Smoothing approximation to the lower order exact penalty function for inequality constrained optimization
- Shujun Lian^{1}Email authorView ORCID ID profile and
- Nana Niu^{1}
https://doi.org/10.1186/s13660-018-1723-x
© The Author(s) 2018
Received: 11 March 2018
Accepted: 5 June 2018
Published: 11 June 2018
Abstract
For inequality constrained optimization problem, we first propose a new smoothing method to the lower order exact penalty function, and then show that an approximate global solution of the original problem can be obtained by solving a global solution of a smooth lower order exact penalty problem. We propose an algorithm based on the smoothed lower order exact penalty function. The global convergence of the algorithm is proved under some mild conditions. Some numerical experiments show the efficiency of the proposed method.
Keywords
MSC
1 Introduction
This problem is widely applied in transportation, economics, mathematical programming, regional science, etc. [1–3], and it has received extensive attention on a related problem, for example, variational inequalities, equilibrium problem, minimizers of convex functions, etc. (see, e.g., [4–15]).
The remainder of this paper is organized as follows. In Sect. 2, a new smoothing function is proposed. The error estimates are obtained among the optimal objective function values of the smoothed penalty problem, the non-smooth penalty problem, and the original problem. In Sect. 3, the corresponding algorithm is proposed to obtain an approximate solution to (P). The global convergence of the algorithm is proved. In Sect. 4, some numerical experiments are given to illustrate the efficiency of the algorithm. In Sect. 5, some conclusions are presented.
2 A smoothing penalty function
Assumption 2.1
- (1)\(f_{0}(x)\) satisfies the coercive condition$$\lim_{\|x\|\rightarrow+\infty}f_{0}(x)=+\infty. $$
- (2)
The optimal solution set \(G(\mbox{($P$)})\) is a finite set.
Now we consider a new smoothing technique to the lower order penalty function (1.3).
The following figure shows the process of function \(p_{k,\epsilon}(t)\) approaching function \(p_{k}(t)\).
For problems (P), (\(\mathit {LP}^{\prime}\)), and (SP), we have the following conclusion.
Lemma 2.1
Proof
Theorem 2.1
For a positive sequence \(\{\varepsilon_{j}\}\), which converges to 0 as \(j\to\infty\), assume that \(x^{j}\) is an optimal solution to \(\min_{x\in X} \varphi_{q,k,\epsilon_{j}}(x)\) for some given \(q>0\), \(k\in[\frac{1}{2},1)\). If x̅ is an accumulating point of sequence \(\{x^{j}\}\), then x̄ is an optimal solution to \(\min_{x\in X} \varphi_{q,k}(x)\).
Proof
Theorem 2.2
Proof
Under the hypothetical conditions, it holds that \(\varphi_{q,k}(x_{q,k}^{*})\leq\varphi_{q,k}(x)\) and \(\varphi _{q,k,\epsilon}(\bar{x}_{q,k,\epsilon})\leq\varphi_{q,k,\epsilon }(x)\), \(\forall x\in X\).
Corollary 2.1
Proof
Definition 1
Theorem 2.3
Proof
Theorems 2.1 and 2.2 show that an optimal solution of (SP) is also an approximate optimal solution of (\(\mathit {LP}^{\prime}\)) when the error ϵ is sufficiently small. By Theorem 2.3, an optimal solution of (SP) is an approximately optimal solution of (P) if the optimal solution of (SP) is an ϵ-feasible solution of (P).
3 A smoothing method
Based on the discussion in the last section, we can design an algorithm to obtain an approximate optimal solution of (P) by solving (SP).
Algorithm 3.1
- Step 1.
Take \(x^{0}\), \(\epsilon_{0}>0\), \(0< a<1\), \(q_{0}>0\), \(b>1\), \(\epsilon>0\), and \(k\in[\frac{1}{2},1)\), let \(j=0\) and go to Step 2.
- Step 2.
Solve \(\min_{x\in R^{n}} \varphi_{q_{j},k,\epsilon_{j}}(x)\) starting at \(x^{j}\). Let \(x^{j+1}\) be the optimal solution (\(x^{j+1}\) can be obtained by a quasi-Newton method).
- Step 3.
Let \(\epsilon_{j+1}=a\epsilon_{j}\), \(q_{j+1}=b q_{j}\), and \(j=j+1\), then go to Step 2.
Remark
Since \(0< a<1\) and \(b>1\), let \(a^{2k-1}b<1\), as \(j\rightarrow+\infty\), the sequence \(\{\epsilon_{j}\}\) is gradually decreased to 0, the sequence \(\{q_{j}\}\) is gradually increased to +∞ and \(\{\epsilon_{j}^{2k-1}q_{j}\} \) is gradually decreased to 0.
Under some mild conditions, the following conclusion shows the global convergence of Algorithm 3.1.
Theorem 3.1
- (1)
\(\{x^{j+1}\}\) is bounded.
- (2)
Any limit point of \(\{x^{j+1}\}\) is an optimal solution of (P).
Proof
(2) Without loss of generality, we assume \(x^{j+1}\rightarrow x^{*}\) as \(j\rightarrow\infty\).
To prove \(x^{*}\) is the optimal solution of (P), it is only needed to show that \(x^{*}\in X_{0}\) and \(f_{0}(x^{*})\leq f_{0}(x)\), \(\forall x\in X_{0}\).
Next, we show that \(f_{0}(x^{*})\leq f_{0}(x)\), \(\forall x\in X_{0}\).
4 Numerical examples
In this section, we will do some numerical experiments to show the efficiency of Algorithm 3.1.
Example 4.1
Numerical results for Example 4.1 with \(x^{0}=(0,0,0,0)\)
j | \(x^{j+1}\) | \(q_{j}\) | \(\epsilon_{j}\) | \(f_{1}(x^{j+1})\) | \(f_{2}(x^{j+1})\) | \(f_{3}(x^{j+1})\) | \(f_{0}(x^{j+1})\) |
---|---|---|---|---|---|---|---|
0 | (0.185009,0.804369,2.015460,−0.952409) | 1 | 0.01 | −4.797079 | −0.00109 | −2.028111 | −44.225926 |
1 | (0.169902,0.835670,2.008151,−0.965196) | 2 | 0.0001 | −9.748052 | −9.337847 | −1.883271 | −44.231252 |
Numerical results for Example 4.1 with \(x^{0}=(2,0,3.5,0)\)
j | \(x^{j+1}\) | \(q_{j}\) | \(\epsilon_{j}\) | \(f_{1}(x^{j+1})\) | \(f_{2}(x^{j+1})\) | \(f_{3}(x^{j+1})\) | \(f_{0}(x^{j+1})\) |
---|---|---|---|---|---|---|---|
0 | (0.169693,0.835634,2.008291,−0.965082) | 1 | 0.01 | −9.502428 | −8.676884 | −1.883244 | −44.231403 |
Numerical results for Example 4.1 with \(x^{0}=(2,2,2,0.5)\)
j | \(x^{j+1}\) | \(q_{j}\) | \(\epsilon_{j}\) | \(f_{1}(x^{j+1})\) | \(f_{2}(x^{j+1})\) | \(f_{3}(x^{j+1})\) | \(f_{0}(x^{j+1})\) |
---|---|---|---|---|---|---|---|
0 | (0.169691,0.835633,2.008294,−0.965080) | 1 | 0.01 | −9.502279 | −8.676796 | −1.883249 | −44.231403 |
From Tables 1, 2, 3, we know that the obtained approximate optimal solutions are similar, which shows that the numerical result of Algorithm 3.1 does not depend on the section of the starting points for this example. In [18], the objective function value \(f_{0}(x^{*})=-44.23040\) was obtained in the forth iteration. From the numerical results given in [22], we know that the optimal solution is \(x^{*}=(0.1585001,0.8339736,2.014753,-0.959688 )\) with the objective function value \(f_{0}(x^{*})=-44.22965\). In [23], the objective function value obtained in the 25th iteration is \(f_{0}(x^{*})=-44\). Hence, the numerical results obtained by Algorithm 3.1 are better than the numerical results given in [18, 22, 23] for this example.
Example 4.2
Numerical results for Example 4.2 with \(k=\frac{3}{4}\)
j | \(x^{j+1}\) | \(q_{j}\) | \(\epsilon_{j}\) | \(f_{1}(x^{j+1})\) | \(f_{2}(x^{j+1})\) | \(f_{0}(x^{j+1})\) |
---|---|---|---|---|---|---|
0 | (3.4217,2.7082) | 2 | 0.001 | 4.1300 | −0.0053 | −15.2492 |
1 | (0.8022,1.1978) | 20 | 0.000001 | 0.0000 | −0.4066 | −7.1999 |
Numerical results for Example 4.2 with \(k=\frac{3}{5}\)
j | \(x^{j+1}\) | \(q_{j}\) | \(\epsilon_{j}\) | \(f_{1}(x^{j+1})\) | \(f_{2}(x^{j+1})\) | \(f_{0}(x^{j+1})\) |
---|---|---|---|---|---|---|
0 | (4.0607,3.0227) | 2 | 0.001 | 5.0834 | −0.0153 | −16.0434 |
1 | (0.8027,1.1971) | 20 | 0.000001 | −0.0003 | −0.4086 | −7.1992 |
Numerical results for Example 4.2 with \(k=\frac{8}{9}\)
j | \(x^{j+1}\) | \(q_{j}\) | \(\epsilon_{j}\) | \(f_{1}(x^{j+1})\) | \(f_{2}(x^{j+1})\) | \(f_{0}(x^{j+1})\) |
---|---|---|---|---|---|---|
0 | (2.6356,2.3168) | 2 | 0.001 | 2.9523 | −0.0020 | −13.7027 |
1 | (0.8005,1.1995) | 20 | 0.000001 | 0.0000 | −0.4015 | −7.2000 |
From Tables 4, 5, 6, we can see that almost the same approximate optimal solutions are obtained for different k in this example. The objective function value is similar to the objective function value \(f_{0}(x^{*})=-7.2000\) with \(x^{*}=(0.8000,1.2000)\) obtained in the forth iteration in [17].
Example 4.3
Numerical results for Example 4.3 with \(x^{0}=(0,0)\)
j | \(x^{j+1}\) | \(q_{j}\) | \(\epsilon_{j}\) | \(f_{1}(x^{j+1})\) | \(f_{2}(x^{j+1})\) | \(f_{0}(x^{j+1})\) |
---|---|---|---|---|---|---|
0 | (2.329795,3.133729) | 5 | 10^{−2} | −0.047009 | −0.043471 | −5.463524 |
1 | (2.329238,3.173320) | 10 | 10^{−4} | −0.002868 | −0.006501 | −5.502557 |
2 | (2.329452,3.177637) | 20 | 10^{−6} | −0.000302 | −0.001176 | −5.507089 |
3 | (2.329626,3.177558) | 40 | 10^{−8} | −0.001802 | −0.000436 | −5.507185 |
Numerical results for Example 4.3 with \(x^{0}=(1.0,1.5)\)
j | \(x^{j+1}\) | \(q_{j}\) | \(\epsilon_{j}\) | \(f_{1}(x^{j+1})\) | \(f_{2}(x^{j+1})\) | \(f_{0}(x^{j+1})\) |
---|---|---|---|---|---|---|
0 | (2.330261,3.061875) | 5 | 10^{−1} | −0.1226776 | −0.1131323 | −5.392137 |
1 | (2.329664,3.161611) | 15 | 10^{−2} | −0.018055 | −0.016207 | −5.491275 |
2 | (2.329639,3.171941) | 45 | 10^{−3} | −0.007524 | −0.005993 | −5.501580 |
3 | (2.329560,3.177804) | 135 | 10^{−4} | −0.001013 | −0.000503 | −5.507363 |
4 | (2.329593,3.177793) | 405 | 10^{−5} | −0.001297 | −0.000357 | −5.507386 |
5 | (2.329622,3.177781) | 1215 | 10^{−6} | −0.001544 | −0.000234 | −5.507403 |
Numerical results for Example 4.3 with \(x^{0}=(2,0.5)\)
j | \(x^{j+1}\) | \(q_{j}\) | \(\epsilon_{j}\) | \(f_{1}(x^{j+1})\) | \(f_{2}(x^{j+1})\) | \(f_{0}(x^{j+1})\) |
---|---|---|---|---|---|---|
0 | (2.330460,3.179900) | 2 | 10^{−5} | −0.006287 | 0.005832 | −5.510360 |
1 | (2.329672,3.179735) | 20 | 10^{−8} | −0.000001 | 0.001957 | −5.509408 |
2 | (2.329672,3.179735) | 200 | 10^{−11} | −0.000000 | 0.001957 | −5.509407 |
3 | (2.329541,3.178391) | 2000 | 10^{−14} | −0.000000 | −0.000000 | −5.507933 |
In [24], with three different starting points, similar numerical results are given with \({k=\frac{2}{3}}\). The optimal solution \((2.329517, 3.178421)\) is given with the objective function value −5.507938. In [25], the optimal solution \((2.3295, 3.1783)\) is given with the objective function value −5.5079. The numerical results of Example 4.3 are similar to the numerical results of [24] and [25] in this example.
From Tables 7, 8, 9, we can see that we need to adjust the parameters \(q_{0}\), \(\epsilon_{0}\), a, b to get the better numerical results with different k and \(x^{0}\). Usually, \(\epsilon_{0}\) may be 0.5, 0.1, 0.01, 0.001, or smaller digits, and \(a=0.5, 0.1,0.01\), or 0.001. \(q_{0}\) may be 1, 2, 3, 5, 10, 100, or larger digits, and \(b= 2, 3, 5, 10\), or 100.
5 Concluding remarks
In this paper, we proposed a method to smooth the lower order exact penalty function with \(k\in[\frac{1}{2},1)\) for inequality constrained optimization. Furthermore, we proved that the algorithm based on the smoothed penalty functions is globally convergent under mild conditions. The given numerical experiments show that the algorithm is effective.
Declarations
Funding
This work was supported by the National Natural Science Foundation of China (71371107 and 61373027) and the Natural Science Foundation of Shandong Province (ZR2016AM10)
Authors’ contributions
SL and NN drafted the manuscript. SL revised it. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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