Smoothing Approximation to the Square-Order Exact Penalty Functions for Constrained Optimization

Abstract

A method is proposed to smooth the square-order exact penalty function for inequality constrained optimization. It is shown that, under some conditions, an approximately optimal solution of the original problem can be obtained by searching an approximately optimal solution of the smoothed penalty problem. An algorithm based on the smoothed penalty functions is given. The algorithm is shown to be convergent under mild conditions. Two numerical examples show that the algorithm seems efficient.

1. Introduction

Consider the following nonlinear constrained optimization problem: (𝑥)[𝑃]min𝑓s.t.𝑔_𝑖(𝑥)≤0,𝑖=1,2,…,𝑚,𝑥∈𝑅^𝑛, (1) where 𝑓∶𝑅^𝑛→𝑅 and 𝑔_𝑖(𝑥)∶𝑅^𝑛→𝑅, 𝑖∈𝐼={1,2,…,𝑚} are twice continuously differentiable functions. Let 𝐺₀=𝑥∈𝑅^𝑛∣𝑔_𝑖(𝑥)≤0,𝑖=1,2,…,𝑚. (2)

To solve [𝑃], many penalty function methods have been proposed in numerous pieces of literature. One of the popular penalty functions is given by 𝐹𝑥,𝑞=𝑓(𝑥)+𝑞^𝑚_𝑖=1𝑔⁺_𝑖(𝑥)², (3) where 𝑔⁺_𝑖(𝑥)=max{0,𝑔_𝑖(𝑥)}, 𝑖=1,2,…,𝑚. Obviously, it is a continuously differentiable function, but it is not an exact penalty function. If each minimum of the penalty problem is a minimum of the original problem or each minimum of the original problem is a minimum of the penalty problem when the penalty parameter 𝑞 is large enough, the corresponding penalty function is called exact penalty function.

In Zangwill [1], the classical 𝑙₁ exact penalty function is defined as follows: 𝑓𝑥,𝑞=𝑓(𝑥)+𝑞^𝑚_𝑖=1𝑔⁺_𝑖(𝑥). (4) After Zangwill’s development, exact penalty functions have attracted most of the attention (see, e.g., [2–6]). It is known from the theory of ordinary constrained optimization that the 𝑙₁ penalty function is a better candidate for penalization. However, it is not a smooth function and causes some numerical instability problems in its implementation when the value of the penalty parameter 𝑞 becomes larger. Some methods for smoothing the exact penalty function are developed (see, e.g., [7–14]).

In [15, 16], the square-order penalty function 𝜑_𝑞(𝑥)=𝑓(𝑥)+𝑞^𝑚_𝑖=1𝑔⁺_𝑖(𝑥) (5) has been introduced and investigated. The penalty function 𝜑_𝑞(𝑥) is exact but not smooth. Its smoothing has been investigated in [15, 16]. So, it can been applied to solve the problem [𝑃] via a gradient-type or a Newton-type method.

In this paper, a new smoothing function to the square-order penalty function of the form (5) is investigated. The rest of this paper is organized as follows. In Section 2, a new smoothing function to the square-order penalty function is introduced, and some fundamental properties of the smoothing function are discussed. In Section 3, an algorithm is presented to compute an approximate solution to [𝑃] based on the smooth penalty function and is shown to be convergent. In Section 4, two numerical examples are given to show the applicability of the algorithm. In Section 5, we conclude the paper with some remarks.

2. Smoothing Exact Lower Order Penalty Function

Consider the following lower order penalty problem: [𝐿𝑂𝑃]min_{𝑥∈𝑅𝑛}𝜑_𝑞(𝑥). (6) In this paper, we say that the pair (𝑥^∗,𝜆^∗) satisfies the KKT condition if ∇𝑓𝑥^∗=−_𝑖∈𝐼𝜆^∗_𝑖∇𝑔_𝑖𝑥^∗,𝜆^∗_𝑖𝑔_𝑖𝑥^∗=0,𝜆^∗_𝑖≥0,𝑔_𝑖𝑥^∗≤0,𝑖∈𝐼 (7) and that the pair (𝑥^∗,𝜆^∗) satisfies the second-order sufficiency condition [17, page 169] if ∇_𝑥𝐿𝑥^∗,𝜆^∗𝑔=0,_𝑖𝑥^∗𝜆≤0,𝑖∈𝐼,_∗𝑖𝜆≥0,𝑖∈𝐼,_∗𝑖𝑔_𝑖𝑥^∗𝑦=0,𝑖∈𝐼,_𝑇∇²𝐿𝑥^∗,𝜆^∗𝑦>0,forany𝑦∈𝑉𝑥^∗, (8) where 𝐿(𝑥,𝜆)=𝑓(𝑥)+∑^𝑚_𝑖=1𝜆_𝑖𝑔_𝑖(𝑥) and 𝑉𝑥^∗=𝑦∈𝑅^𝑛∣∇^𝑇𝑔_𝑖𝑥^∗𝑦=0,𝑖∈𝐴𝑥^∗,∇^𝑇𝑔_𝑖𝑥^∗𝑦≤0,𝑖∈𝐵𝑥^∗,𝐴𝑥^∗=𝑖∈𝐼∣𝑔_𝑖𝑥^∗=0,𝜆^∗_𝑖,𝐵𝑥>0_∗=𝑖∈𝐼∣𝑔_𝑖𝑥^∗=0,𝜆^∗_𝑖=0. (9) In order to establish the exact penalization, we need the following assumptions.

Assumption 1. 𝑓(𝑥) satisfies the following coercive condition: lim_{‖𝑥‖→+∞}𝑓(𝑥)=+∞. (10)

Under Assumption 1, there exists a box 𝑋 such that 𝐺([𝑃])⊂int(𝑋), where 𝐺([𝑃]) is the set of global minima of problem [𝑃] and int(𝑋) denotes the interior of the set 𝑋. Consider the following problem:(𝑥)𝑃min𝑓_′s.t.𝑔_𝑖(𝑥)≤0,𝑖=1,…,𝑚,𝑥∈𝑋. (11) Let 𝐺([𝑃^′]) denote the set of global minima of problem [𝑃^′]. Then 𝐺([𝑃^′])=𝐺([𝑃]).

Assumption 2. The set 𝐺([𝑃]) is a finite set.

Then we consider the penalty problem of the form 𝐿𝑂𝑃^′min_𝑥∈𝑋𝜑_𝑞(𝑥). (12)

Let 𝑝(𝑢)=(max{0,𝑢})^1/2; that is, 𝑝(𝑢)=𝑢^1/2if𝑢>0,0otherwise, (13) then 𝜑_𝑞(𝑥)=𝑓(𝑥)+𝑞^𝑚_𝑖=1𝑝𝑔_𝑖(𝑥). (14) For any 𝜖>0, let 𝑝_𝜖(𝑢)=20if𝑢≤0,3𝜖⁻²𝑢^5/2−13𝜖⁻³𝑢^7/2𝑢if0<𝑢≤𝜖,_1/2−23𝜖^1/2if𝑢>𝜖. (15) It follows that 𝑝^′_𝜖(𝑢)=50if𝑢≤0,3𝜖⁻²𝑢^3/2−76𝜖⁻³𝑢^5/21if0<𝑢≤𝜖,2𝑢^−1/2if𝑢>𝜖. (16) It is easy to see that 𝑝_𝜖(𝑢) is continuously differentiable on 𝑅. Furthermore, we can obtain that 𝑝_𝜖(𝑢)→𝑝(𝑢) as 𝜖→0.

Figure 1 shows the behavior of 𝑝(𝑢) (represented by the real line), 𝑝_0.1(𝑢) (represented by the real line with plus sign), 𝑝_0.01(𝑢) (represented by the dash and dot line), and 𝑝_0.001(𝑢) (represented by broken line).

Figure 1 

The behavior of 𝑝𝜖(𝑢) and 𝑝(𝑢).

Let 𝜑_𝑞,𝜖(𝑥)=𝑓(𝑥)+𝑞^𝑚_𝑖=1𝑝_𝜖𝑔_𝑖(𝑥). (17) Then 𝜑_𝑞,𝜖(𝑥) is continuously differentiable on 𝑅^𝑛. Consider the following smoothed optimization problem: [𝑆𝑃]min_𝑥∈𝑋𝜑_𝑞,𝜖(𝑥). (18)

Lemma 3. For any 𝑥∈𝑋, 𝜖>0, −23𝑚𝑞𝜖^1/2≤𝜑_𝑞,𝜖(𝑥)−𝜑_𝑞(𝑥)≤43𝑚𝑞𝜖^1/2. (19)

Proof. Note that 𝑝𝑔_𝑖(𝑥)−𝑝_𝜖𝑔_𝑖(𝑥)=0if𝑔_𝑖(𝑥)𝑔≤0,_𝑖(𝑥)^1/2−23𝜖⁻²𝑔_𝑖(𝑥)^5/2+13𝜖⁻³𝑔_𝑖(𝑥)^7/2if0<𝑔_𝑖(𝑥)2≤𝜖,3𝜖^1/2if𝑔_𝑖(𝑥)>𝜖. (20)
When 𝑔_𝑖(𝑥)∈(0,𝜖], we have −23𝜖^1/2≤𝑝𝑔_𝑖(𝑥)−𝑝_𝜖𝑔_𝑖(𝑥≤4)3𝜖^1/2. (21)
Then −23𝑚𝑞𝜖^1/2≤𝜑_𝑞,𝜖(𝑥)−𝜑_𝑞(𝑥)≤43𝑚𝑞𝜖^1/2. (22)

As a direct result of Lemma 3, we have the following result.

Theorem 4. Let {𝜖_𝑗}→0 be a sequence of positive numbers, and assume that 𝑥^𝑗 is a solution to min_𝑥∈𝑋𝜑_{𝑞,𝜖𝑗}(𝑥) for some 𝑞>0. Let 𝑥 be an accumulating point of the sequence {𝑥^𝑗}. Then 𝑥 is an optimal solution to min_𝑥∈𝑋𝜑_𝑞(𝑥).

Proof. Because 𝑥^𝑗 is a solution to min_𝑥∈𝑋𝜑_{𝑞,𝜖𝑗}(𝑥), we have 𝜑_{𝑞,𝜖𝑗}𝑥^𝑗≤𝜑_{𝑞,𝜖𝑗}(𝑥). (23) By Lemma 3, we have 𝜑_𝑞𝑥^𝑗≤𝜑_{𝑞,𝜖𝑗}𝑥^𝑗+23𝑚𝑞𝜖_𝑗^1/2,𝜑_{𝑞,𝜖𝑗}(𝑥)≤𝜑_𝑞(𝑥)+43𝑚𝑞𝜖_𝑗^1/2. (24) It follows that 𝜑_𝑞𝑥^𝑗≤𝜑_{𝑞,𝜖𝑗}𝑥^𝑗+23𝑚𝑞𝜖_𝑗^1/2≤𝜑_{𝑞,𝜖𝑗}(𝑥)+23𝑚𝑞𝜖_𝑗^1/2≤𝜑_𝑞(𝑥)+43𝑚𝑞𝜖_𝑗^1/2+23𝑚𝑞𝜖_𝑗^1/2=𝜑_𝑞(𝑥)+2𝑚𝑞𝜖_𝑗^1/2. (25) Let 𝑗→0; we have 𝜑_𝑞(𝑥)≤𝜑_𝑞(𝑥). (26)
We complete the proof.

Theorem 5. Let 𝑥^∗_𝑞∈𝑋 be an optimal solution of problem [𝐿𝑂𝑃^′] and 𝑥_𝑞,𝜖∈𝑋 an optimal solution of problem [𝑆𝑃] for some 𝑞>0 and 𝜖>0. Then −23𝑚𝑞𝜖^1/2≤𝜑_𝑞𝑥^∗_𝑞−𝜑_𝑞,𝜖𝑥_𝑞,𝜖≤43𝑚𝑞𝜖^1/2. (27) If both 𝑥^∗_𝑞 and 𝑥_𝑞,𝜖 are feasible, then 0≤𝑓𝑥_𝑞,𝜖−𝑓𝑥^∗_𝑞≤23𝑚𝑞𝜖^1/2. (28)

Proof. By Lemma 3, we have −23𝑚𝑞𝜖^1/2≤𝜑_𝑞𝑥^∗_𝑞−𝜑_𝑞,𝜖𝑥^∗_𝑞≤𝜑_𝑞𝑥^∗_𝑞−𝜑_𝑞,𝜖𝑥_𝑞,𝜖≤𝜑_𝑞𝑥_𝑞,𝜖−𝜑_𝑞,𝜖𝑥_𝑞,𝜖≤43𝑚𝑞𝜖^1/2. (29) Specially, if both 𝑥^∗_𝑞 and 𝑥_𝑞,𝜖 are feasible, we have 𝑓𝑥^∗_𝑞≤𝑓𝑥_𝑞,𝜖 (30) by 𝜑_𝑞(𝑥^∗_𝑞)≤𝜑_𝑞(𝑥_𝑞,𝜖).
It follows that 0≤𝑓𝑥_𝑞,𝜖−𝑓𝑥^∗_𝑞. (31) On the other hand, by (14), (15), (17), and (19), we have 𝑓𝑥_𝑞,𝜖−𝑓𝑥^∗_𝑞=𝜑_𝑞,𝜖𝑥_𝑞,𝜖−𝜑_𝑞𝑥^∗_𝑞≤23𝑚𝑞𝜖^1/2. (32)
We complete the proof.

Theorem 6. Supposing that Assumptions 1 and 2 hold, and that, for any 𝑥^∗∈𝐺([𝑃]), there exists a 𝜆∈𝑅^𝑚₊ such that the pair (𝑥^∗,𝜆^∗) satisfies the second-order sufficiency condition (8). Let 𝑥^∗∈𝑋 be a global solution of problem [𝑃] and 𝑥_𝑞,𝜖∈𝑋 a global solution of problem [𝑆𝑃] for 𝜖>0. Then there exists 𝑞^∗>0 such that for any 𝑞>𝑞^∗, −23𝑚𝑞𝜖^1/2≤𝑓𝑥^∗−𝜑_𝑞,𝜖𝑥_𝑞,𝜖≤43𝑚𝑞𝜖^1/2, (33) where 𝑞^∗ is defined in Corollary  2.3 in [16].

Proof. By Corollary  2.3 in [16], we have that 𝑥^∗ is a global solution of problem [𝐿𝑂𝑃^′]. Then, by Theorem 5, we have −23𝑚𝑞𝜖^1/2≤𝜑_𝑞𝑥^∗_𝑞−𝜑_𝑞,𝜖𝑥_𝑞,𝜖≤43𝑚𝑞𝜖^1/2. (34)
Since ∑^𝑚_𝑖=1𝑝(𝑔_𝑖(𝑥^∗))=0, we have 𝜑_𝑞𝑥^∗=𝑓𝑥^∗+𝑞^𝑚_𝑖=1𝑝𝑔_𝑖𝑥^∗=𝑓𝑥^∗. (35)
We complete the proof.

Theorems 4 and 5 mean that an approximate solution to [𝑆𝑃] is also an approximate solution to [𝐿𝑂𝑃^′]. Furthermore, by Theorem 6, an optimal solution to [𝑆𝑃] is an approximately optimal solution to [𝑃]. Now we present a penalty function algorithm to solve [𝑃].

3. A Smoothing Method

We propose the following algorithm to solve [𝑃].

Algorithm 7. Consider the following.
Step  1. Choose a point 𝑥⁰. Given 𝜖₀>0, 𝑞₀>0, 0<𝜂<1, and 𝑁>1, let 𝑗=0, and go to Step  2.
Step  2. Use 𝑥^𝑗 as the starting point to solve min_𝑥∈𝑅^𝑛𝜑_{𝑞𝑗,𝜖𝑗}(𝑥). Let 𝑥^∗_𝑗 be the optimal solution obtained (𝑥^∗_𝑗 is obtained by a quasi-Newton method and a finite difference gradient). Go to Step  3.
Step  3. Let 𝑞_𝑗+1=𝑁𝑞_𝑗, 𝜖_𝑗+1=𝜂𝜖_𝑗, 𝑥^𝑗+1=𝑥^∗_𝑗 and 𝑗=𝑗+1; then go to Step  2.

Remark 8. From 0<𝜂<1 and 𝑁>1, we can easily obtain that the sequence {𝜖_𝑗} is decreasing to 0 and the sequence {𝑞_𝑗} is increasing to +∞ as 𝑗→+∞.
Now we prove the convergence of the algorithm under mild conditions.

Theorem 9. Suppose that, for any 𝑞∈[𝑞₀,+∞), 𝜖∈(0,𝜖₀], the set argmin_{𝑥∈𝑅𝑛}𝜑_𝑞,𝜖(𝑥)≠∅. (36) Let {𝑥^∗_𝑗} be the sequence generated by Algorithm 7. If {𝑥^∗_𝑗} has limit point, then any limit point of {𝑥^∗_𝑗} is the solution of [𝑃].

Proof. Let 𝑥 be any limit point of {𝑥^∗_𝑗}. Then there exists a natural number set 𝐽∈𝑁, such that 𝑥^∗_𝑗→𝑥, 𝑗∈𝐽. If we can prove that (i) 𝑥∈𝐺₀ and (ii) 𝑓(𝑥)≤inf_𝑥∈𝐺0𝑓(𝑥) hold, then 𝑥 is the optimal solution of [𝑃].
(i) Suppose, to the contrary, that 𝑥∉𝐺₀; then there exist 𝛿₀>0, 𝑖₀∈𝐼, and the subset 𝐽₁⊂𝐽 such that 𝑔_𝑖0𝑥^∗_𝑗≥𝛿₀ (37) for any 𝑗∈𝐽₁.
If 𝜖_𝑗≥𝑔_𝑖0(𝑥^∗_𝑗)≥𝛿₀, it follows from Step  2 in Algorithm 7 and (15) that 𝑓𝑥^∗_𝑗+23𝑞_𝑗𝜖_𝑗⁻²𝛿₀^5/2−13𝑞_𝑗𝜖_𝑗⁻³𝛿₀^7/2≤𝜑_{𝑞𝑗,𝜖𝑗}𝑥^∗_𝑗≤𝜑_{𝑞𝑗,𝜖𝑗}(𝑥)=𝑓(𝑥) (38) for any 𝑥∈𝐺₀, which contradicts with 𝜖_𝑗→0 and 𝑞_𝑗→+∞.
If 𝑔_𝑖0(𝑥^∗_𝑗)≥𝛿₀>𝜖_𝑗 or 𝑔_𝑖0(𝑥^∗_𝑗)>𝜖_𝑗≥𝛿₀, it follows from Step  2 in Algorithm 7 and (15) that 𝑓𝑥^∗_𝑗+𝑞_𝑗𝛿₀^1/2−23𝜖_𝑗^1/2≤𝜑_{𝑞𝑗,𝜖𝑗}𝑥^∗_𝑗≤𝜑_{𝑞𝑗,𝜖𝑗}(𝑥)=𝑓(𝑥) (39) for any 𝑥∈𝐺₀, which contradicts with 𝜖_𝑗→0 and 𝑞_𝑗→+∞.
Then we have 𝑥∈𝐺₀.
(ii) For any 𝑥∈𝐺₀, it holds that 𝑓𝑥^∗_𝑗≤𝜑_{𝑞𝑗,𝜖𝑗}𝑥^∗_𝑗≤𝜑_{𝑞𝑗,𝜖𝑗}(𝑥)=𝑓(𝑥); (40) then 𝑓(𝑥)≤inf_𝑥∈𝐺0𝑓(𝑥) holds.
This completes the proof.

4. Numerical Examples

In this section, we solve two numerical examples to show the applicability of Algorithm 7 on Fortran.

Example 1 (see [18, Example 4.1]). We can see the following: min𝑓(𝑥)=𝑥²₁+𝑥²₂−cos17𝑥₁−cos17𝑥₂+3s.t.𝑔₁(𝑥)=𝑥₁−2²+𝑥²₂−1.6²𝑔≤0₂(𝑥)=𝑥²₁+𝑥₂−3²−2.7²≤00≤𝑥₁≤20≤𝑥₂≤2. (41)
Starting point 𝑥⁰=(0,0), 𝑞₀=5.0, 𝜖₀=0.1, 𝜂=0.5, and 𝑁=10, we obtain the results by Algorithm 7 shown in Table 1.
Furthermore, the algorithms based on the penalty function (3) or the exact penalty function (4) are described as follows.

Table 1 

Numerical results for Example 1 by Algorithm 7.

Algorithm 10. Consider the following.
Step  1. Choose a point 𝑥⁰, and a stopping tolerance 𝜖>0. Given 𝜖₀>0, 𝑞₀>0, 0<𝜂<1, and 𝑁>1, let 𝑗=0, and go to Step  2.
Step  2. Use 𝑥^𝑗 as the starting point to solve min_𝑥∈𝑅^𝑛𝐹(𝑥,𝑞_𝑗). Let 𝑥^∗_𝑗 be the optimal solution obtained (𝑥^∗_𝑗 is obtained by a quasi-Newton method and a finite difference gradient). Go to Step  3.
Step  3. Let 𝑞_𝑗+1=𝑁𝑞_𝑗, 𝜖_𝑗+1=𝜂𝜖_𝑗, 𝑥^𝑗+1=𝑥^∗_𝑗, and 𝑗=𝑗+1; then go to Step  2.

Algorithm 11. Consider the following.
Step  1. Choose a point 𝑥⁰ and a stopping tolerance 𝜖>0. Given 𝜖₀>0, 𝑞₀>0, 0<𝜂<1, and 𝑁>1, let 𝑗=0, and go to Step  2.
Step  2. Use 𝑥^𝑗 as the starting point to solve min_𝑥∈𝑅^𝑛𝑓(𝑥,𝑞_𝑗). Let 𝑥^∗_𝑗 be the optimal solution obtained (𝑥^∗_𝑗 is obtained by a quasi-Newton method and a finite difference gradient). Go to Step  3.
Step  3. Let 𝑞_𝑗+1=𝑁𝑞_𝑗, 𝜖_𝑗+1=𝜂𝜖_𝑗, 𝑥^𝑗+1=𝑥^∗_𝑗, and 𝑗=𝑗+1; then go to Step  2.

Let 𝑥⁰=(0,0), 𝑞₀=0.1, 𝜖₀=0.1, 𝜂=0.1, and 𝑁=5; numerical results by Algorithm 10 are shown in Table 2.

Table 2 

Numerical results for Example 1 by Algorithm 10.

Let 𝑥⁰=(0,0), 𝑞₀=0.1, 𝜖₀=0.1, 𝜂=0.1, and 𝑁=20; numerical results by Algorithm 11 are shown in Table 3.

Table 3 

Numerical results for Example 1 by Algorithm 11.

This example is a nonconvex problem with 22 local optimal solutions in the interior of the feasible region. By Sun and Li [18], we know that 𝑥^∗=(0.7255,0.3993) is a global minimum with global optimal value 𝑓^∗=1.8376. It is clear from Table 1 that the obtained approximately optimal solution is 𝑥^∗=(0.7245065,0.3990242) with corresponding objective function value 1.837684.

From Tables 1–3, one can see that Algorithm 11 converges faster than Algorithms 7 and 10, but the solution generated by Algorithm 11 is the worst. Algorithm 10 is the slowest one, and the solution generated by Algorithm 10 is worse than the solution generated by Algorithm 7.

Example 2 (see the Rosen-Suzki problem in [15]). We can see the following: min𝑓(𝑥)=𝑥²₁+𝑥²₂+2𝑥²₃+𝑥²₄−5𝑥₁−5𝑥₂−21𝑥₃+7𝑥₄s.t.𝑔₁(𝑥)=2𝑥²₁+𝑥²₂+𝑥²₃+2𝑥₁+𝑥₂+𝑥₄𝑔−5≤0₂(𝑥)=𝑥²₁+𝑥²₂+𝑥²₃+𝑥²₄+𝑥₁−𝑥₂+𝑥₃−𝑥₄𝑔−8≤0₃(𝑥)=𝑥²₁+2𝑥²₂+𝑥²₃+2𝑥²₄−𝑥₁−𝑥₄−10≤0. (42)
Let 𝑥⁰=(1,1,1,1), 𝑞₀=2.0, 𝜖₀=1.0, 𝜂=0.1, and 𝑁=2; the results by Algorithm 7 are shown in Table 4.
Let 𝑥⁰=(1,1,1,1), 𝑞₀=0.1, 𝜖₀=1.0, 𝜂=0.1, and 𝑁=5; numerical results by Algorithm 10 are shown in Table 5.
Let 𝑥⁰=(1,1,1,1), 𝑞₀=2.0, 𝜖₀=1.0, 𝜂=0.1, and 𝑁=2; the results by Algorithm 11 are shown in Table 6.

Table 4 

Numerical results for Example 2 by Algorithm 7.

Table 5 

Numerical results for Example 2 by Algorithm 10.

Table 6 

Numerical results for Example 2 by Algorithm 11.

It is clear from Table 4 that the obtained approximately optimal solution is 𝑥^∗=(0.1585001,0.8339736,2.014753,−0.959688) with corresponding objective function value −44.22965. From [15], the obtained approximately optimal solution is 𝑥^∗=(0.169234,0.835656,2.008690,−0.964901) with corresponding objective function value −44.233582.

From Tables 4–6, one can see that Algorithm 11 converges faster than Algorithms 7 and 10, but the solution generated by Algorithm 11 is the worst. Algorithm 10 is the slowest one, and the solution generated by Algorithm 10 is worse than the solution generated by Algorithm 7.

From Tables 1–6, one can see that Algorithm 7 yields some approximate solutions to [𝑃] that have a better objective function value in comparison with Algorithms 10 and 11.

5. Conclusion

In this paper, we propose a method for smoothing the nonsmooth square-order exact penalty function for inequality constrained optimization. Error estimations are obtained among the optimal objective function values of the smoothed penalty problem, of the nonsmooth penalty problem, and of the original optimization problem. The algorithm based on the smoothed penalty functions is shown to be convergent under mild conditions.

According to the numerical results given in Section 4, one may draw that the smoothing penalty function 𝜑_𝑞(𝑥) yields some better convergence results for computing an approximate solution to [𝑃] than 𝐹(𝑥,𝑞) and 𝑓(𝑥,𝑞).

Finally, we give some advices on how to choose a parameter in the algorithm. According to our experience, initially, 𝑞₀ may be 0.1, 1, 5, 10, 100, 1000, or 10000, 𝑁 = 2, 5, 10, or 100, and the iteration formula 𝑞=𝑁𝑞. The initial value of 𝜖₀ may be 10, 5, 1, 0.5, or 0.1, 𝜂 = 0.5, 0.1, 0.05, or 0.01, and the iteration formula 𝜖=𝜂𝜖.

Acknowledgments

This work is supported by National Natural Science Foundation of China (10971118 and 71371107) and the Foundation of Shandong Province (J10LG04 and ZR2012AL07).