀; by the choice of 𝑟_𝑘, 𝜌_𝑘, 𝜆_𝑘 in Algorithm 2.1 and the properties of 𝜙, we obtain that inf_{𝑥∈𝑅𝑛}𝑓_𝑟𝑘𝑥,𝜆_𝑘,𝜌_𝑘≤𝑓_𝑟𝑘𝑥,𝜆_𝑘,𝜌_𝑘=𝑓(𝑥)+𝜌_𝑘2𝑟_𝑘Ω𝜙𝑔(𝑥,𝑠)𝑟_𝑘(𝑠)+𝜆𝑑𝜇^𝑘𝜌_𝑘²−𝜆²_𝑘2𝜌_𝑘≤𝑓(𝑥)+𝜌_𝑘2𝑟_𝑘Ω𝜙(0)(𝑠)+𝜆𝑑𝜇^𝑘𝜌_𝑘²−𝜆²_𝑘2𝜌_𝑘=𝑓(𝑥)+𝜌_𝑘2𝑟²_𝑘𝜙(0)𝜇(Ω)²+𝜆_𝑘𝑟_𝑘𝜙(0)𝜇(Ω)≤𝑓(𝑥)+𝜌₀𝑟²₀𝜙(0)𝜇(Ω)²+√𝑟₀𝜙(0)𝜇(Ω). (3.5) This indicates that inf_𝑥∈𝑅^𝑛𝑓_𝑟𝑘(𝑥,𝜆_𝑘,𝜌_𝑘) has an upper bound. It is in contradiction with inf_𝑥∈𝑅^𝑛𝑓_𝑟𝑘(𝑥,𝜆_𝑘,𝜌_𝑘)→+∞(𝑘∈𝐾,𝑘→+∞).

By using Lemma 3.1, we have the following Lemma 3.2.

Lemma 3.2. Suppose that the point sequence {𝑥_𝑘} is generated by Algorithm 2.1. Then for every accumulation point 𝑥^∗ of {𝑥_𝑘}, one has 𝑥^∗∈𝑅₀.

Theorem 3.3. Suppose that the sequence {𝑥_𝑘} is generated by Algorithm 2.1, then(i)lim_𝑘→∞𝑓(𝑥_𝑘)=lim_𝜀→0⁺𝜃_𝑓(𝜀);(ii)lim_𝑘→∞𝑓_𝑟𝑘(𝑥_𝑘,𝜆_𝑘,𝜌_𝑘)=lim_𝜀→0⁺𝜃_𝑓(𝜀);(iii)lim_𝑘→∞(𝜌_𝑘/2)[𝑟_𝑘∫_Ω𝜙(𝑔(𝑥_𝑘,𝑠)/𝑟_𝑘)𝑑𝜇(𝑠)+𝜆_𝑘/𝜌_𝑘]²−𝜆²_𝑘/2𝜌_𝑘=0.

Proof. Since 𝜃_𝑓(𝜀) is monotonically decreasing with respect to 𝜀>0 and has below bound, we know lim_𝜀→0⁺𝜃_𝑓(𝜀) exists and is finite. By Algorithm 2.1, we have 𝜀_𝑘↓0. Then lim_𝑘→+∞𝜃_𝑓𝜀_𝑘=lim_𝜀→0+𝜃_𝑓(𝜀). (3.6) Taking 𝛿_𝑘>0 and 𝛿_𝑘→0(𝑘→+∞), by the definition of infimum, there exists 𝑧_𝑘∈𝑅_𝜀𝑘 such that 𝑓𝑧_𝑘≤𝜃_𝑓𝜀_𝑘+𝛿_𝑘. (3.7) Since 𝑧_𝑘∈𝑅_𝜀𝑘, that is, ∫_Ω[𝑔(𝑧_𝑘,𝑠)]₊𝑑𝜇(𝑠)≤𝜀_𝑘.
On the other hand, by Lemma 3.1, for any 𝜀>0, when 𝑘 is sufficiently large, we have 𝑥_𝑘∈𝑅_𝜀. (3.8) Since 𝜙(𝑡) satisfies conditions (a) and (c), we obtain lim_𝑡→+∞𝜙(𝑡)−𝜙(0)𝑡=lim_𝑡→+∞𝜙(𝑡)𝑡=1,lim_𝑡→0+𝜙(𝑡)−𝜙(0)𝑡=lim_𝑡→0+𝜙^′(𝑡)=𝜙^′(0). (3.9) Therefore, there exists 𝑀>0 such that 𝜙(𝑡)≤𝑀𝑡+𝜙(0), (3.10) for any 𝑡≥0. As stated previously, by the choice of 𝑟_𝑘, 𝜀_𝑘, 𝜌_𝑘, and 𝜆_𝑘 in Algorithm 2.1, (3.7), (3.8), and (3.10) derive that for any 𝜀>0, 𝜃_𝑓(𝜀)≤𝑓𝑥_𝑘≤𝑓𝑥_𝑘+𝜌_𝑘2𝑟_𝑘Ω𝜙𝑔𝑥_𝑘,𝑠𝑟_𝑘(𝑠)+𝜆𝑑𝜇^𝑘𝜌_𝑘²−𝜆²_𝑘2𝜌_𝑘≤inf_{𝑥∈𝑅𝑛}𝑓_𝑟𝑘𝑥,𝜆_𝑘,𝜌_𝑘+𝜀_𝑘≤𝑓𝑧_𝑘+𝜌_𝑘2𝑟_𝑘Ω𝜙𝑔𝑧_𝑘,𝑠𝑟_𝑘(𝑠)+𝜆𝑑𝜇^𝑘𝜌_𝑘²−𝜆²_𝑘2𝜌_𝑘+𝜀_𝑘≤𝜃_𝑓𝜀_𝑘+𝛿_𝑘+𝜌_𝑘2𝑟_𝑘Ω𝜙𝑔𝑧_𝑘,𝑠𝑟_𝑘(𝑠)+𝜆𝑑𝜇^𝑘𝜌_𝑘²−𝜆²_𝑘2𝜌_𝑘+𝜀_𝑘≤𝜃_𝑓𝜀_𝑘+𝛿_𝑘+𝜌_𝑘2𝑀𝜀_𝑘+𝑟_𝑘𝜙(0)𝜇(Ω)+𝜆_𝑘𝜌_𝑘²−𝜆²_𝑘2𝜌_𝑘+𝜀_𝑘≤𝜃_𝑓𝜀_𝑘+𝛿_𝑘+𝜌_𝑘2max𝑟_𝑘,𝜀_𝑘𝑀+𝜙(0)𝜇(Ω)²+𝜆_𝑘max𝑟_𝑘,𝜀_𝑘𝑀+𝜙(0)𝜇(Ω)+𝜀_𝑘≤𝜃_𝑓𝜀_𝑘+𝛿_𝑘+𝜌₀2max𝑟₀,𝜀₀𝑀+𝜙(0)𝜇(Ω)²max𝑟_𝑘,𝜀_𝑘+𝑀+𝜙(0)𝜇(Ω)max√𝑟_𝑘,√𝜀_𝑘+𝜀_𝑘. (3.11) From the above inequalities and (3.6), noticing that 𝑟_𝑘𝜀_𝑘→0(𝑘→∞), for any 𝜀>0, we have 𝜃_𝑓(𝜀)≤liminf_𝑘→+∞𝑓𝑥_𝑘≤limsup_𝑘→+∞𝑓𝑥_𝑘≤limsup_𝑘→+∞𝑓𝑥_𝑘+𝜌_𝑘2𝑟_𝑘Ω𝜙𝑔𝑥_𝑘,𝑠𝑟_𝑘(𝑠)+𝜆𝑑𝜇^𝑘𝜌_𝑘²−𝜆²_𝑘2𝜌_𝑘≤lim_𝑘→+∞𝜃_𝑓𝜀_𝑘=lim_𝜀→0+𝜃_𝑓(𝜀). (3.12) Then lim_𝜀→0⁺𝜃_𝑓(𝜀)=lim_𝑘→+∞𝑓(𝑥_𝑘)=limsup_𝑘→+∞{𝑓(𝑥_𝑘𝜌)+^𝑘2[𝑟_𝑘∫_Ω𝜙(𝑔(𝑥_𝑘,𝑠)/𝑟_𝑘)𝑑𝜇(𝑠)+𝜆_𝑘/𝜌_𝑘]²−𝜆²_𝑘/2𝜌_𝑘}. So the conclusions (i)–(iii) hold.

Now, we prove the global convergence of Algorithm 2.1.

Corollary 3.4. Suppose that the point sequence {𝑥_𝑘} is generated by Algorithm 2.1. Then every accumulation point of {𝑥_𝑘} is the optimal solution of the problem (1.1).

Proof. Let 𝑥^∗ be an accumulation point of {𝑥_𝑘}; from Lemma 3.2, we have 𝑥^∗∈𝑅₀. (3.13) By the conclusion (i) of Theorem 3.3 and (3.13), we obtain lim_𝜀→0+𝜃_𝑓(𝜀)=lim_𝑘→∞𝑓𝑥_𝑘=𝑓𝑥^∗≥𝜃_𝑓(0). (3.14) Then we get 𝑓(𝑥^∗)=𝜃_𝑓(0), because (3.14) and 𝜃_𝑓(𝜀) are upper semi-continuous at the point 𝜀=0.

By using Theorem 3.3, we have the following Corollary 3.5.

Corollary 3.5. lim_𝑘→∞𝑓(𝑥_𝑘)=𝜃_𝑓(0) if and only if 𝜃_𝑓(𝜀) is lower semi-continuous at the point 𝜀=0.

4. Numerical Results

To give some insight into the behavior of the algorithm presented in this paper. It is implemented in Matlab 7.0.4 and runs are made on AMD Athlon(tm) 64×2 Dual Core Processor 4800+ with CPU 2.50 GHz and 1.87 GB memory. Tables 1 and 2 show the computational results of the corresponding problems with the following items:  𝑘: number of iterations;  𝐱₀: starting point;  𝜙(𝑡): smoothing function;  𝐱_𝑘: the final iteration point;  𝜆_𝑘: the final Lagrangian multiplier;  𝑓(𝑥_𝑘): the function value of 𝑓(𝑥) at the final 𝑥_𝑘.

Table 1 

Numerical results of Example 4.1.

Table 2 

Numerical results of Example 4.2.

The parameters used in the Algorithm 2.1 are specified as follows: 𝑟₀=1,𝜀₀=1,𝜌₀=1,𝜆₀𝜆=1,_𝑘+1=min𝜆_𝑘+𝜌_𝑘𝑟_𝑘Ω𝜙𝑔𝑥_𝑘,𝑠𝑟_𝑘𝑑𝜇(𝑠),10³. (4.1)

Example 4.1 (see [18]). Consider the following: min1.21exp𝑥₁+exp𝑥₂s.t.𝑠−exp𝑥₁+𝑥₂≤0,∀𝑠∈[0,1]. (4.2) We choose the starting point 𝐱₀=(0,0). This example has the optimal solution 𝑥^∗=(−ln1.1,ln1.1).

Example 4.2 (see [18]). Consider the following: min^𝑛_𝑖=1𝑥_𝑖/𝑖s.t.tan𝑠−^𝑛_𝑖=1𝑥_𝑖𝑠^𝑖−1≤0,∀𝑠∈[0,1], (4.3) for 𝑛=3, 6, and 8. We choose zero vectors as the starting points.

Throughout the computational experiments, we use trust region method for solving an unconstrained optimization subproblem at each step. For the corresponding trust region subproblem, we directly use the trust function in Matlab toolbox. The test results of Example 4.1 are summarized in Table 1. We test the three cases for 𝜙(𝑡)=log(1+𝑒^𝑡), 𝜙(𝑡)=(1/2)(𝑡+√𝑡²+4) and 𝜙(𝑡)=𝑒^𝑡, which are, respectively, used as the smoothing approximation functions. 𝑘 denotes the number of the iteration, 𝜆_𝑘 denotes the approximate Lagrangian multiplier at the final iteration, and 𝑥_𝑘 and 𝑓(𝑥_𝑘) are the approximate solution and the objective function at the final iteration. For Example 4.2, we test the results when 𝑛=3, 𝑛=6, and 𝑛=8 in Table 2. Numerical results demonstrate that augmented Lagrangian algorithm established in this paper is a practical and effective method for solving semi-infinite programming problem.

Acknowledgments

This work was supported by National Natural Science Foundation under Grants 10971118, 10901096, and 11271226, the Scientific Research Fund for the Excellent Middle-Aged and Youth Scientists of Shandong Province under Grant BS2012SF027, and the Natural Science Foundation of Shandong Province under Grant ZR2009AL019.