A Nonmonotone Line Search Filter Algorithm for the System of Nonlinear Equations

Abstract

We present a new iterative method based on the line search filter method with the nonmonotone strategy to solve the system of nonlinear equations. The equations are divided into two groups; some equations are treated as constraints and the others act as the objective function, and the two groups are just updated at the iterations where it is needed indeed. We employ the nonmonotone idea to the sufficient reduction conditions and filter technique which leads to a flexibility and acceptance behavior comparable to monotone methods. The new algorithm is shown to be globally convergent and numerical experiments demonstrate its effectiveness.

1. Introduction

We consider the following system of nonlinear equations: 𝑐_𝑖(𝑥)=0,𝑖=1,2,…,𝑚,(1.1) where each 𝑐_𝑖∶𝑅^𝑛→𝑅(𝑖=1,2,…,𝑚) is a twice continuously differentiable function. It is one of the most basic problems in mathematics and has lots of applications in many scientific fields such as physics, chemistry, and economics.

In the context of solving nonlinear equations, a well-known method is the Newton method, which is known to exhibit local and second order convergence near a regular solution, but its global behavior is unpredictable. To improve the global properties, some important algorithms [1] for nonlinear equations proceed by minimizing a least square problem: minℎ(𝑥)=𝑐(𝑥)^𝑇𝑐(𝑥),(1.2) which can be also handled by the Newton method, while Powell [2] gives a counterexample to show a dissatisfactory fact that the iterates generated by the above least square problem may converge to a nonstationary point of ℎ(𝑥).

However, as we all know, there are several difficulties in utilizing the penalty functions as a merit function to test the acceptability of the iterates. Hence, the filter, a new concept first introduced by Fletcher and Leyffer [3] for constrained nonlinear optimization problems in a sequential quadratic programming (SQP) trust-region algorithm, replaces the merit fuctions avoiding the penalty parameter estimation and the difficulties related to the nondifferentiability. Furthermore, Fletcher et al. [4, 5] give the global convergence of the trust-region filter-SQP method, then Ulbrich [6] gets its superlinear local convergence. Consequently, filter method has been actually applied in many optimization techniques, for instance the pattern search method [7], the SLP method [8], the interior method [9], the bundle approaches [10, 11], and so on. Also combined with the trust-region search technique, Gould et al. extended the filter method to the system of nonlinear equations and nonlinear least squares in [12], and to the unconstrained optimization problem with multidimensional filter technique in [13]. In addition, Wächter and Biegler [14, 15] presented line search filter methods for nonlinear equality-constrained programming and the global and local convergence were given.

In fact, filter method exhibits a certain degree of nonmonotonicity. The idea of nonmonotone technique can be traced back to Grippo et al. [16] in 1986, combined with the line search strategy. Due to its excellent numerical exhibition, many nonmonotone techniques have been developed in recent years, for example [17, 18]. Especially in [17], a nonmonotone line search multidimensional filter-SQP method for general nonlinear programming is presented based on the Wächter and Biegler methods [14, 15].

Recently, some other ways were given to attack the problem (1.1) (see [19–23]). There are two common features in these papers; one is the filter approach is utilized, and the other is that the system of nonlinear equations is transformed into a constrained nonlinear programming problem and the equations are divided into two groups; some equations are treated as constraints and the others act as the objective function. And two groups of equations are updated at every iteration in those methods. For instance combined with the filter line search technique [14, 15], the system of nonlinear equations in [23] becomes the following optimization problem with equality constraints: sm.t.in𝑐_𝑗_𝑖∈𝑆1𝑐^2𝑖(𝑥)(𝑥)=0,𝑗∈𝑆₂.(1.3) The choice of two sets 𝑆₁ and 𝑆₂ are given as follows: for some positive constant 𝑛₀>0, it is defined that 𝑐^2𝑖₁(𝑥_𝑘)≥𝑐^2𝑖₂(𝑥_𝑘)≥⋯≥𝑐^2𝑖_𝑚(𝑥_𝑘), then 𝑆₁={𝑖_k∣𝑘≤𝑛₀} and 𝑆₂={𝑖_𝑘∣𝑘≥𝑛₀+1}.

In this paper we present an algorithm to solve the system of nonlinear equations, combining the nonmonotone technique and line search filter method. We also divide the equations into two groups; one contains the equations that are treated as equality constraints and the square of other equations is regarded as objective function. But different from those methods in [19–23], we just update the two groups at the iterations where it is needed indeed, which can make the scale of the calculation decrease in a certain degree. Another merit of our paper is to employ the nonmonotone idea to the sufficient reduction conditions and filter which leads to a flexibility and acceptance behavior comparable to monotone methods. Moreover, in our algorithm two groups of equations cannot be changed after an f-type iteration, thus in the case that |𝒜|<∞, the two groups are fixed after finite number of iterations. And the filter should not be updated after an f-type iteration, so naturally the global convergence is discussed, respectively, according to whether the number of updated filter is infinite or not. Furthermore, the global convergent property is induced under some reasonable conditions. In the end, numerical experiments show that the method in this paper is effective.

The paper is outlined as follows. In Section 2, we describe and analyze the nonmonotone line search filter method. In Section 3 we prove the global convergence of the proposed algorithm. Finally, some numerical tests are given in Section 4.

2. A Nonmonotone Line Search Filter Algorithm

Throughout this paper, we use the notations 𝑚_𝑘(𝑥)=‖𝑐_𝑆1(𝑥)‖²²=∑_𝑖∈𝑆1𝑐^2𝑖(𝑥) and 𝜃_𝑘(𝑥)=‖𝑐_𝑆2(𝑥)‖²²=∑_𝑖∈𝑆2𝑐^2𝑖(𝑥). In addition, we denote the set of indices of those iterations in which the filter has been augmented by 𝒜⊆ℕ.

The linearization of the KKT condition of (1.3) at the 𝑘th iteration 𝑥_𝑘 is as follows: ⎛⎜⎜⎝𝐵_𝑘𝐴^𝑘𝑆₂𝐴^𝑘𝑆₂^𝑇0⎞⎟⎟⎠⎛⎜⎜⎝𝜆𝑠^𝑘+𝑘⎞⎟⎟⎠=−⎛⎜⎜⎝𝑔_𝑘𝑐^𝑘𝑆₂⎞⎟⎟⎠,(2.1) where 𝐵_𝑘 is the Hessian or approximate Hessian matrix of 𝐿(𝑥,𝜆)=𝑚_𝑘(𝑥)+𝜆^𝑇𝑐_𝑆2(𝑥), 𝐴^𝑘𝑆₂=∇𝑐_𝑆2(𝑥_𝑘) and 𝑔(𝑥_𝑘)=∇𝑚_𝑘(𝑥_𝑘). Then the iterate formation is 𝑥_𝑘(𝛼_𝑘,𝑙)=𝑥_𝑘+𝛼_𝑘,𝑙𝑠_𝑘, where 𝑠_𝑘 is the solution of (2.1) and 𝛼_𝑘,𝑙∈(0,1] is a step size chosen by line search.

Now we describe the nonmonotone Armijo rule. Let 𝑀 be a nonnegative integer. For each 𝑘, let 𝑚(𝑘) satisfy 𝑚(0)=1,0≤𝑚(𝑘)≤min{𝑚(𝑘−1)+1,𝑀} for 𝑘≥1. For fixed constants 𝛾_𝑚,𝛾_𝜃∈(0,1), we might consider a trial point to be acceptable, if it leads to sufficient progress toward either goal, that is, if 𝜃_𝑘𝑥_𝑘𝛼_𝑘,𝑙≤1−𝛾_𝜃max𝜃_𝑘𝑥_𝑘,^{𝑚(𝑘)−1}_𝑟=0or𝑚_𝑘𝜆_𝑘𝑟𝜃_𝑘−𝑟𝑥_𝑘−𝑟𝑥_𝑘𝛼_𝑘,𝑙≤max𝑚_𝑘𝑥_𝑘,^{𝑚(𝑘)−1}_𝑟=0𝜆_𝑘𝑟𝑚_𝑘−𝑟𝑥_𝑘−𝑟−𝛾_𝑚𝜃_𝑘𝑥_𝑘,(2.2) where 𝜆_𝑘𝑟∈(0,1), ∑^{𝑚(𝑘)−1𝑟=0}𝜆_𝑘𝑟=1.

For the convenience we set 𝑚(𝑥_𝑘)=max{𝑚_𝑘(𝑥_𝑘),∑^{𝑚(𝑘)−1𝑟=0}𝜆_𝑘𝑟𝑚_𝑘−𝑟(𝑥_𝑘−𝑟)}, and 𝜃(𝑥_𝑘)=max{𝜃_𝑘(𝑥_𝑘),∑^{𝑚(𝑘)−1𝑟=0}𝜆_𝑘𝑟𝜃_𝑘−𝑟(𝑥_𝑘−𝑟)}. In order to avoid the case of convergence to a feasible but nonoptimal point, we consider the following switching condition: 𝑔^𝑇𝑘𝑠_𝑘<−𝜉𝑠^𝑇𝑘𝐵_𝑘𝑠_𝑘,−𝛼_𝑘,𝑙𝑔^𝑇𝑘𝑠_𝑘>𝜃_𝑘𝑥_𝑘^𝑠𝜃,(2.3) with 𝜉∈(0,1],𝑠_𝜃∈(0,1). If the switching condition holds, the trial point 𝑥_𝑘(𝛼_𝑘,𝑙) has to satisfy the Armijo nonmonotone reduction condition, 𝑚_𝑘𝑥_𝑘𝛼_𝑘,𝑙≤𝑚𝑥_𝑘+𝜏₃𝛼_𝑘,𝑙𝑔^𝑇𝑘𝑠_𝑘,(2.4) where 𝜏₃∈(0,1/2) is a fixed constant.

To ensure the algorithm cannot cycle, it maintains a filter, a “taboo region” ℱ_𝑘⊆[0,∞]×[0,∞] for each iteration 𝑘. The filter contains those combinations of constraint violation value 𝜃 and the objective function value 𝑚, that are prohibited for a successful trial point in iteration 𝑘. During the line search, a trial point 𝑥_𝑘(𝛼_𝑘,𝑙) is rejected, if (𝜃(𝑥_𝑘(𝛼_𝑘,𝑙)),𝑚(𝑥_𝑘(𝛼_𝑘,𝑙)))∈ℱ_𝑘. We then say that the trial point is not acceptable to the current filter, which is also called 𝑥_𝑘(𝛼_𝑘,𝑙)∈ℱ_𝑘.

If a trial point 𝑥_𝑘(𝛼_𝑘,𝑙)∉ℱ_𝑘 satisfies the switching condition (2.3) and the reduction condition (2.4), then this trial point is called an f-type point, and accordingly this iteration is called an f-type iteration. An f-type point should be accepted as 𝑥_𝑘+1 with no updating of the filter, that is ℱ_𝑘+1=ℱ_𝑘.(2.5)

While if a trial point 𝑥_𝑘(𝛼_𝑘,𝑙)∉ℱ_𝑘 does not satisfy the switching condition (2.3), but this trial point satisfies (2.2), we call it an h-type point, or accordingly an h-type iteration. An h-type point should be accepted as 𝑥_𝑘+1 with updating of the filter, that is ℱ_𝑘+1=ℱ_𝑘(𝜃,𝑚)∈𝑅²∶𝜃≥1−𝛾_𝜃𝜃𝑥_𝑘,𝑚≥𝑚𝑥_𝑘−𝛾_𝑚𝜃_𝑘𝑥_𝑘.(2.6)

In some cases it is not possible to find a trial step size that satisfies the above criteria. We approximate a minimum desired step size using linear models of the involved functions. For this, we define 𝛼^𝑘min=⎧⎪⎪⎪⎨⎪⎪⎪⎩min1−1−𝛾^𝜃𝜃𝑥_𝑘𝜃_𝑘𝑥_𝑘,𝑚𝑥_𝑘−𝑚_𝑘𝑥_𝑘−𝛾_𝑚𝜃_𝑘𝑥_𝑘𝑔^𝑇𝑘𝑠_𝑘,𝜃^𝑘𝑥_𝑘^𝑠𝜃if𝑔_𝑇𝑘−𝑔^𝑇𝑘𝑠_𝑘,1−1−𝛾_𝜃𝑠_𝑘<−𝜉𝑠^𝑇𝑘𝐵_𝑘𝑠_𝑘,𝜃𝑥_𝑘𝜃_𝑘𝑥_𝑘,otherwise.(2.7) If the nonmonotone line search encounters a trial step size with 𝛼_𝑘,𝑙<𝛼^𝑘min, the algorithm reverts to a feasibility restoration phase. Here, we try to find a new iterate which is acceptable to the current filter and for which (2.2) holds, by reducing the constraint violation with some iterative method.

The corresponding algorithm can be written as follows.

Algorithm 2.1. Step 1. Initialization: choose an initial guess 𝑥₀, 𝜌₁,𝜌₂∈(0,1), 𝜌₁<𝜌₂, and 𝜖>0. Compute 𝑔₀, 𝑐_𝑖(𝑥₀), 𝑆⁰¹, 𝑆⁰², and 𝐴_𝑘 for 𝑖∈𝑆⁰². Set 𝑀>0, 𝑚(0)=1, 𝑘=0, and ℱ₀=∅.
Step 2. If ‖𝑐(𝑥_𝑘)‖≤𝜖 then stop.
Step 3. Compute (2.1) to obtain 𝑠_𝑘. If there exists no solution to (2.1), go to Step 8. If ‖𝑠_𝑘‖≤𝜖 then stop.
Step 4. Use nonmonotone line search. Set 𝑙=0 and 𝛼_𝑘,𝑙=1.  Step 4.1. If 𝛼_𝑘,𝑙<𝛼^𝑘min, where the 𝛼^𝑘min is obtained by (2.7), go to Step 8. Otherwise we get 𝑥_𝑘(𝛼_𝑘,𝑙)=𝑥_𝑘+𝛼_𝑘,𝑙𝑠_𝑘. If 𝑥_𝑘(𝛼_k,𝑙)∈ℱ_𝑘, go to Step 4.3.  Step 4.2. Check sufficient decrease with respect to current iterate.  Step 4.2.1. If the switching condition (2.3) and the nonmonotone reduction condition (2.4) hold, set ℱ_𝑘+1=ℱ_𝑘 and go to Step 5. While only the switching condition (2.3) are satisfied, go to Step 4.3.  Step 4.2.2. The switching conditions (2.3) are not satisfied. If the nonmonotone filter condition (2.2) holds, set 𝑥_𝑘+1=𝑥_𝑘+𝛼_𝑘,𝑙𝑠_𝑘, augment the filter using (2.6) and go to Step 6. Otherwise, go to Step 4.3. Step 4.3. Choose 𝛼_𝑘,𝑙+1∈[𝜌₁𝛼_𝑘,𝑙,𝜌₂𝛼_𝑘,𝑙]. Let 𝑙=𝑙+1 and go to Step 4.1.
Step 5. Set 𝑥_𝑘+1=𝑥_𝑘+𝛼_𝑘,𝑙𝑠_𝑘, 𝑆^1𝑘+1=𝑆^𝑘1 and 𝑆^2𝑘+1=𝑆^𝑘2. Go to Step 7.
Step 6. Compute 𝑆^1𝑘+1 and 𝑆^2𝑘+1 by (1.3). If (𝜃_𝑘+1(𝑥_𝑘+1),𝑚_𝑘+1(𝑥_𝑘+1))∈ℱ_𝑘+1, set 𝑆^1𝑘+1=𝑆^𝑘1 and 𝑆^2𝑘+1=𝑆^𝑘2.
Step 7. Compute 𝑔_𝑘+1, 𝐵_𝑘+1, 𝐴_𝑘+1 and 𝑚(𝑘+1)=min{𝑚(𝑘)+1,𝑀}. Let 𝑘=𝑘+1 and go to Step 2.
Step 8 (restoration stage). Find 𝑥^𝑟𝑘=𝑥_𝑘+𝛼^𝑟𝑘𝑠^𝑟𝑘 such that 𝑥^𝑟𝑘 is acceptable to 𝑥_𝑘 and (𝜃_𝑘(𝑥^𝑟𝑘),𝑚_𝑘(𝑥^𝑟𝑘))∉ℱ_𝑘. Set 𝑥_𝑘+1=𝑥^𝑟𝑘 and augment the filter by (2.6). Let 𝑘=𝑘+1, 𝑚(𝑘)=1 and go to Step 2.

In a restoration algorithm, the infeasibility is reduced and it is, therefore, desired to decrease the value of 𝜃_𝑘(𝑥). The direct way is to utilize the Newton method or the similar ways to attack 𝜃_𝑘(𝑥+𝑠)=0. We now give the restoration algorithm.

Restoration Algorithm
 Step R1. Let 𝑥^0𝑘=𝑥_𝑘, 𝐻₀=𝐸_𝑛, Δ^0𝑘=Δ⁰, 𝑔_𝜃=∇𝜃_𝑘(𝑥), 𝑗=0, 𝜂₁=0.25, 𝜂₂=0.75.  Step R2. If 𝑥^𝑗𝑘 is acceptable to 𝑥_𝑘 and (𝜃_𝑘(𝑥^𝑟𝑘),𝑚_𝑘(𝑥^𝑟𝑘))∉ℱ_𝑘, then let 𝑥^𝑟𝑘=𝑥^𝑗𝑘 and stop.  Step R3. Compute min𝑔^𝑇𝜃𝑠+12𝑠^𝑇𝐻_𝑗𝑠s.t.‖𝑠‖≤Δ^𝑗𝑘(2.8) to get 𝑠^𝑗𝑘. Let 𝑟^𝑗𝑘=(𝜃_𝑘(𝑥^𝑗𝑘)−𝜃_𝑘(𝑥^𝑗𝑘+𝑠^𝑗𝑘))/(−𝑔^𝑇𝜃𝑠^𝑗𝑘−(1/2)𝑠^𝑗𝑘𝑇𝐻_𝑗𝑠^𝑗𝑘).  Step R4. If 𝑟^𝑗𝑘≤𝜂₁, set Δ^𝑘𝑗+1=(1/2)Δ^𝑗𝑘; If 𝑟^𝑗𝑘≥𝜂₂, set Δ^𝑘𝑗+1=2Δ^𝑗𝑘; otherwise, Δ^𝑘𝑗+1=Δ^𝑗𝑘. Let 𝑥^𝑘𝑗+1=𝑥^𝑗𝑘+𝑠^𝑗𝑘, 𝐻_𝑗 be updated to 𝐻_𝑗+1, 𝑗=𝑗+1 and go to Step R2.

The above restoration algorithm is an SQP method for 𝜃_𝑘(𝑥+𝑠)=0. Of course, there are other restoration algorithms, such as the Newton method, interior point restoration algorithm, SLP restoration algorithm, and so on.

3. Global Convergence of Algorithm

In this section, we present a proof of global convergence of Algorithm 2.1. We first state the following assumptions in technical terms.

Assumptions. (A1) All points 𝑥^𝑘 that are sampled by algorithm lie in a nonempty closed and bounded set 𝑋.
(A2) The functions 𝑐_𝑖(𝑥), 𝑗=1,2,…,𝑚 are all twice continuously differentiable on an open set containing 𝑋.
(A3) There exist two constants 𝑏≥𝑎>0 such that the matrices sequence {𝐵_𝑘} satisfies 𝑎‖𝑠‖²≤𝑠^𝑇𝐵_𝑘𝑠≤𝑏‖𝑠‖² for all 𝑘 and 𝑠∈𝑅^𝑛.
(A4) (𝐴^𝑘𝑠₂)^𝑇 has full column rank and ‖𝑠_𝑘‖≤𝛾_𝑠 for all 𝑘 with a positive constant 𝛾_𝑠.

In the remainder of this section, we will not consider the case where Algorithm 2.1 terminates successfully in Step 2, since in this situation the global convergence is trivial.

Lemma 3.2. Under Assumption A1, there exists the solution to (2.1) with exact (or inexact) line search which satisfies the following descent conditions: ||𝑚||𝜃_𝑘𝑥_𝑘+𝛼𝑠_𝑘−(1−2𝛼)𝜃_𝑘𝑥_𝑘||≤𝜏₁𝛼²‖‖𝑠_𝑘‖‖²,(3.1)_𝑘𝑥_𝑘+𝛼𝑠_𝑘−𝑚_𝑘𝑥_k−𝛼𝑔^𝑇𝑘𝑠_𝑘||≤𝜏₂𝛼²‖‖𝑠_𝑘‖‖²,(3.2) where 𝛼∈(0,1), 𝜏₁ and 𝜏₂ are all positive constants independent of 𝑘.

Proof. By virtue of the Taylor expansion of 𝑐^2𝑖(𝑥_𝑘+𝛼𝑠_𝑘) with 𝑖∈𝑆₂, we obtain |||𝑐_2𝑖𝑥_𝑘+𝛼𝑠_𝑘−𝑐^2𝑖𝑥_𝑘−2𝛼𝑐_𝑖𝑥_𝑘∇𝑐_𝑖𝑥_𝑘^𝑇𝑠_𝑘=|||𝑐_2𝑖|||=|||1𝑥_𝑘+𝛼𝑠_𝑘−𝑐^2𝑖𝑥_𝑘−2𝑐_𝑖𝑥_𝑘∇𝑐_𝑖𝑥_𝑘^𝑇𝛼𝑠_𝑘|||2𝛼𝑠^𝑘^𝑇2𝑐^𝑖𝑥_𝑘=|||𝛼₂+𝜁𝛼𝑠_𝑘∇𝑐^2𝑖𝑥_𝑘+𝜁𝛼𝑠_𝑘+2∇𝑐_𝑖𝑥_𝑘+𝜁𝛼𝑠_𝑘∇𝑐_𝑖𝑥_𝑘+𝜁𝛼𝑠_𝑘^𝑇𝛼𝑠^𝑘|||𝑠^𝑇𝑘𝑐^𝑖𝑥_𝑘+𝜁𝛼𝑠_𝑘∇c^2𝑖𝑥_𝑘+𝜁𝛼𝑠_𝑘+∇𝑐_𝑖𝑥_𝑘+𝜁𝛼𝑠_𝑘∇𝑐_𝑖𝑥_𝑘+𝜁𝛼𝑠_𝑘^𝑇𝑠^𝑘≤1|||𝑚𝜏¹𝛼²‖‖𝑠_𝑘‖‖²,(3.3) where the last inequality can be done by Assumption A1 and 𝜁∈[0,1]. Furthermore, from (2.1) we immediately obtain 𝑐_𝑖(𝑥_𝑘)+∇𝑐_𝑖(𝑥_𝑘)^𝑇𝑠_𝑘=0, that is, −2𝛼𝑐^2𝑖(𝑥_𝑘)−2𝛼𝑐_𝑖(𝑥_𝑘)∇𝑐_𝑖(𝑥_𝑘)^𝑇𝑠_𝑘=0. With |𝑆₂|≤𝑚, thereby, ||𝜃_𝑘=𝑥_𝑘|||||+𝛼_𝑖∈𝑆2𝑠_𝑘−(1−2𝛼)𝜃_𝑘𝑥_𝑘||≤𝑐^2𝑖𝑥_𝑘+𝛼𝑠_𝑘−(1−2𝛼)𝑐^2𝑖𝑥_𝑘_𝑖∈𝑆2|||||=_𝑖∈𝑆2||𝑐^2𝑖𝑥_𝑘+𝛼𝑠_𝑘−(1−2𝛼)𝑐^2𝑖𝑥_𝑘|||||𝑐_2𝑖𝑥_𝑘+𝛼𝑠_𝑘−𝑐^2𝑖𝑥_𝑘−2𝛼𝑐_𝑖𝑥_𝑘∇𝑐_𝑖𝑥_𝑘^𝑇𝑠_𝑘≤𝑚⋅1|||𝑚𝜏¹𝛼²‖‖𝑠_𝑘‖‖²≤𝜏₁𝛼²‖‖𝑠_𝑘‖‖²,(3.4) then the first inequality consequently holds.
According to the Taylor expansion of ∑_𝑖∈𝑆1(𝑐^2𝑖(𝑥_𝑘+𝛼𝑠_𝑘)) (i.e., 𝑚_𝑘(𝑥_𝑘+𝛼𝑠_𝑘)), we then have |||||_𝑖∈𝑆1𝑐^2𝑖𝑥_𝑘+𝛼𝑠_𝑘−_𝑖∈𝑆1𝑐^2𝑖𝑥_𝑘−𝛼𝑔^𝑇𝑘𝑠_𝑘|||||=|||||12𝛼²𝑠_𝑘^𝑇∇²_𝑖_∈𝑆1𝑐^2𝑖𝑥_𝑘+𝜚𝛼𝑠_𝑘𝑠_𝑘|||||≤𝜏₂𝛼²‖‖𝑠_𝑘‖‖²,(3.5) where the last inequality follows from Assumption A1 and 𝜚∈[0,1]. That is to say, ||𝑚_𝑘𝑥_𝑘+𝛼𝑠_𝑘−𝑚_𝑘𝑥_𝑘−𝛼𝑔^𝑇𝑘𝑠_𝑘||≤𝜏₂𝛼²‖‖𝑠_𝑘‖‖²,(3.6) which is just (3.2).

Lemma 3.3. Let {𝑥_𝑘𝑖} be a subsequence of iterates for which (2.3) holds and has the same 𝑆₁ and 𝑆₂. Then there exists some 𝛼∈(0,1] such that 𝑚_𝑘𝑖𝑥_𝑘𝑖+𝛼𝑠_𝑘𝑖≤𝑚_𝑘𝑖𝑥_𝑘𝑖+𝛼𝜏₃𝑔^𝑇𝑘_𝑖𝑠_𝑘𝑖.(3.7)

Proof. Because {𝑥_𝑘𝑖} have the same 𝑆₁ and 𝑆₂, it follows that 𝑚_𝑘𝑖(𝑥) are fixed and by (2.3) 𝑑_𝑘𝑖 is a decent direction. Hence there exists some 𝛼∈(0,1] satisfying (3.7).

Theorem 3.4. Suppose that {𝑥_𝑘} is an infinite sequence generated by Algorithm 2.1 and |𝒜|<∞, one has lim_𝑘→∞‖‖‖𝑐_{𝑘𝑆𝑘2}‖‖‖+‖‖𝑠_𝑘‖‖=0,(3.8) namely, every limit point is the 𝜖 solution to (1.1) or a local infeasible point. If the gradients of 𝑐_𝑖(𝑥_𝑘) are linear independent for all 𝑘 and 𝑖=1,2,…,𝑚, then the solution to SNE is obtained.

Proof. From |𝒜|<∞, we know the filter updates in a finite number, then there exists 𝐾∈ℕ, for 𝑘>𝐾 the filter does not update. As h-type iteration and restoration algorithm all need the updating of the filter, so for 𝑘>𝐾 our algorithm only follows the f-type iterations. We then have that for all 𝑘>𝐾 both conditions (2.3) and (2.4) are satisfied for 𝑥_𝑘+1=𝑥_𝑘+𝛼_𝑘𝑠_𝑘 and 𝑚_𝑘(𝑥)=𝑚_𝐾(𝑥).
Then by (2.4) we get 𝑚_𝑘(𝑥_𝑘+1)≤max{𝑚_𝑘(𝑥_𝑘),∑^{𝑚(𝑘)−1𝑟=0}𝜆_𝑘𝑟𝑚_𝑘−𝑟(𝑥_𝑘−𝑟)}+𝜏₃𝛼_𝑘𝑔^𝑇𝑘𝑠_𝑘. We first show that for all 𝑘≥𝐾+1, it holds 𝑚_𝑘𝑥_𝑘<𝑚𝑥_𝐾+𝜆𝜏_{3𝑘−2𝑟}_=𝐾𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟+𝜏₃𝛼_𝑘−1𝑔^𝑇𝑘−1𝑠_𝑘−1<𝑚𝑥_𝐾+𝜆𝜏_{3𝑘−1𝑟}_=𝐾𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟,(3.9) where 𝑚(𝑥_𝐾)=max{𝑚_𝐾(𝑥_𝐾),∑^{𝑚(𝑘)−1𝑟=0}𝜆_𝐾𝑟𝑚_𝐾−𝑟(𝑥_𝐾−𝑟)}. We prove (3.9) by induction.
If 𝑘=𝐾+1, we have 𝑚_𝐾+1(𝑥_𝐾+1)<𝑚(𝑥_𝐾)+𝜏₃𝛼_𝑘𝑔^𝑇𝑘𝑠_𝑘<𝑚(𝑥_𝐾)+𝜆𝜏₃𝛼_𝑘𝑔^𝑇𝑘𝑠_𝑘. Suppose that the claim is true for 𝐾+1,𝐾+2,…,𝑘, then we consider two cases.
Case 1. If max{𝑚_𝑘(𝑥_𝑘),∑^{𝑚(𝑘)−1𝑟=0}𝜆_𝑘𝑟𝑚_𝑘−𝑟(𝑥_𝑘−𝑟)}=𝑚_𝑘(𝑥_𝑘), it is clear that 𝑚_𝑘+1𝑥_𝑘+1<𝑚_𝑘𝑥_𝑘+𝜏₃𝛼_𝑘𝑔^𝑇𝑘𝑠_𝑘<𝑚𝑥_𝐾+𝜆𝜏_{3𝑘−1𝑟}_=𝐾𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟+𝜏₃𝛼_𝑘𝑔^𝑇𝑘𝑠_𝑘≤𝑚𝑥_𝐾+𝜆𝜏_3𝑘𝑟_=𝐾𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟.(3.10)
Case 2. If max{𝑚_𝑘(𝑥_𝑘),∑^{𝑚(𝑘)−1𝑟=0}𝜆_𝑘𝑟𝑚_𝑘−𝑟(𝑥_𝑘−𝑟)}=∑^{𝑚(𝑘)−1𝑟=0}𝜆_𝑘𝑟𝑚_𝑘−𝑟(𝑥_𝑘−𝑟), let 𝑢=𝑚(𝑘)−1. By the fact that  ∑^𝑢𝑡=0𝜆_𝑘𝑡=1, 𝜆≤𝜆_𝑘𝑡<1, we have 𝑚_𝑘+1𝑥_𝑘+1<^𝑢_𝑡=0𝜆_𝑘𝑡𝑚_𝑘−𝑡𝑥_𝑘−𝑡+𝜏₃𝛼_𝑘𝑔^𝑇𝑘𝑠_𝑘<^𝑢_𝑡=0𝜆_𝑘𝑡𝑚𝑥_𝐾+𝜆𝜏_{3𝑘−𝑡−2𝑟}_=𝐾𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟+𝜏₃𝛼_{𝑘−𝑡−1}𝑔^{𝑇𝑘−𝑡−1}𝑠_{𝑘−𝑡−1}+𝜏³𝛼_𝑘𝑔^𝑇𝑘𝑠_𝑘=𝜆_𝑘0𝑚𝑥_𝐾+𝜆𝜏_{3𝑘−𝑢−2𝑟}_=𝐾𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟+𝜆𝜏_{3𝑘−2𝑟=}_{𝑘−𝑢−1}𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟+𝜏₃𝛼_𝑘−1𝑔^𝑇𝑘−1𝑠_𝑘−1+𝜆_𝑘1𝑚𝑥_𝐾+𝜆𝜏_{3𝑘−𝑢−2𝑟}_=𝐾𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟+𝜆𝜏_{3𝑘−3𝑟=}_{𝑘−𝑢−1}𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟+𝜏₃𝛼_𝑘−2𝑔^𝑇𝑘−2𝑠_𝑘−2+⋯+𝜆_𝑘𝑢𝑚𝑥_𝐾+𝜆𝜏_{3𝑘−𝑢−2𝑟}_=𝐾𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟+𝜏₃𝛼_{𝑘−𝑢−1}𝑔^{𝑇𝑘−𝑢−1}𝑠_{𝑘−𝑢−1}+𝜏³𝛼_𝑘𝑔^𝑇𝑘𝑠_𝑘<𝑚𝑥_𝐾+𝜆𝜏_{3𝑘−𝑢−2𝑟}_=𝐾𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟+𝜆𝜏_{3𝑘−1𝑟=}_{𝑘−𝑢−1}𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟+𝜏₃𝛼_𝑘𝑔^𝑇𝑘𝑠_𝑘=𝑚𝑥_𝐾+𝜆𝜏_{3𝑘−1𝑟}_=𝐾𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟+𝜏₃𝛼_𝑘𝑔^𝑇𝑘𝑠_𝑘<𝑚𝑥_𝐾+𝜆𝜏_3𝑘𝑟_=𝐾𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟.(3.11) Moreover, since 𝑚_𝑘(𝑥_𝑘) is bounded below as 𝑘→∞, we get ∑^{𝑘𝑟=𝐾}𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟<∞, that is, lim_𝑘→∞𝛼_𝑟𝑔^𝑇𝑟𝑠_𝑟=0. By Lemma 3.3, there exists a 𝛼∈(0,1] such that 𝛼_𝑘≥𝛼. Then together with 𝑔^𝑇𝑘𝑠_𝑘<−𝜉𝑠^𝑇𝑘𝐵_𝑘𝑠_𝑘 and Assumption A1, we have lim_𝑘→∞‖𝑠_𝑘‖=0. From −𝛼_𝑘,𝑙𝑔^𝑇𝑘𝑠_𝑘>[𝜃_𝑘(𝑥_𝑘)]^𝑠𝜃 it is easy to obtain that lim_𝑘→∞𝜃_𝑘(𝑥_𝑘)=0. This completes the proof.

Lemma 3.5. Under Assumptions A1 and A2, if 𝑔^𝑇𝑘𝑠_𝑘≤−𝜀₀ for a positive constant 𝜀₀ independent of 𝑘(∈𝑎𝑠𝑢𝑏𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒) and for all 𝛼∈(0,1] and 𝛼≥𝛼^min𝑘,𝑙 with (𝜃_𝑘(𝑥_𝑘),𝑚_𝑘(𝑥_𝑘))∉ℱ_𝑘, then there exists 𝛾₁, 𝛾₂>0 so that (𝜃_𝑘(𝑥_𝑘+𝛼𝑠_𝑘),𝑚_𝑘(𝑥_𝑘+𝛼𝑠_𝑘))∉ℱ_𝑘 for all 𝑘(∈𝑎𝑠𝑢𝑏𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒) and 𝛼≤min{𝛾₁,𝛾₂𝜃_𝑘(𝑥_𝑘)}.

Proof. Choose 𝛾₁=𝜀₀/𝜏₂𝛾^2𝑠, then 𝛼≤𝛾₁ implies that −𝛼𝜀₀+𝜏₂𝛼²𝛾^2𝑠≤0. So we note from (3.2) that 𝑚_𝑘𝑥_𝑘+𝛼𝑠_𝑘≤𝑚_𝑘𝑥_𝑘+𝛼𝑔^𝑇𝑘𝑠_𝑘+𝜏₂𝛼²‖‖𝑠_𝑘‖‖²≤𝑚_𝑘𝑥_𝑘−𝛼𝜀₀+𝜏₂𝛼²𝛾^2𝑠≤𝑚_𝑘𝑥_𝑘.(3.12)
Let 𝛾₂=2/𝜏₁𝛾^2𝑠, then 𝛼≤𝛾₂𝜃_𝑘(𝑥_𝑘) implies that −2𝛼𝜃_𝑘(𝑥_𝑘)+𝜏₁𝛼²𝛾^2𝑠≤0. So from (3.1), we obtain 𝜃_𝑘𝑥_𝑘+𝛼𝑠_𝑘≤𝜃_𝑘𝑥_𝑘−2𝛼𝜃_𝑘𝑥_𝑘+𝜏₂𝛼²‖‖𝑠_𝑘‖‖²≤𝜃_𝑘𝑥_𝑘−2𝛼𝜃_𝑘𝑥_𝑘+𝜏₁𝛼²𝛾^2𝑠≤𝜃_𝑘𝑥_𝑘.(3.13)
We further point a fact according to the definition of filter. If (𝜃,𝑚)∉ℱ_𝑘 and 𝜃≤𝜃, 𝑚≤𝑚, we obtain (𝜃,𝑚)∉ℱ_𝑘. Thus from (𝜃_𝑘(𝑥_𝑘),𝑚_𝑘(𝑥_𝑘))∉ℱ_𝑘, 𝑚_𝑘(𝑥_𝑘+𝛼𝑠_𝑘)≤𝑚_𝑘(𝑥_𝑘), and 𝜃_𝑘(𝑥_𝑘+𝛼𝑠_𝑘)≤𝜃_𝑘(𝑥_𝑘), we have (𝜃_𝑘(𝑥_𝑘+𝛼𝑠_𝑘),𝑚_𝑘(𝑥_𝑘+𝛼𝑠_𝑘))∉ℱ_𝑘.

Lemma 3.6. If 𝑔^𝑇𝑘𝑠_𝑘≤−𝜀₀ for a positive constant 𝜀₀ independent of 𝑘(∈𝑎𝑠𝑢𝑏𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒), then there exists a constant 𝛼>0, for all 𝑘(∈𝑎𝑠𝑢𝑏𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒) and 𝛼≤𝛼 such that 𝑚_𝑘𝑥_𝑘+𝛼𝑠_𝑘−max𝑚_𝑘𝑥_𝑘,^{𝑚(𝑘)−1}_𝑟=0𝜆_𝑘𝑟𝑚_𝑘−𝑟𝑥_𝑘−𝑟≤𝜏₃𝛼𝑔^𝑇𝑘𝑠_𝑘.(3.14)

Proof. Let 𝛼=(1−𝜏₃)𝜀₀/𝜏₂𝛾^2𝑠. In view of (3.2), ‖𝑠_𝑘‖≤𝛾_𝑠 and 𝛼≤𝛼, we know 𝑚_𝑘𝑥_𝑘+𝛼𝑠_𝑘−max𝑚_𝑘𝑥_𝑘,^{𝑚(𝑘)−1}_𝑟=0𝜆_𝑘𝑟𝑚_𝑘−𝑟𝑥_𝑘−𝑟−𝛼𝑔^𝑇𝑘𝑠_𝑘≤𝑚_𝑘𝑥_𝑘+𝛼𝑠_𝑘−𝑚_𝑘𝑥_𝑘−𝛼𝑔^𝑇𝑘𝑠_𝑘≤𝜏₂𝛼²‖‖𝑠_𝑘‖‖²≤𝜏₂𝛼𝛼𝛾^2𝑠=1−𝜏₃𝛼𝜀₀≤−1−𝜏₃𝛼𝑔^𝑇𝑘𝑠_𝑘,(3.15) which shows that the assertion of the lemma follows.

Theorem 3.7. Suppose that {𝑥_𝑘} is an infinite sequence generated by Algorithm 2.1 and |𝒜|=∞. Then there exists at least one accumulation which is the 𝜖 solution to (1.1) or a local infeasible point. Namely, one has lim_𝑘→∞inf‖‖‖𝑐^𝑘𝑆_𝑘2‖‖‖+‖‖𝑠_𝑘‖‖=0.(3.16) If the gradients of 𝑐_𝑖(𝑥_𝑘) are linear independent for all 𝑘 and 𝑖=1,2,…,𝑚, then the solution to (1.1) is obtained.

Proof. We prove that lim_{𝑘→∞,𝑘∈𝒜}𝜃_𝑘(𝑥_𝑘)=0 first.
Suppose by contradiction that there exits an infinite subsequence {𝑘_𝑖} of 𝒜 such that 𝜃_𝑘𝑖(𝑥_𝑘𝑖)≥𝜀 for some 𝜀>0. At each iteration 𝑘_𝑖, (𝜃_𝑘𝑖(𝑥_𝑘𝑖),𝑚_𝑘𝑖(𝑥_𝑘𝑖)) is added to the filter which means that no other (𝜃,𝑚) can be added to the filter at a later stage within the area: 𝜃𝑥_𝑘𝑖−𝛾_𝜃𝜀,𝜃𝑥_𝑘𝑖×𝑚𝑥_𝑘𝑖−𝛾_𝑚𝜀,𝑚𝑥_𝑘𝑖,(3.17) and the area of the each of these squares is at least 𝛾_𝜃𝛾_𝑚𝜀².
By Assumption A1 we have ∑^𝑛𝑖=1𝑐^2𝑖(𝑥_𝑘)≤𝑀^max. Since 0≤𝑚_𝑘(𝑥_𝑘)≤𝑚_𝑘(𝑥_𝑘)+𝜃_𝑘(𝑥_𝑘)=∑^𝑛𝑖=1𝑐^2𝑖(𝑥_𝑘) and 0≤𝜃_𝑘(𝑥_𝑘)≤𝑚_𝑘(𝑥_𝑘)+𝜃_𝑘(𝑥_𝑘)=∑^𝑛𝑖=1𝑐^2𝑖(𝑥_𝑘), then (𝜃,𝑚) associated with the filter are restricted to ℬ=0,𝑀^max×0,𝑀^max.(3.18)
Thereby ℬ is completely covered by at most a finite number of such areas in contraction to the infinite subsequence {𝑘_𝑖} satisfying 𝜃_𝑘𝑖(𝑥_𝑘𝑖)≥𝜀. Therefore, lim_{𝑘→∞,𝑘∈𝒜}𝜃_𝑘(𝑥_𝑘)=0.
By Assumption A1 and |𝒜|=∞, there exits an accumulation point 𝑥, that is, lim_𝑖→∞𝑥_𝑘𝑖=𝑥, 𝑘_𝑖∈𝒜. It follows from lim_{𝑘→∞,𝑘∈𝒜}𝜃_𝑘(𝑥_𝑘)=0, that lim_𝑖→∞𝜃_𝑘𝑖𝑥_𝑘𝑖=0,(3.19) which implies lim_𝑖→∞‖𝑐^𝑘𝑖_{𝑆𝑘𝑖2}‖=0. If lim_𝑖→∞‖𝑠_𝑘𝑖‖=0, then (3.16) is true. Otherwise, there exists a subsequence {𝑥_𝑘𝑖𝑗} of {𝑥_𝑘𝑖} and a constant 𝜀₁>0 so that for all 𝑘_𝑖𝑗, ‖‖𝑠_𝑘𝑖𝑗‖‖≥𝜀₁.(3.20) The choice of {𝑥_𝑘𝑖𝑗} implies 𝑘_𝑖𝑗∈𝒜forall𝑘_𝑖𝑗.(3.21) According to ‖𝑠_𝑘𝑖𝑗‖≥𝜀₁, Assumption A1 as well as 𝜉∈(0,1), we have 𝑔^𝑇𝑘_𝑖𝑗𝑠_𝑘𝑖𝑗+𝜉𝑠^𝑇𝑘_𝑖𝑗𝐵_𝑘𝑖𝑗𝑠_𝑘𝑖𝑗=(𝜉−1)𝑠^𝑇𝑘_𝑖𝑗𝐵_𝑘𝑖𝑗𝑠_𝑘𝑖𝑗−𝜆^+𝑘_𝑖𝑗^𝑇𝑐^𝑘𝑖𝑗_{𝑆𝑘𝑖𝑗2}≤(𝜉−1)𝑎‖‖𝑠_𝑘𝑖𝑗‖‖²+𝑐₁≤(𝜉−1)𝑎𝜀₂₁‖‖‖𝑐_{𝑘𝑖𝑗𝑆𝑘𝑖𝑗2}‖‖‖+𝑐₁‖‖‖𝑐_{𝑘𝑖𝑗𝑆𝑘𝑖𝑗2}‖‖‖.(3.22) Since 𝜉−1<0 and ‖𝑐^𝑘𝑖𝑗_{𝑆𝑘𝑖𝑗2}‖→0 as 𝑗→∞, we obtain 𝑔^𝑇𝑘_𝑖𝑗𝑠_𝑘𝑖𝑗≤−𝜉𝑠^𝑇𝑘_𝑖𝑗𝐵_𝑘𝑖𝑗𝑠_𝑘𝑖𝑗,(3.23) for sufficiently large 𝑗. Similarly, we have 𝛼𝑔^𝑇𝑘_𝑖𝑗𝑠_𝑘𝑖𝑗+𝜃_𝑘𝑥_𝑘^𝑠𝜃≤−𝛼𝑠^𝑇𝑘_𝑖𝑗𝐵_𝑘𝑖𝑗𝑠_𝑘𝑖𝑗+𝑐₁‖‖‖𝑐^𝑘𝑖𝑗_{𝑘𝑖𝑗2}‖‖‖+𝜃_𝑘𝑥_𝑘^𝑠𝜃≤−𝛼𝑎𝜀²¹+𝑐₁‖‖‖𝑐^𝑘𝑖𝑗_{𝑘𝑖𝑗2}‖‖‖+𝜃_𝑘𝑥_𝑘^𝑠𝜃,(3.24) and thus −𝛼𝑔^𝑇𝑘_𝑖𝑗𝑠_𝑘𝑖𝑗≥𝜃_𝑘𝑥_𝑘^𝑠𝜃,(3.25) for sufficiently large 𝑗. This means the condition (2.3) is satisfied for sufficiently large 𝑗. Therefore, the reason for accepting 𝑥_𝑘+1 must been that 𝑥_𝑘+1 satisfies nonmonotone Armijo condition (2.4). In fact let 𝜀₀=𝜉𝑎𝜀²¹, then 𝑔^𝑇𝑘_𝑖𝑗𝑠_𝑘𝑖𝑗≤−𝜉𝑠^𝑇𝑘_𝑖𝑗𝐵_𝑘𝑖𝑗𝑠_𝑘𝑖𝑗≤−𝜉𝑎𝜀²¹=−𝜀₀; by Lemma 3.6 we obtain nonmonotone Armijo condition (2.4) is satisfied. Consequently, the filter is not augmented in iteration 𝑘_𝑖𝑗 which is a contraction to (3.21). The whole proof is completed.

4. Numerical Experiments

In this section, we test our algorithm on some typical test problems. In the whole process, the program is coded in MATLAB and we assume the error tolerance 𝜖 in this paper is always 1.0𝑒−5. The selected parameter values are 𝛾_𝜃=0.1, 𝛾_𝑚=0.1, 𝑠_𝜃=0.9, 𝜌₁=0.25, 𝜌₂=0.75, and 𝑀=3. In the following tables, the notations NIT, NOF, and NOG mean the number of iterates, number of functions, and number of gradients, respectively.

Example 4.1. Find a solution of the nonlinear equations system as follows: ⎛⎜⎜⎝𝑥+3𝑦²(𝑥−1.0)𝑦⎞⎟⎟⎠=⎛⎜⎜⎝00⎞⎟⎟⎠.(4.1)
The only solution of Example 4.1 is (𝑥^∗,𝑦^∗)=(0,0). Define the line Γ={(1,𝑦)∶𝑦∈ℝ}. If the starting point (𝑥₀,𝑦₀)∈Γ, the Newton method [24] are confined to Γ. We choose two starting points which belong to Γ in the experiments and then the (𝑥^∗,𝑦^∗) is obtained. Table 1 shows the results.

Table 1 

Numerical results of Example 4.1.

Example 4.2. Consider the system of nonlinear equations: 𝑐𝑐₁(𝑥)=𝑥³¹−𝑥³²+𝑥³³−1,₂𝑐(𝑥)=𝑥²¹+𝑥²²−𝑥²³−1,₃(𝑥)=𝑥₁+𝑥₂+𝑥₃−3.(4.2)
The solution to Example 4.2 is 𝑥^∗=(1,1,1)^𝑇. The numerical results of Example 4.2 are given in Table 2.

Table 2 

Numerical results of Example 4.2.

Example 4.3. Find a solution of the nonlinear equations system: ⎛⎜⎜⎝10𝑥𝑥(𝑥+0.1)+2𝑦²⎞⎟⎟⎠=⎛⎜⎜⎝00⎞⎟⎟⎠.(4.3)
The unique solution is (𝑥^∗,𝑦^∗)=(0,0). It has been proved in [2] that, under initial point (𝑥₀,𝑦₀)=(3,1), the iterates converge to the point 𝑧=(1.8016,0.0000), which is not a stationary point. Utilizing our algorithm, a sequence of points converging to (𝑥^∗,𝑦^∗) is obtained. The detailed numerical results for Example 4.3 are listed in Table 3.

Table 3 

Numerical results of Example 4.3.

Example 4.4. Consider the following system of nonlinear equations: 𝑐𝑐₁(𝑥)=𝑥²¹+𝑥₁𝑥₂+2𝑥²²−𝑥₁−𝑥₂−2,₂(𝑥)=2𝑥²¹+𝑥₁𝑥₂+3𝑥²²−𝑥₁−𝑥₂−4.(4.4)
There are three solutions of above example, (1,1)^𝑇, (−1,1)^𝑇, and (1,−1)^𝑇. The numerical results of Example 4.4 are given in Table 4.

Table 4 

Numerical results of Example 4.4.

Example 4.5. Consider the system of nonlinear equations: 𝑐_𝑖(𝑥)=−(𝑁+1)+2𝑥_𝑖+^𝑁_𝑗=_{1,𝑗≠𝑖}𝑐𝑥_𝑗,𝑖=1,2,…,𝑁−1,(4.5)_𝑁(𝑥)=−1+^𝑁_𝑗=1𝑥_𝑗,(4.6) with the initial point 𝑥^𝑖(0)=0.5, 𝑖=1,2,…,𝑁. The solution to Example 4.5 is 𝑥^∗=(1,1,…,1)^𝑇. The numerical results of Example 4.5 are given in Table 5.

Table 5 

Numerical results of Example 4.5.

Refer to these above problems, running the Algorithm 2.1 with different starting points yields the results in the corresponding tables, which, summarized, show that our proposed algorithm is practical and effective. From the computation efficiency, we should point out our algorithm is competitive with the method in [22]. The results in Table 5 in fact show that our method also succeeds well to solve the cases when more equations are active.

Constrained optimization approaches attacking the system of nonlinear equations are exceedingly interesting and are further developed by using the nonmonotone line search filter strategy in this paper. Moreover, the local property of the algorithm is a further topic of interest.

Acknowledgment

The research is supported by the National Natural Science Foundation of China (no. 11126060) and Science & Technology Program of Shanghai Maritime University (no. 20120060).