A Newton-Type Algorithm for Solving Problems of Search Theory

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.


Introduction
We consider the problem where  = {x ∈ R  + | e  x = }, a, b ∈ R  ++ ,  > 0, and e ∈ R  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  with probability a  , and −b  is proportional to the difficulty of searching inside the box.If the searcher spends x  time units looking inside box , then he/she will find the object with probability 1 − exp(−b  x  ).The problem described by (1) represents the optimum search strategy if the available time is limited to  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][5][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  = 10000.
The rest of this paper is organized as follows.The Newtontype 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  denotes transpose, R  (resp., R) denotes the space of -dimensional real column vectors (resp., real numbers), and R  + and R ++ denote the nonnegative and positive orthants of R  and R, respectively.Let Φ  denote the derivative of the function Φ.We define  := {1, 2, . . ., }.For any vector x ∈ R  , we denote by diag{x  :  ∈ } the diagonal matrix whose th diagonal element is x  and vec{x  :  ∈ } the vector x.The symbol ‖ ⋅ ‖ stands for the 2-norm.We denote by  the solution set of Problem (1).For any ,  > 0,  = () (resp.,  = ()) means / is uniformly bounded (resp., tends to zero) as  → 0.

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.
where  is the generalized Jacobian of  in the sense of Clarke [13].
(4) Definition 3. Let  ̸ = 0 be a parameter.Function   (x) is called a smoothing function of a semismooth function (x) if it is continuously differentiable everywhere and there is a constant c > 0 independent of  such that        (x) −  (x)      ≤ c, ∀x.
The Fischer-Burmeister function [7] is one of the wellknown NCP functions: Clearly, the Fischer-Burmeister function defined by ( 6) is not smooth, but it is strongly semismooth [14].Let  : R 3 → R be the perturbed Fischer-Burmeister function defined by It is obvious that for any  > 0,  is differentiable everywhere and for each  ≥ 0, we have In particular, (0, , ) = (, ) for all (, ) ∈ R 2 .Namely,  defined by ( 7) is a smoothing function of  defined by ( 6).
According to Kuhn-Tucker theorem, the problem described by (1) can be transformed to e  x = , where  ∈ R. Define According to the Fischer-Burmeister function defined by ( 6), we formulate (9) as the following semismooth equation: Based on the perturbed Fischer-Burmeister function defined by (7), we obtain the following smooth equation: where Clearly, if y * = (0,  * , 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  in the following lemma, which will be used in the sequel.
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 = (, , x) ∈ R +2 and  ∈ (0, 1).Step 0. Choose ,  ∈ (0, 1) and  0 > 0.
The following theorem proves that Algorithm 5 is well defined.Theorem 5. Algorithm 5 is well defined.If it finitely terminates at th iteration then x  is an optimal solution to the problem described by (1).Otherwise, it generates an infinite sequence {y  = (  ,   , x  )} with   > 0 and   ≥    0 .
Proof.If   > 0 then Lemma 4 shows that the matrix   (y  ) is nonsingular.Hence, Step 2 is well defined at the th iteration.For any 0 <  ≤ 1, define  () :=  (y  + Δy  ) −  (y  ) −   (y  ) Δy  . ( It follows from (21) that Hence, for any 0 <  ≤ 1, we have From Lemma 4,  is continuously differentiable around y  .Thus, (23) implies that On the other hand, (20) yields This inequality shows that Step 3 is well defined at the th iteration.In addition, by (24), Steps 3 and 4 in Algorithm 5, we have holds since 0 <   ≤ 1 and   > 0. Consequently, from  0 > 0 and the above statements, we obtain that Algorithm 5 is well defined.
It is obvious that if Algorithm 5 finitely terminates at th iteration then (y  ) = 0, which implies that   = 0 and (  , x  ) satisfies (9).Hence, x  is an optimal solution to the problem described by (1).

Convergence Analysis
In this section we establish the convergence property for Algorithm 5. We show that the sequence {y  = (  ,   , x  )} 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  } is superlinearly convergent.
We next analyze the rate of convergence for Algorithm 5.By Theorem 6, we know that Algorithm 5 generates a bounded iteration sequence {y  } and it has at least one accumulation point.The following lemma will be used in the sequel.
Lemma 7. Suppose that y * = ( * ,  * , x * ) is an accumulation point of the iteration sequence {y  } generated by Algorithm 5. Let  be a matrix in ( * , x * ).Then the matrix  is nonsingular.

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  = 0.75,  = 0.25,  0 = 0.001, and  = min{1/‖(y 0 )‖, 0.99}.In Step 1, we used ‖(y  )‖ ≤ 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  was randomly generated in the interval [10,20], b  was randomly generated in the interval [1,2] for each  ∈ , and  was randomly generated in the interval [50, 51].In the second randomly generated example, a  with ∑  =1 a  = 1 and b  was randomly generated in the interval [0, 1] for each  ∈ , and  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 ‖(y  )‖ at the final iteration.
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, , among  independent activities, where a  (1−exp(−b  x  )) is the return from the th activity, that is, effort   , to yield the maximum total return.Note that here a  is the potential attainable and   is the rate of attaining the potential from effort x  .When no effort is devoted to the th activity, the value of x  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  and b  as in Table 3.
For this problem Algorithm 5 obtained the maximum total return 4.8952 × 10 6 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  is the prior probability with ∑ 5 =1 a  = 1 of an object of search being in the th region and 1 − exp(−b  x  ) is the probability of finding an object known to be in the th region with x  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  = 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.

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.

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Definition 2. A locally Lipschitz function  : R  → R  is called semismooth at x ∈ R  if  is directionally differentiable at x and for all  ∈ (x + d

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