 Research
 Open Access
A new filter QPfree method for the nonlinear inequality constrained optimization problem
 Youlin Shang^{1},
 ZhengFen Jin^{1}View ORCID ID profile and
 Dingguo Pu^{2}Email author
https://doi.org/10.1186/s1366001818513
© The Author(s) 2018
 Received: 18 April 2018
 Accepted: 12 September 2018
 Published: 11 October 2018
Abstract
In this paper, a filter QPfree infeasible method with nonmonotone line search is proposed for minimizing a smooth optimization problem with smooth inequality constraints. This proposed method is based on the solution of nonsmooth equations, which are obtained by the Lagrangian multiplier method and the function of the nonlinear complementarity problem for the Karush–Kuhn–Tucker optimality conditions. Especially, each iteration of this method can be viewed as a perturbation of a Newton or quasiNewton iteration on both the primal and dual variables for the solution of the Karush–Kuhn–Tucker optimality conditions. What is more, it is considered to use the function of the nonlinear complementarity problem in the filter, which makes the proposed algorithm avoid the incompatibility. Then the global convergence of the proposed method is given. And under some mild conditions, the superlinear convergence rate can be obtained. Finally, some preliminary numerical results are shown to illustrate that the proposed filter QPfree infeasible method is quite promising.
Keywords
 Nonlinear constrained optimization
 Filter method
 QPfree method
 Nonmonotone line search
MSC
 90C20
 90C30
 90C33
1 Introduction
In this paper, we mainly consider solving the nonlinear optimization problem (NLP) with the inequality constraints, where the objective function and the constrained functions are Lipschitz continuously differentiable functions. We give the Lagrangian function associated with this problem, then the Karush–Kuhn–Tucker (KKT) optimality conditions for our solved problem can be obtained.
It is well known that the KKT optimality conditions is a mixed nonlinear complementarity problem (NCP). And this NCP has attracted much attention due to its various applications [1–3] such as the economic equilibrium problem, the restructuring problems of electricity and gas markets, and so on. Of course, there are many efficient methods for solving the NCP, which can be seen in [4–7]. One popular way to solve the NCP is to construct a Newton method for solving the related nonlinear equations, which is a reformulation of the KKT optimality condition. Another way is to use the filter method to directly solve the NLP with the inequality constraints. Recently Pu, Li, and Xue [8] proposed a new quadratic programming (QP)free infeasible method for minimizing a smooth function subject to some inequality constraints. This method is based on the solution of nonsmooth equations which are obtained by the multiplier and the Fischer–Burmeister NCP function for the KKT conditions. They proved that the method had a superlinear convergence rate under some mild conditions.
Fletcher and Leyffer [9] proposed a filter method for solving the NLP problem, which was an alternative to the traditional merit functions approach. Provided that there is a sufficient decrease in the objective function or the constraints violation function, it was shown that the trial points generated from solving a sequence of trust region QP subproblems are accepted. In addition, the computational results reported in [9, 10] are also very encouraging. For more related methods, one can refer to [11–16].
Stimulated by the progress in these two aspects, in this paper, we propose a nonmonotone filter QPfree infeasible method for minimizing a smooth function subject to smooth inequality constraints. This proposed iterative method is based on the solution of nonsmooth equations, which are obtained by the multiplier and some NCP functions for the KKT first order optimality conditions. And each iteration of this method can be viewed as a perturbation of a Newton or quasiNewton iteration on both the primal and dual variables for the solution of the KKT optimality conditions. Specifically, we use the filter on the linear search with a nonmonotone acceptance mechanism [17, 18]. Moreover, we also consider to use the NCP function in the filter. Thus our algorithm can avoid the incompatibility, which may appear in the filter SQP algorithm. We also give the global convergence and the superlinear convergence rate of the proposed method under some mild conditions. Finally, we take some numerical tests to illustrate the effectiveness of the proposed filter QPfree infeasible method.
The rest of this paper is organized as follows. In Sect. 2, we give some preliminaries and the formulation of the solved problem. Then we propose an infeasible filter QPfree method. In Sect. 3, we show that the proposed method is well defined and establish its global convergence and superlinear convergence rate under some mild conditions. Some numerical tests are given in Sect. 4. Finally, we give some brief conclusions in Sect. 5.
2 Preliminaries and algorithm
In this section, we firstly introduce the formulation of the solved problem. Then we give some preliminaries for structuring a new filter QPfree method. Finally, we present the structure of our proposed method in detail.
2.1 Preliminaries
Let \(\phi_{i}(x, \mu)=\psi(g_{i}(x), \mu_{i})\), \(1\le i\le m\). Given the above formulation of problem (1), we can denote \(\Phi (x, \mu)=((\nabla_{x}L(x, \mu))^{T}, (\Phi_{1}(x, \mu))^{T})^{T}\), where \(\Phi_{1}(x, \mu)=( \phi_{1}(x, \mu), \ldots , [4] \phi_{m}(x, \mu) )^{T}\).
Clearly, the KKT optimality conditions (2) can be equivalently reformulated as the nonsmooth equations \(\Phi(x, \mu)=0\).
2.2 Algorithm
In this subsection, we give the process and the framework of the filter QPfree method for solving problem (1). We firstly give some closed forms for preparing the method.
Remark 1
2.3 Implementation
 A1.:

The level set \(\{xf(x)\le f(x^{0})\}\) is bounded, and for sufficiently large k, \(\\mu^{k}+\lambda^{k0}+\lambda^{k1}\< \bar{\mu}\).
 A2.:

f and \(g_{i}\) are Lipschitz continuously differentiable, and for all y, \(z\in R^{n+m}\),where \(m_{0}>0\) is the Lipschitz constant.$$\bigl\Vert \nabla L(y)\nabla L(z) \bigr\Vert \le m_{0} \Vert yz \Vert , \qquad \bigl\Vert \Phi(y)\Phi(z) \bigr\Vert \le m_{0} \Vert yz \Vert , $$
 A3.:

\(H^{k}\) is positive definite and there exist positive numbers \(m_{1}\) and \(m_{2}\) such that \(m_{1}\d\^{2}\le d^{T}H^{k}d\le m_{2}\d\^{2}\) for all \(d\in R^{n}\) and all k.
Lemma 1
If \(\Phi^{k} \neq0\), then \(V^{k} \) is nonsingular.
Proof
Lemma 2
\(d^{k0}=0\) if and only if \(\nabla f^{k}=0\), and \(d^{k0}=0\) implies \(\bar{\lambda}^{k0}=0\) and \(\lambda^{k0}=0\).
It is clear that the following lemma holds, with reference to [8].
Lemma 3
Proof
Lemma 4
Proof
We define that if \((g_{i}^{k},\mu_{i}^{k})\neq(0,0)\), then \((\bar{\xi}_{i}^{k0},\bar{\eta}_{i}^{k0})=(\xi_{i}^{k},\eta_{i}^{k})\), otherwise \(\bar{\xi}_{i}^{k0}(\nabla g_{i}^{k})^{T}d^{k0}+\bar{\eta}_{i}^{k0} \lambda_{i}^{k0}= \phi_{i}'((x^{k},\mu^{k}), (d^{k0}, \lambda^{k0}))\), where \(\phi_{i}'((x^{k},\mu^{k}), (d^{k0}, \lambda^{k0}))\) is the direction derivative of \(\phi_{i}(x,\mu )\) at \((x^{k},\mu^{k})\) in the direction \((d^{k0}, \lambda^{k0})\).
Lemma 5
Proof
Lemma 6
\(d^{k0}=0\) if and only if \(\nabla f^{k}=0\), and \(d^{k0}=0\) implies \(\bar{\lambda}^{k0}=0\) and \(\lambda^{k0}=0\).
Proof
Clearly, \(\bar{\lambda}^{k0}=0\), \(\lambda^{k0}=0\), and \((\nabla f^{k},0)=(V^{k})^{1}(0,0)=(0,0)\). □
From Lemmas 3–6, we know that, if \(\Phi _{1}^{k}\neq0\), then \((d^{k},\lambda^{k})\) is the decreasing direction of \(\\Phi^{k}\\); if \(d^{k0}\neq0\), then \(d^{k}\) is the decreasing direction of \(f^{k}\). If \(\Phi_{1}^{k}=0\) and \(d^{k0}=0\), then \((x^{k},\mu^{k})\) is a KKT point. We consider four cases for linear searches.
Case 1. \(k1\) iteration has a Φstep and \(\Phi_{1}^{k}=0\). In this case, \(p^{k}_{\max}=p^{k1}_{\max}\) and \(\min\{p^{k} j_{\max}j\in F^{k}>0\}\). Clearly, we can find \(\alpha _{k}\) such that \(\hat{x}^{k+1}\) satisfies (9).
Case 4. The \((k1)\) iteration has an fstep and \(d^{k0}=0\). In this case, if \(\Phi_{1}^{k}=0\), then \((x^{k}, \mu^{k} )\) is a KKT point, otherwise \(x^{k}\) may be an infeasible stationary point.
If there are no such \(x^{k+1}\) and \(\mu^{k+1}\) or \(\alpha_{k}\) too small, we use the backtracking technology or use the feasibility restoration phase to find \(x^{k+1}\) and \(\mu^{k+1}\) so that it is acceptable that the filter and the \(QP(x^{k+1})\) subproblem are compatible.
3 Convergence
 A4.:

For all k and some \(\alpha_{\min}>0\), \(\alpha_{k}>\alpha_{\min}>0\).
It implies from (3) and (4) that \(p^{k}_{\max}>0\) is monotonically nonincreasing and, if \(\\Phi_{1}(x^{k})\ \to0\), then \(p^{k}_{\max}\to0\).
Lemma 7
Proof
Lemma 8
Consider an infinite sequence iterations on which \(\{f^{k},\\Phi_{1}(x^{k})\^{2}\}\) entered into the filter, where \(\\Phi _{1}(x^{k})\>0\) and \(\{f^{k}\}\) is bounded below. It follows that \(\Phi_{1}(x^{k})\to0 \).
Theorem 1
If \((x^{*}, \mu^{*})\) is an accumulation point of \(\{( x^{k},\mu^{k})\}\), then \(x^{*}\) is a KKT point of problem (1).
 A5.:

The Mangasarian–Fromovitz (MF) qualification condition is satisfied at \(x^{*}\), i.e., \(\{\nabla g_{i}(x^{*})\}\) are linear independent for all \(i\in I=\{i g_{i}(x^{*})=0\}\), and there exists a direction such that \(d^{T}\nabla g_{i}(x^{*})<0\), \(i\in I=\{i g_{i}(x^{*})=0\}\), where \(i\in I=\{i g_{i}(x^{*})=0\}\), where \(x^{*}\) is an accumulation point of \(\{x^{k}\}\) and a KKT point of problem (1).
 A6.:

The sequence of \(\{H^{k}\}\) satisfies$$\frac{ \Vert (H^{k}\nabla_{x}^{2}L(x^{k},\mu^{k} ))d^{k1} \Vert }{ \Vert d^{k1} \Vert }\to0. $$
 A7.:

The strict complementarity condition holds at each KKT point \((x^{*},\mu^{*} )\).
It follows that \(\phi^{k}\) is differentiable at each KKT point \((x^{*},\mu^{*} )\).
Assumption A7 implies that Φ is continuously differentiable at each KKT point \((x^{*},\mu^{*} )\). As Lemma 1, we have that the following lemmas hold.
Lemma 9
Assume A1–A7 hold, then \(\{ \ (V^{k})^{1}\ \}\) and \(\{ \(\hat{V}^{k})^{1}\ \}\) are bounded. Furthermore, if \(V^{*}\) is an accumulation matrix of \(\{ V^{k}\}\), then \(V^{*}\) is nonsingular.
Proof
By Theorem 1, \(\Phi^{*}=0\) and \(c^{k}\to0\). Without loss of generality, we may assume that \(( x^{k},\mu^{k})\to(x^{*}, \mu^{*})\), \(H^{k}\to H^{*}\), \(\operatorname{diag}(\xi^{k})\to\operatorname{diag}(\xi ^{*})\), and \(\operatorname{diag} (\eta^{k})\to \operatorname{diag}(\eta^{*})\). By the definitions of \(\xi_{i}^{k}\) and \(\eta _{i}^{k}\), we know that \((\xi_{i}^{*})^{2}+(\eta_{i}^{*})^{2}\neq0\). \(H^{k}\to H^{*}\) implies that \(H^{*}\) is positive definite.
On the other hand, suppose to the contrary that there exists a subsequence \(\{(x^{k(i)}, \lambda^{k(i)})\}\) such that \(\( V^{k(i)})^{1}\\to\infty\) as \(k(i)\to\infty\) and \((x^{k(i)}, \lambda^{k(i)}) \to(x^{*}, \lambda^{*})\). We can choose \(k(i)\) properly such that \(V^{k(i)}\to V^{*}\) including \(\xi^{k(i)}\to\xi^{*}\) and \(\eta^{k(i)}\to\eta^{*}\). Clearly, \((\xi_{j}^{*})^{2}+ (\eta_{j}^{*})^{2}\ge32\sqrt{2}>0\) and \(V^{*}\in\partial \Phi^{*}\). But \(V^{*}\) is nonsingular by the above proof, which contradicts the assumption \(\( V^{k(i)})^{1}\\to\infty\). Hence, \(\{ \( V^{k(i)})^{1}\ \}\) is bounded. \(\Phi^{k}\to0\) implies \(\lim_{k\to\infty}V^{k}=\lim_{k\to\infty}\hat{V}^{k}\), we can also obtain that \(\{ \( \hat{V}^{k})^{1}\ \}\) is bounded. This lemma holds. □
Assumption A5 shows that \((x^{k}, \mu^{k})\) is a Newton direction with a high order perturbation. We obtain the following lemma.
Lemma 10
For sufficiently large k, \(x^{k+1}=x^{k}+d^{k1}\) and \(\mu^{k+1}=\mu^{k}+\lambda^{k1}\).
Furthermore, Lemma 10 implies that the following theorem holds.
Theorem 2
Assume A1–A7 hold. Let Algorithm 1 (NFQPIM) be implemented to generate a sequence \(\{(x^{k},\mu^{k})\}\), \((x^{*}, \mu^{*})\) be an accumulation point of \(\{(x^{k},\lambda^{k})\}\). Then \((x^{*}, \mu^{*})\) is a KKT point of problem (1), and \((x^{k},\mu^{k})\) converges to \((x^{*}, \mu^{*})\) superlinearly.
4 Numerical tests
Numerical results on the NFQPIM for some constrained optimization problems
Problem  n  m  NIT  NG  NF 

hs001  2  1  65  43  40 
hs002  2  1  15  19  18 
hs003  2  1  3  5  4 
hs004  2  2  4  6  5 
hs005  2  4  9  11  9 
hs006  2  1  5  9  8 
hs007  2  1  16  25  21 
hs008  2  2  3  6  5 
hs009  2  1  10  13  12 
hs010  2  1  27  43  33 
hs011  2  1  13  23  15 
hs012  2  1  13  19  16 
hs013  2  1  3  6  4 
hs014  2  2  4  7  4 
hs015  2  2  7  11  7 
hs016  2  5  5  8  7 
hs017  2  5  14  19  16 
hs018  2  6  20  24  23 
hs019  2  6  4  7  6 
hs020  2  5  5  8  6 
hs021  2  5  4  7  5 
hs022  2  2  5  8  5 
hs023  2  9  3  4  4 
hs024  2  5  7  13  10 
hs025  3  6  4  6  6 
hs026  3  1  24  33  27 
hs027  3  1  28  34  31 
hs028  3  1  6  12  9 
hs029  3  1  21  37  31 
hs030  3  7  9  11  10 
hs031  3  7  12  17  14 
hs032  3  5  8  14  12 
hs033  3  6  11  16  24 
hs034  3  8  8  12  10 
hs035  3  4  7  11  10 
hs036  3  7  10  13  11 
hs037  3  8  13  18  15 
hs038  4  8  83  111  98 
hs039  4  2  21  35  32 
hs040  4  3  11  16  14 
hs041  4  9  13  17  15 
hs042  4  2  11  14  12 
hs043  4  3  15  23  20 
hs044  4  10  16  23  21 
hs045  5  10  5  7  7 
hs046  5  10  29  37  33 
hs047  5  3  26  33  30 
hs048  5  2  10  15  13 
hs049  5  2  36  46  40 
hs050  5  3  35  43  39 
hs051  5  3  9  13  11 
hs052  5  3  7  14  12 
hs053  5  3  5  7  7 
hs054  6  13  9  13  10 
hs055  6  14  8  14  11 
hs056  7  4  12  14  12 
hs057  2  3  6  9  8 
hs059  2  7  28  33  30 
hs060  3  7  13  23  21 
hs061  3  2  59  68  58 
hs062  3  7  24  33  28 
hs063  3  5  15  23  20 
hs064  3  4  36  43  37 
hs065  3  7  21  29  27 
hs066  3  8  13  23  19 
hs067  3  41  65  53  47 
hs099  7  16  65  43  40 
hs100  7  4  65  43  40 
hs101  7  20  65  43  40 
hs102  7  20  65  43  40 
hs103  7  20  65  43  40 
hs104  8  22  65  43  40 
hs105  8  17  65  43  40 
hs106  8  22  65  43  40 
hs107  9  14  65  43  40 
hs108  9  14  33  30  
hs110  10  20  25  43  40 
hs111  10  23  61  73  68 
hs112  10  13  51  73  67 
hs113  10  8  55  65  61 
hs114  10  31  56  73  68 
hs116  13  41  123  143  140 
hs117  15  20  511  63  60 
hs118  15  59  67  81  77 
hs119  16  40  68  83  78 
5 Conclusions
In this paper, we developed a nonmonotone filter QPfree infeasible method for minimizing a smooth optimization problem with inequality constraints. This proposed method is based on the solution of nonsmooth equations which are obtained by the multiplier and some NCP functions for the KKT firstorder optimality conditions. At each iteration of the proposed method, it was a perturbation of a Newton or quasiNewton iteration on both the primal and dual variables for the solution of the KKT optimality conditions. Moreover, we used the filter on linear searches with a nonmonotone acceptance mechanism. We also showed that the proposed method had a global convergence and a superlinear convergence rate. Finally, the numerical results illustrated that the proposed method was efficient. However, how to apply this method to the real optimal problem will be studied in the near future.
Declarations
Acknowledgements
We would like to thank the anonymous referees and the associate editor for their useful comments and suggestions which improved this paper greatly.
Funding
The work of Y. Shang is supported by the National Natural Science Foundation of China Grant No.11471102. The work of D. Pu is supported by the National Natural Science Foundation of China Grant No. 11371281. The work of Z.F. Jin is supported by the National Natural Science Foundation of China Grant No. 61772174, Plan for Scientific Innovation Talent of Henan Province Grant No. 174200510011.
Authors’ contributions
All authors contributed equally to this work. 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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