A Newton-Type Algorithm for Solving Problems of Search Theory

Abstract

In the survey of the continuous nonlinear resource allocation problem, Patriksson pointed out that Newton-type algorithms have not been proposed for solving the problem of search theory in the theoretical perspective. In this paper, we propose a Newton-type algorithm to solve the problem. We prove that the proposed algorithm has global and superlinear convergence. Some numerical results indicate that the proposed algorithm is promising.

1. Introduction

We consider the problem max^𝑛_{𝐱∈𝑋𝑖=1}𝐚_𝑖1−exp−𝐛_𝑖𝐱_𝑖, (1) where 𝑋={𝐱∈ℝ^𝑛₊∣𝐞^𝑇𝐱=𝑐}, 𝐚,𝐛∈ℝ^𝑛₊₊, 𝑐>0, and 𝐞∈ℝ^𝑛 is the vector of ones. The problem described by (1) is called the theory of search by Koopman [1] and Patriksson [2]. It has the following interpretation: an object is inside box 𝑖 with probability 𝐚_𝑖, and −𝐛_𝑖 is proportional to the difficulty of searching inside the box. If the searcher spends 𝐱_𝑖 time units looking inside box 𝑖, then he/she will find the object with probability 1−exp(−𝐛_𝑖𝐱_𝑖). The problem described by (1) represents the optimum search strategy if the available time is limited to 𝑐 time units. Such problems in the form of (1) arise, for example, in searching for a lost object, in distribution of destructive effort such as a weapons allocation problem [3], in drilling for oil, and so forth [2]. Patriksson [2] surveyed the history and applications as well as algorithms of Problem (1); see [2, Sections 2.1.4, 2.1.5, and 3.1.2]. Patriksson pointed out that Newton-type algorithms have not been theoretically analyzed for the problem described by (1) in the references listed in [2].

Recently, related problems and methods were considered in many articles, for example, [4–6]. For example, a projected pegging algorithm was proposed in [5] for solving convex quadratic minimization. However, the question proposed by Patriksson [2] was not answered in the literature. In this paper, we design a Newton-type algorithm to solve the problem described by (1). We show that the proposed algorithm has global and superlinear convergence.

According to the Fischer-Burmeister function [7], the problem described by (1) can be transformed to a semismooth equation. Based on the framework of the algorithms in [8, 9], a smoothing Newton-type algorithm is proposed to solve the semismooth equation. It is shown that the proposed algorithm can generate a bounded iteration sequence. Moreover, the iteration sequence superlinearly converges to an accumulation point which is a solution to the problem described by (1). Numerical results indicate that the proposed algorithm has good performance even for 𝑛=10000.

The rest of this paper is organized as follows. The Newton-type algorithm is proposed in Section 2. The global and superlinear convergence is established in Section 3. Section 4 reports some numerical results. Finally, Section 5 gives some concluding remarks.

The following notation will be used throughout this paper. All vectors are column ones, the subscript 𝑇 denotes transpose, ℝ^𝑛 (resp., ℝ) denotes the space of 𝑛-dimensional real column vectors (resp., real numbers), and ℝ^𝑛₊ and ℝ₊₊ denote the nonnegative and positive orthants of ℝ^𝑛 and ℝ, respectively. Let Φ^′ denote the derivative of the function Φ. We define 𝑁∶={1,2,…,𝑛}. For any vector 𝐱∈ℝ^𝑛, we denote by diag{𝐱_𝑖∶𝑖∈𝑁} the diagonal matrix whose 𝑖th diagonal element is 𝐱_𝑖 and vec{𝐱_𝑖∶𝑖∈𝑁} the vector 𝐱. The symbol ‖⋅‖ stands for the 2-norm. We denote by 𝑆 the solution set of Problem (1). For any 𝛼,  𝛽>0,  𝛼=𝑂(𝛽) (resp., 𝛼=𝑜(𝛽)) means 𝛼/𝛽 is uniformly bounded (resp., tends to zero) as 𝛽→0.

2. Algorithm Description

In this section, we formulate the problem described by (1) as a semismooth equation and develop a smoothing Newton-type algorithm to solve the semismooth equation.

We first briefly recall the concepts of NCP, semismooth and smoothing functions [10–12].

Definition 1. A function 𝜙∶ℝ²→ℝ is called an NCP function if 𝜙(𝑢,𝑣)=0⟺𝑢≥0,𝑣≥0,𝑢𝑣=0. (2)

Definition 2. A locally Lipschitz function 𝐹∶ℝ^𝑛→ℝ^𝑚 is called semismooth at 𝐱∈ℝ^𝑛 if 𝐹 is directionally differentiable at 𝐱 and for all 𝑄∈𝜕𝐹(𝐱+𝐝) and 𝐝→𝟎, 𝐹(𝐱+𝐝)−𝐹(𝐱)−𝑄𝐝=𝑜(‖𝐝‖), (3) where 𝜕𝐹 is the generalized Jacobian of 𝐹 in the sense of Clarke [13].
𝐹 is called strongly semismooth at 𝐱∈ℝ^𝑛 if 𝐹 is semismooth at 𝐱 and for all 𝑄∈𝜕𝐹(𝐱+𝐝) and 𝐝→𝟎, 𝐹(𝐱+𝐝)−𝐹(𝐱)−𝑄𝐝=𝑂‖𝐝‖². (4)

Definition 3. Let 𝜇≠0 be a parameter. Function 𝐹_𝜇(𝐱) is called a smoothing function of a semismooth function 𝐹(𝐱) if it is continuously differentiable everywhere and there is a constant 𝑐̃>0 independent of 𝜇 such that 𝐹_𝜇(𝐱)−𝐹(𝐱)≤𝑐𝜇̃,∀𝐱. (5)

The Fischer-Burmeister function [7] is one of the well-known NCP functions: 𝜙(𝑢,𝑣)=𝑢+𝑣−√𝑢²+𝑣². (6) Clearly, the Fischer-Burmeister function defined by (6) is not smooth, but it is strongly semismooth [14]. Let 𝜑∶ℝ³→ℝ be the perturbed Fischer-Burmeister function defined by 𝜑𝜇,𝑢,𝑣=𝑢+𝑣−𝑢²+𝑣²+𝜇². (7) It is obvious that for any 𝜇>0,  𝜑 is differentiable everywhere and for each 𝜇≥0, we have 𝜑𝜇,𝑢,𝑣−𝜙(𝑢,𝑣)≤𝜇,∀(𝑢,𝑣)∈ℝ². (8) In particular, 𝜑(0,𝑢,𝑣)=𝜙(𝑢,𝑣) for all (𝑢,𝑣)∈ℝ². Namely, 𝜑 defined by (7) is a smoothing function of 𝜙 defined by (6).

According to Kuhn-Tucker theorem, the problem described by (1) can be transformed to 𝐞^𝑇𝐱𝐱=𝑐,_𝑖≥0,𝑠−𝐚_𝑖𝐛_𝑖exp−𝐛_𝑖𝐱_𝑖𝐱≥0,∀𝑖∈𝑁,_𝑖𝑠−𝐚_𝑖𝐛_𝑖exp−𝐛_𝑖𝐱_𝑖=0,∀𝑖∈𝑁, (9) where 𝑠∈ℝ.

Define Ψ()⋮𝑠,𝐱∶=𝜙𝐱_𝑖,𝑠−𝐚_𝑖𝐛_𝑖exp−𝐛_𝑖𝐱_𝑖⋮_𝑖∈𝑁,𝐻()𝐞𝑠,𝐱∶=^𝑇Ψ().𝐱−𝑐𝑠,𝐱 (10) According to the Fischer-Burmeister function defined by (6), we formulate (9) as the following semismooth equation: 𝐻(𝑠,𝐱)=0. (11) Based on the perturbed Fischer-Burmeister function defined by (7), we obtain the following smooth equation: 𝐺(𝐲)𝜇∶=𝐺𝜇,𝑠,𝐱∶=𝐞^𝑇Φ𝐱−𝑐𝜇,𝑠,𝐱=0, (12) where Φ⋮𝜇,𝑠,𝐱∶=𝜑(𝜇,𝐱_𝑖,𝑠−𝐚_𝑖𝐛_𝑖exp(−𝐛_𝑖𝐱_𝑖⋮))_𝑖∈𝑁. (13) Clearly, if 𝐲^∗=(0,𝑠^∗,𝐱^∗) is a solution to (12) then 𝐱^∗ is an optimal solution to the problem described by (1).

We give some properties of the function 𝐺 in the following lemma, which will be used in the sequel.

Lemma 4. Let 𝐺 be defined by (12). Then 𝐺 is semismooth on ℝ^𝑛+2 and continuously differentiable at any 𝐲=(𝜇,𝑠,𝐱)∈ℝ₊₊×ℝ^𝑛+1 with its Jacobian 𝐺^′(𝐲)=10𝟎00𝐞^𝑇Φ^′_𝜇Φ^′_𝑠Φ^′_𝐱, (14) where Φ^′_𝜇−𝜇∶=vec𝑤_𝑖,Φ∶𝑖∈𝑁_′𝑠∶=vec1−𝑠−𝐚_𝑖𝐛_𝑖exp−𝐛_𝑖𝐱_𝑖𝑤_𝑖,Φ∶𝑖∈𝑁_′𝐱𝐱∶=diag1−^𝑖𝑤_𝑖+𝐚_𝑖𝐛²_𝑖exp−𝐛_𝑖𝐱_𝑖Φ^′_𝑠𝑖∶𝑖∈𝑁, (15) with 𝑤_𝑖∶=(𝐱_𝑖)²+(𝑠−𝐚_𝑖𝐛_𝑖exp(−𝐛_𝑖𝐱_𝑖))²+𝜇², 𝑖∈𝑁. Moreover, the matrix 𝐺^′(𝐲) is nonsingular on ℝ₊₊×ℝ^𝑛+1.

Proof. 𝐺 is semismooth on ℝ^𝑛+2 due to the strong semismoothness of 𝜙(𝑢,𝑣). 𝐺 is continuously differentiable on ℝ₊₊×ℝ^𝑛+1. For any 𝜇>0, (14) can be obtained by a straightforward calculation from (12). Clearly, we have for any 𝜇>0, 𝐱0<1−^𝑖𝑤_𝑖<2,0<Φ^′_𝑠𝑖𝐚<2,_𝑖𝐛²_𝑖exp−𝐛_𝑖𝐱_𝑖>0,∀𝑖∈𝑁, (16) which implies that (Φ^′_𝐱)_𝑖𝑖>0 for all 𝑖∈𝑁. Let 10𝟎00𝐞^𝑇Φ^′_𝜇Φ^′_𝑠Φ^′_𝐱𝑢𝑣𝐳=0. (17) Then, we have 𝑢=0 and ^𝑛_𝑖=1𝐳_𝑖=0,(Φ^′_𝑠)_𝑖𝑣+Φ^′_𝐱𝑖𝑖𝐳_𝑖=0,∀𝑖∈𝑁. (18) The second equality in (18) implies 𝐳_𝑖=−Φ^′_𝑠𝑖Φ^′_𝐱𝑖𝑖𝑣,𝑖∈𝑁. (19) Since (Φ^′_𝑠)_𝑖>0 and (Φ^′_𝐱)_𝑖𝑖>0 for 𝑖∈𝑁, the first equality in (18) yields 𝑣=0 and hence 𝐳=𝟎. Therefore, the matrix 𝐺^′(𝐲) defined by (14) is nonsingular for 𝜇>0.

We now propose a smoothing Newton-type algorithm for solving the smooth equation in (12). It is a modified version of the smoothing Newton method proposed in [8]. The main difference is that we add a different perturbed item in Newton equation, which allows the algorithm to generate a bounded iteration sequence. Let 𝐲=(𝜇,𝑠,𝐱)∈ℝ^𝑛+2 and 𝛾∈(0,1). Define a function 𝜌∶ℝ^𝑛+2→ℝ₊ by 𝜌(𝐲)∶=𝛾‖𝐺(𝐲)‖min{1,‖𝐺(𝐲)‖}. (20)

Algorithm  5

Step  0. Choose 𝛿,𝜎∈(0,1) and 𝜇⁰>0. Let 𝐮∶=(𝜇⁰,0,𝟎)∈ℝ₊₊×ℝ×ℝ^𝑛. Let 𝑠⁰∈ℝ and 𝐱⁰∈ℝ^𝑛 be arbitrary points. Let 𝐲⁰∶=(𝜇⁰,𝑠⁰,𝐱⁰). Choose 𝛾∈(0,1) such that 𝛾‖𝐺(𝐲⁰)‖<1 and 𝛾𝜇⁰<1. Set 𝑘∶=0.

Step  1. If 𝐺(𝐲^𝑘)=0, stop. Otherwise, let 𝜌_𝑘∶=𝜌(𝐲^𝑘).

Step  2. Compute Δ𝐲^𝑘∶=(Δ𝜇^𝑘,Δ𝑠^𝑘,Δ𝐱^𝑘)∈ℝ^𝑛+2 by 𝐺𝐲^𝑘+𝐺^′𝐲^𝑘Δ𝐲^𝑘=𝜌_𝑘𝐮. (21)Step  3. Let 𝑚_𝑘 be the smallest nonnegative integer such that 𝐺𝐲^𝑘+𝛿^𝑚𝑘Δ𝐲^𝑘≤1−𝜎1−𝛾𝜇⁰𝛿^𝑚𝑘𝐺𝐲^𝑘. (22) Let 𝜆_𝑘∶=𝛿^𝑚𝑘.

Step  4. Set 𝐲^𝑘+1∶=𝐲^𝑘+𝜆_𝑘Δ𝐲^𝑘 and 𝑘∶=𝑘+1. Go to Step  1.

The following theorem proves that Algorithm  5 is well defined.

Theorem 5. Algorithm  5 is well defined. If it finitely terminates at 𝑘th iteration then 𝐱^𝑘 is an optimal solution to the problem described by (1). Otherwise, it generates an infinite sequence {𝐲^𝑘=(𝜇^𝑘,𝑠^𝑘,𝐱^𝑘)} with 𝜇^𝑘>0 and 𝜇^𝑘≥𝜌_𝑘𝜇⁰.

Proof. If 𝜇^𝑘>0 then Lemma 4 shows that the matrix 𝐺^′(𝐲^𝑘) is nonsingular. Hence, Step  2 is well defined at the 𝑘th iteration. For any 0<𝛼≤1, define 𝑅(𝛼)∶=𝐺𝐲^𝑘+𝛼Δ𝐲^𝑘−𝐺𝐲^𝑘−𝛼𝐺^′𝐲^𝑘Δ𝐲^𝑘. (23) It follows from (21) that Δ𝜇^𝑘=−𝜇^𝑘+𝜌_𝑘𝜇⁰. (24) Hence, for any 0<𝛼≤1, we have 𝜇^𝑘+𝛼Δ𝜇^𝑘=(1−𝛼)𝜇^𝑘+𝛼𝜌_𝑘𝜇⁰>0. (25) From Lemma 4, 𝐺 is continuously differentiable around 𝐲^𝑘. Thus, (23) implies that ‖𝑅(𝛼)‖=𝑜(𝛼). (26) On the other hand, (20) yields 𝜌_𝑘≤𝛾𝐺𝐲^𝑘,𝜌_𝑘≤𝛾𝐺𝐲^𝑘2. (27) Therefore, for any sufficiently small 𝛼∈(0,1], 𝐺𝐲^𝑘+𝛼Δ𝐲^𝑘≤‖𝑅(𝛼)‖+(1−𝛼)𝐺𝐲^𝑘+𝛼𝜌_𝑘𝜇⁰≤(1−𝛼)𝐺𝐲^𝑘+𝛼𝛾𝜇⁰𝐺𝐲^𝑘(𝛼)=+𝑜1−1−𝛾𝜇⁰𝛼𝐺𝐲^𝑘+𝑜(𝛼), (28) where the first inequality follows from (21) and (23), and the second one follows from (26) and (27). Inequality in (28) implies that there exists a constant 0<𝛼≤1 such that 𝐺𝐲^𝑘+𝛼Δ𝐲^𝑘≤1−𝜎1−𝛾𝜇⁰𝛼𝐺𝐲^𝑘,∀𝛼∈(0,𝛼]. (29) This inequality shows that Step  3 is well defined at the 𝑘th iteration. In addition, by (24), Steps  3 and 4 in Algorithm  5, we have 𝜇^𝑘+1=1−𝜆_𝑘𝜇^𝑘+𝜆_𝑘𝜌_𝑘𝜇⁰>0 (30) holds since 0<𝜆_𝑘≤1 and 𝜇^𝑘>0. Consequently, from 𝜇⁰>0 and the above statements, we obtain that Algorithm  5 is well defined.
It is obvious that if Algorithm  5 finitely terminates at 𝑘th iteration then 𝐺(𝐲^𝑘)=0, which implies that 𝜇^𝑘=0 and (𝑠^𝑘,𝐱^𝑘) satisfies (9). Hence, 𝐱^𝑘 is an optimal solution to the problem described by (1).
Subsequently, we assume that Algorithm  5 does not finitely terminate. Let {𝐲^𝑘=(𝜇^𝑘,𝑠^𝑘,𝐱^𝑘)} be the sequence generated by the algorithm. It follows that 𝜇^𝑘>0. We want to prove that {𝐲^𝑘} satisfies 𝜇^𝑘≥𝜌_𝑘𝜇⁰ through the induction method. Clearly, 𝜌(𝐲⁰)≤𝛾‖𝐺(𝐲⁰)‖<1, which yields 𝐲⁰∈Ω. Assume that 𝜇^𝑘≥𝜌_𝑘𝜇⁰; then (24) yields 𝜇^𝑘+1−𝜌_𝑘+1𝜇⁰=𝜇^𝑘+𝜆_𝑘Δ𝜇^𝑘=1−𝜆_𝑘𝜇^𝑘+𝜆_𝑘𝜌_𝑘𝜇⁰−𝜌_𝑘+1𝜇⁰≥1−𝜆_𝑘𝜌_𝑘𝜇⁰+𝜆_𝑘𝜌_𝑘𝜇⁰−𝜌_𝑘+1𝜇⁰≥𝜇⁰𝜌_𝑘−𝜌_𝑘+1. (31) Clearly, 𝜌_𝑘=𝛾𝐺𝐲^𝑘2,if𝐺𝐲^𝑘𝛾𝐺𝐲<1,_𝑘,otherwise. (32) It follows from (20) and (22) that 𝐺𝐲^𝑘+1≤𝐺𝐲^𝑘,𝜌_𝑘+1≤𝛾𝐺𝐲^𝑘+1,𝜌_𝑘+1≤𝛾𝐺(𝐲^𝑘+1)². (33) Hence, combining (31), (32), and (33), we obtain that 𝜇^𝑘+1≥𝜌_𝑘+1𝜇⁰ which gives the desired result.

3. Convergence Analysis

In this section we establish the convergence property for Algorithm  5. We show that the sequence {𝐲^𝑘=(𝜇^𝑘,𝑠^𝑘,𝐱^𝑘)} generated by Algorithm  5 is bounded and its any accumulation point yields an optimal solution to the problem described by (1). Furthermore, we show that the sequence {𝐲^𝑘} is superlinearly convergent.

Theorem 6. The sequence {𝐲^𝑘=(𝜇^𝑘,𝑠^𝑘,𝐱^𝑘)} generated by Algorithm  5 is bounded. Let 𝐲^∗=(𝜇^∗,𝑠^∗,𝐱^∗) denote an accumulation point of {𝐲^𝑘}. Then 𝜇^∗=0, and 𝐱^∗ is the optimal solution of the problem described by (1).

Proof. By Theorem 5, 𝜇^𝑘+1=1−𝜆_𝑘𝜇^𝑘+𝜆_𝑘𝜌_𝑘𝜇⁰≤1−𝜆_𝑘𝜇^𝑘+𝜆_𝑘𝜇^𝑘=𝜇^𝑘, (34) which implies that {𝜇^𝑘} is monotonically decreasing and hence converges. It follows from (22) that 𝐺𝐲^𝑘+1≤𝐺𝐲^𝑘,∀𝑘. (35) Hence, {‖𝐺(𝐲^𝑘)‖} also converges. Consequently, there exists a constant 𝑀>0 such that ‖𝐺(𝐲^𝑘)‖≤𝑀 for all 𝑘. This implies that {𝜇^𝑘} and {𝐱^𝑘} are bounded, and that for any 𝑘 and 𝑖∈𝑁, 𝐱^𝑘_𝑖+𝑠^𝑘−𝐚_𝑖𝐛_𝑖exp−𝐛_𝑖𝐱^𝑘_𝑖−𝐱^𝑘_𝑖²+𝑠^𝑘−𝐚_𝑖𝐛_𝑖exp−𝐛_𝑖𝐱^𝑘_𝑖²+𝜇^𝑘2≤𝑀. (36) This shows that {𝑠^𝑘} is also bounded. Consequently, {𝐲^𝑘} is bounded. Let 𝐲^∗=(𝜇^∗,𝑠^∗,𝐱^∗) be an accumulation point of {𝐲^𝑘}. We assume, without loss of generality, that {𝐲^𝑘} converges to 𝐲^∗. Then, we have lim_𝑘→∞𝐺𝐲^𝑘=𝐺𝐲^∗,lim_𝑘→∞𝜇^𝑘=𝜇^∗. (37) By (20), lim_𝑘→∞𝜌_𝑘=𝜌_∗∶=𝛾𝐺𝐲^∗min1,𝐺𝐲^∗. (38) Suppose ‖𝐺(𝐲^∗)‖>0 by contradiction. Then 𝜌_∗>0. From Theorem 5 and (22), we have 𝜇^∗≥𝜌_∗𝜇⁰>0,lim_𝑘→∞𝜆_𝑘=0. (39) Hence, from Step  3 of Algorithm  5, we obtain 𝐺𝐲^𝑘+𝜆_𝑘𝛿Δ𝐲^𝑘>1−𝜎1−𝛾𝜇⁰𝜆_𝑘𝛿𝐺𝐲^𝑘. (40) Taking 𝑘→∞ in (40) and then combining with (21), we have −𝐺𝐲^∗2+𝜌_∗𝐺𝐲^∗𝑇𝐮≥−𝜎1−𝛾𝜇⁰𝐺(𝐲^∗)², (41) which yields 𝜌_∗𝜇⁰≥1−𝜎1−𝛾𝜇⁰𝐺𝐲^∗. (42) Since 𝜌_∗≤𝛾‖𝐺(𝐲^∗)‖, (42) implies (1−𝜎)1−𝛾𝜇⁰≤0, (43) which contradicts 𝜎<1 and 𝛾𝜇⁰<1. Therefore, ‖𝐺(𝐲^∗)‖=0 and then 𝜇^∗=0. Consequently, 𝐱^∗ is the optimal solution of the problem described by (1).

We next analyze the rate of convergence for Algorithm  5. By Theorem 6, we know that Algorithm  5 generates a bounded iteration sequence {𝐲^𝑘} and it has at least one accumulation point. The following lemma will be used in the sequel.

Lemma 7. Suppose that 𝐲^∗=(𝜇^∗,𝑠^∗,𝐱^∗) is an accumulation point of the iteration sequence {𝐲^𝑘} generated by Algorithm  5. Let 𝑄 be a matrix in 𝜕𝐻(𝑠^∗,𝐱^∗). Then the matrix 𝑄 is nonsingular.

Proof. Define two index sets: 𝑁₁∶=𝑖∈𝑁∶𝐱^∗_𝑖²+𝑠^∗−𝐚_𝑖𝐛_𝑖exp−𝐛_𝑖𝐱^∗_𝑖²,𝑁≠0₂∶=𝑗∈𝑁∶𝐱^∗_𝑗=0,𝑠^∗=𝐚_𝑗𝐛_𝑗. (44) By a direct computation, we get 𝑄=0𝐞^𝑇𝜶𝐴𝜷𝐵, (45) where 𝐴 is an |𝑁₁|×𝑛 matrix with all elements being zero except the (𝑖,𝑖)-th as 𝐴_𝑖 for 𝑖∈𝑁₁, 𝐵 is an |𝑁₂|×𝑛 matrix with all elements being zero except the (𝑗,𝑗)-th as 𝐵_𝑗 for 𝑗∈𝑁₂, and 𝑠𝜶=vec1−^∗−𝐚_𝑖𝐛_𝑖exp−𝐛_𝑖𝐱^∗_𝑖𝐱^∗_𝑖²+𝑠^∗−𝐚_𝑖𝐛_𝑖exp−𝐛_𝑖𝐱^∗_𝑖²∶𝑖∈𝑁₁,𝐴_𝑖𝐱=1−^∗𝑖𝐱^∗_𝑖²+𝑠^∗−𝐚_𝑖𝐛_𝑖exp−𝐛_𝑖𝐱^∗_𝑖²+𝐚_𝑖𝐛²_𝑖exp−𝐛_𝑖𝐱^∗_𝑖𝜶_𝑖,𝑖∈𝑁₁,𝜷_𝑗,𝐵_𝑗∈𝑢,𝑣+𝐚_𝑗𝐛²_𝑗𝑢∶(𝑢−1)²+(𝑣−1)²≤1,𝑗∈𝑁₂. (46) Obviously, 0<𝜶_𝑖<2,0<𝐴_𝑖<21+𝐚_𝑖𝐛²_𝑖,𝑖∈𝑁₁,𝜷_𝑗≥0,𝐵_𝑗>0,𝑗∈𝑁₂. (47) Let 𝑄(𝑤,𝐳)=0. Then we have _𝑖∈𝑁1𝐳_𝑖+_𝑗∈𝑁2𝐳_𝑗𝜶=0;_𝑖𝑤+𝐴_𝑖𝐳_𝑖=0,𝑖∈𝑁₁;𝜷_𝑗𝑤+𝐵_𝑗𝐳_𝑗=0,𝑗∈𝑁₂, (48) which implies 𝐳_𝑖𝜶=−^𝑖𝐴_𝑖𝑤,𝑖∈𝑁₁;𝐳_𝑗𝜷=−^𝑗𝐵_𝑗𝑤,𝑗∈𝑁₂. (49) Therefore, _𝑖∈𝑁1𝜶_𝑖𝐴_𝑖+_𝑗∈𝑁2𝜷_𝑗𝐵_𝑗𝑤=0, (50) which, together with (47), yields 𝑤=0. Thus, (49) implies 𝐳=𝟎, and hence the matrix 𝑄 is nonsingular.

Theorem 8. Let {𝐲^𝑘} be the iteration sequence generated by Algorithm  5. Then {𝐲^𝑘} superlinearly converges to 𝐲^∗, that is, ‖𝐲^𝑘+1−𝐲^∗‖=𝑜(‖𝐲^𝑘−𝐲^∗‖).

Proof. By Theorem 6, {𝐲^𝑘} is bounded and then let 𝐲^∗=(𝜇^∗,𝑠^∗,𝐱^∗) be its any accumulation point. Hence, 𝐺(𝐲^∗)=0 and all matrices 𝑄∈𝜕𝐺(𝐲^∗) are nonsingular from Lemma 7. By Proposition 3.1 in [12], 𝐺^′𝐲^𝑘−1=𝑂(1) (51) for all 𝐲^𝑘 sufficiently close to 𝐲^∗. From Lemma 4, we know that 𝐺 is semismooth at 𝐲^∗. Hence, for all 𝐲^𝑘 sufficiently close to 𝐲^∗, 𝐺𝐲^𝑘−𝐺^′𝐲^𝑘𝐲^𝑘−𝐲^∗=𝑜𝐲^𝑘−𝐲^∗. (52) Since 𝐺 is locally Lipschitz continuous near 𝐲^∗. Therefore, for all 𝐲^𝑘 sufficiently close to 𝐲^∗, 𝐺(𝐲^𝑘)²=𝑂𝐲^𝑘−𝐲^∗2, (53) which implies that 𝜌_𝑘𝜇₀≤𝛾𝜇⁰𝐺𝐲^𝑘2=𝑂𝐺𝐲^𝑘2=𝑂𝐲^𝑘−𝐲^∗2. (54) This inequality, together with (51) and (52), yields 𝐲^𝑘+Δ𝐲^𝑘−𝐲^∗=𝐲^𝑘+𝐺^′𝐲^𝑘−1−𝐺𝐲^𝑘+𝜌_𝑘𝐮−𝐲^∗≤𝐺^′𝐲^𝑘−1𝐺𝐲^𝑘−𝐺^′𝐲^𝑘𝐲^𝑘−𝐲^∗+𝜌_𝑘𝜇⁰=𝑜𝐲^𝑘−𝐲^∗. (55) Following the proof of Theorem 3.1 in [15], we obtain ‖𝐲^𝑘−𝐲^∗‖=𝑂(‖𝐺(𝐲^𝑘)‖) for all 𝐲^𝑘 sufficiently close to 𝐲^∗. This implies, from (55), that for all 𝐲^𝑘 sufficiently close to 𝐲^∗, 𝐺𝐲^𝑘+Δ𝐲^𝑘=𝑂𝐲^𝑘+Δ𝐲^𝑘−𝐲^∗=𝑜𝐲^𝑘−𝐲^∗=𝑜𝐺𝐲^𝑘. (56) Since lim_𝑘→∞‖𝐺(𝐲^𝑘)‖=0, it follows from (56) that 𝜆_𝑘=1 can satisfy (22) when 𝐲^𝑘 is sufficiently close to 𝐲^∗. Therefore, for all 𝐲^𝑘 sufficiently close to 𝐲^∗, we have 𝐲^𝑘+1=𝐲^𝑘+Δ𝐲^𝑘, (57) which, together with (55), proves ‖𝐲^𝑘+1−𝐲^∗‖=𝑜(‖𝐲^𝑘−𝐲^∗‖). Namely, {𝐲^𝑘} superlinearly converges to 𝐲^∗.

4. Computational Experiments

In this section, we report some numerical results to show the viability of Algorithm  5. First, we compare the numerical performance of Algorithm  5 and the algorithm in [5] on two randomly generated problems. Second, we apply Algorithm  5 to solve two real world examples. Throughout the computational experiments, the parameters used in Algorithm  5 were 𝛿=0.75,  𝜎=0.25,  𝜇⁰=0.001, and 𝛾=min{1/‖𝐺(𝐲⁰)‖,0.99}. In Step  1, we used ‖𝐺(𝐲^𝑘)‖≤10⁻⁸ as the stopping rule. The vector of ones is all the starting points.

Firstly, problems in the form of (1) with 100, 500, 1000, 5000, and 10000 variables were computed. In the first randomly generated example, 𝐚_𝑖 was randomly generated in the interval [10,20], 𝐛_𝑖 was randomly generated in the interval [1,2] for each 𝑖∈𝑁, and 𝑐 was randomly generated in the interval [50,51]. In the second randomly generated example, 𝐚_𝑖 with ∑^𝑛_𝑖=1𝐚_𝑖=1 and 𝐛_𝑖 was randomly generated in the interval [0,1] for each 𝑖∈𝑁, and 𝑐 was also randomly generated in the interval [0,1]. Each problem was run 30 times. The numerical results are summarized in Tables 1 and 2, respectively. Here, Dim denotes the number of variable; AT [5] denotes the average run time in seconds used by the algorithm in [5]. In particular, we list more items for Algorithm  5, Inter denotes the average number of iterations, CPU (sec.) is the average run time, and Gval denotes the average values of ‖𝐺(𝐲^𝑘)‖ at the final iteration.

Table 1 

Problem (1) with a, b,𝑐 randomly generated in different intervals.

Table 2 

Problem (1) with with a, b,𝑐 randomly generated in [0,1].

The numerical results reported in Tables 1 and 2 show that the proposed algorithm solves the test problems much faster than the algorithm in [5] when the size of problem is large.

Secondly, we apply Algorithm  5 to solve two real world problems. The first example is described in [16]. This problem is how to allocate a maximum amount of total effort, 𝑐, among 𝑛 independent activities, where 𝐚_𝑖(1−exp(−𝐛_𝑖𝐱_𝑖)) is the return from the 𝑖th activity, that is, effort 𝑥_𝑖, to yield the maximum total return. Note that here 𝐚_𝑖 is the potential attainable and 𝑏_𝑖 is the rate of attaining the potential from effort 𝐱_𝑖. When no effort is devoted to the 𝑖th activity, the value of 𝐱_𝑖 is zero. This example usually arises in the marketing field; the activities may correspond to different products, or the same product in different marketing areas, in different advertising media, and so forth. In this example we wish to allocate one million dollars among four activities with values of 𝐚_𝑖 and 𝐛_𝑖 as in Table 3.

Table 3 

For this problem Algorithm  5 obtained the maximum total return 4.8952×10⁶ after 25 iterations and elapsing 0.0625 second CPU time. The total effort one million dollars was allocated, 0.5764 million and 0.4236 million to activities A and B, respectively.

The second example is to search an object in 5 regions, where 𝐚_𝑖 is the prior probability with ∑⁵_𝑖=1𝐚_𝑖=1 of an object of search being in the 𝑖th region and 1−exp(−𝐛_𝑖𝐱_𝑖) is the probability of finding an object known to be in the 𝑖th region with 𝐱_𝑖 time units. The data of this example, listed in Table 4, come from Professor R. Jiang of Jiaozhou Bureau of Water Conservancy: the available time is 𝑐=30 time unite, and after 16 iterations and elapsing 0.0781 second CPU time Algorithm  5 computed the result 𝐱=(0,16.6037,0,13.3963,0). This shows that we will spend 17 time units in Region II and 13 time units in Region IV to find water.

Table 4 

5. Conclusions

In this paper we have proposed a Newton-type algorithm to solve the problem of search theory. We have shown that the proposed algorithm has global and superlinear convergence. Some randomly generated problems and two real world problems have been solved by the algorithm. The numerical results indicate that the proposed algorithm is promising.

Acknowledgment

This work was supported by NSFC Grant (11271221).