An Improved Nonmonotone Filter Trust Region Method for Equality Constrained Optimization

and Applied Analysis 3 2.3. The Improved Accepted Condition for d k . Borrowed from the usual trust region idea, we also need to define the following predicted reduction for the violation function h(x) = ‖c(x)‖ 2 predc k = h (x k ) − 󵄩󵄩󵄩󵄩 c k + A T

The algorithm that we discuss belongs to the class of trust region methods and, more specifically, to that filter methods suggested first by Fletcher and Leyffer [1], in which the use of a penalty function, a common feature of the large majority of the algorithms for constrained optimization, is replaced by the technique so-called "filter." Subsequently, global convergences of the trust region filter SQP methods were established by Fletcher et al. [2,3], and Ulbrich [4] got its superlinear local convergence.Especially, the step framework in [3] is similar in spirit to the composite-step SQP methods pioneered by Vardi [5], Byrd et al. [6], and Omojokun [7].Consequently, filter method has been actually applied in many optimization techniques, for instance, the pattern search method [8], the SLP method [9], the interior method [10], the bundle approaches [11,12], the system of nonlinear equations and nonlinear least squares [13], multidimensional filter method [14], line search filter methods [15,16], and so on.
In fact, filter method exhibits a certain degree of nonmonotonicity.The idea of nonmonotone technique can be traced back to Grippo et al. [17] in 1986, combined with the line search strategy.Due to its excellent numerical exhibition, over the last decades, the nonmonotone technique has been used in trust region method to deal with unconstrained and constrained optimization problems [18][19][20][21][22][23][24][25].More recently, the nonmonotone trust region method without penalty function has also been developed for constrained optimization [26][27][28][29].Especially, in [29] the nonmonotone idea is employed to the filter technique and restoration phase, a common feature of the large majority of filter methods, is not needed.
Based on [29], motivated by above ideas and methods, in this paper we present an improved filter algorithm that combines the nonmonotone and trust region techniques for solving nonlinear equality constrained optimization.Our method improves previous results and gives a more flexible mechanism, weakens some needed conditions, and has the merits as follows.
(i) An improved nonnomotone filter technique is presented.Filter technique is viewed as a biobjective 2 Abstract and Applied Analysis optimization problem that minimizes objective function  and violation function ℎ.In [29], the same nonmonotone measure is utilized for both  and ℎ at the th iteration.In the paper we improve it and define a new measure for ℎ, which may make the discussion and consideration of the nonmonotone properties of  and ℎ more freely.
(ii) The restoration phase is not needed.The restoration procedure has to be considered in most of filter methods.We employ the nonmonotone idea to the filter technique, so as [29] the restoration phase is not needed.
(iii) A more relaxed accepted condition for trial step is considered.Compared to general trust region methods, in [29] the condition for the accepted trial step is relaxed.In the paper we improve it and then the accepted condition for trial step is more relaxed.
(iv) A crucial criterion is weakened in the algorithm.By introducing a new parameter  ∈ (0, 1], we improve the crucial criterion for implementation of the method in [29].It is obvious that our criterion becomes same with [29] if  = 1, but new criterion can be easier to satisfy, so this crucial criterion has been weakened when setting  < 1 in the initialization of our algorithm.
The presentation is organized as follows.Section 2 introduces some preliminary results and improvements on filter technique and some conditions.Section 3 develops a modified nonmonotone filter trust region algorithm, whose global convergence is shown in Section 4. The results of numerical experience with the proposed method are discussed in the last section.

Preliminary and Improvements
In this section, we first recall some definitions and preliminary results about the techniques of composite-step SQP type trust region method and fraction of Cauchy decrease condition.And then the improvements on filter technique and some conditions are given.

Fraction of Cauchy Decrease Condition.
To introduce the corresponding results, we consider the unconstrained optimization problem: min{() |  ∈   }, where  :   →  is continuously differentiable.At iteration point   , it obtains a trial step   by solving the following quadratic program subproblem: where   is a symmetric matrix which is either the Hessian matrix of  at   or an approximation to it, and Δ  > 0 is a trust region radius.
To assure the global convergence, the step is only required to satisfy a fraction of Cauchy decrease condition.It means  must predict via the quadratic model function () at least as much as a fraction of the decreased given by the Cauchy step on (), that is, there exists a constant  > 0 fixed across all iterations, such that where  cp is the steepest descent step for () inside the trust region.And we have the following lemma.
where Δ  is a trust region radius, and the actual reduction To evaluate the descent properties of the step for the objective function, we use the predicted reduction of () and the actual reduction of () In general trust region method, the step   will be accepted if where  ∈ (0, 1) is a fixed constant.
In [29], considering nonmonotone technique, the condition ( 13) is replaced by where rared   is the relaxed actual reduction of (), that is, rared But in this paper, we consider more relaxed accepted condition by increasing ared where  0 is a small positive number.

The Improved Nonmonotone Filter Technique.
In order to obtain next iterate, it needs to determine the step; this procedure that decides which trial step is accepted is "filter method." For the optimization problem with equality constraints a promising trial step should either reduce the constraint violation ℎ or the objective function value .Since ℎ() = ‖()‖ 2 , it is easy to see that ℎ() = 0 if and only if  is a feasible point.So in the traditional filter method, a point  is called acceptance to the filter if and only if where 0 <  <  < 1, F denotes the filter set.Different from above criteria of filter idea, with nonmonotone technique, in [29] a point  is called to be acceptable to the filter if and only if either ℎ () ≤  max where 1), and there exists a positive constant  such that   ≥ .
Observe that () is used in both conditions (19), while in this paper we wish to reduce the relationship of the nonmonotone properties of  and ℎ.We consider the nonmonotone properties of  and ℎ, respectively, and call a trial point  is acceptable to the filter if and only if either ℎ () ≤  max where = is a subsequence of {ℎ(  )}, and 1 ≤ () ≤ min{( − 1) + 1, }.
Similar to the traditional filter methods, if (20) is satisfied, it is called an ℎ-type iteration and the filter set F  needs to be updated at each successful ℎ-type iteration.

The Weakened Criterion for Implementation of Algorithm.
We replace the crucial criterion pred where  ∈ (0, 1].It is obvious that this new criterion becomes the same with [29] if  = 1, but it can be easier to satisfy, so the criterion has been weakened when setting  < 1 in the initialization of our algorithm.
Step 6. Remark 3. At the beginning of each iteration, we always set Δ  ≥ Δ min , which will avoid too small trust region radius.
Remark 5.In the above algorithm, let  be a positive integer.
For each , let () satisfying and we also find () satisfy so the nonmonotonicity is showed as  > 1.

Global Convergence of Algorithm
In this section, we will prove the global convergence properties of Algorithm 2 under the following assumptions.From the above assumptions, it is easier to suppose that there exist five constants  1 ,  2 ,  3 ,  4 ,  5 such that  +  0 } for some  ∈ (0, 1).Thus, the trial step is accepted for all Δ  ≤ .
Lemma 8 has provided the implementation of Algorithm 2. By mechanism of Algorithm 2, it is obvious that there exists a constant Δ > 0, such that Δ  ≥ Δ for sufficiently large .
In the remainder of this paper we denote the set of indices of those iterations in which the filter has been augmented by A, that is to say, if the th iteration is ℎ-type iteration, we have  + 1 ∈ A. By this definition, if (20) holds for infinite iterations, then |A| = ∞; otherwise, we can say |A| ̸ = ∞.
Lemma 9. Let {  } be an infinite sequence generated by Algorithm 2.
Proof.By mechanism of Algorithm 2, we can assume that for all , it follows Then we first show that for all , it holds Next we prove (38) by induction.
Lemma 10.Suppose that the assumptions hold.If Algorithm 2 does not terminate finitely, let {  } be an infinite sequence generated by Algorithm 2 Proof.Suppose to contrary that there exist constant  > 0 and  > 0 such that ‖ ĝ ‖ > 2 for all  > .Then similar to the proof of Lemma As in the proof of Lemma 8, there exists  ∈ (0, 1) such that rared In common with the proof of Lemma 9, we have Proof.In view of convenience, denote where  +  − () + 1 ≤ () ≤  + .From the algorithm, we know  ++1 = ℎ( +1 ) and it holds Since ( + 1) ≤ () + 1, we have Proof.The conclusion follows immediately by Lemmas 9, 10, 11, and 12. Thus, the whole proof is completed.
During the numerical experiments, updating of   is done by where In the following tables, the notations mean as follows: (i) SC: the problems from Schittkowski [32]; (ii) HS: the problems from Hock and Schittkowski [31]; (iii) n: the number of variables; (iv) m: the number of inequality constraints; (v) NIT: the number of iterations; (vi) NF: the number of evaluations of the objective functions; (vii) NG: the number of evaluations of scalar constraint functions; (viii) Algorithm 1: our method in this paper; (ix) Algorithm 2: the method proposed in [29]; (x) Algorithm 3: the method proposed in [33].
The detailed numerical results are summarized in Tables 1  and 2. Now we give a brief analysis for numerical test results.From Table 1, we can see that our algorithm executes well for these typical problems taken from [31,32].From the computation efficiency in Table 2, by solving same problems in [31,32] we should point out our algorithm is competitive with the some existed nonmonotone filter type methods to solve equality constrained optimization, for example, [29,33].However, in our method an improved nonmonotone filter technique is proposed, which may make the discussion and consideration of the nonmonotone properties of  and ℎ more freely.Furthermore, we consider a more relaxed accepted condition for trial step and a weakened crucial criterion is presented.It is easier to meet those new criteria, so our method is much more flexible.All results summarized show that our algorithm is promising and numerically effective.
to obtain    and    , and set   =    +    .If   +   is acceptable to the filter, go to Step 4, otherwise go to Step 5.
≥ , then go to Step 5, otherwise go to Step 6.