﻿Modified Piyavskii’s Global One-Dimensional Optimization of a Differentiable Function

Applied Mathematics
Vol.3 No.10A(2012), Article ID:24105,15 pages DOI:10.4236/am.2012.330187

Modified Piyavskii’s Global One-Dimensional Optimization of a Differentiable Function

Rachid Ellaia, Mohamed Zeriab Es-Sadek*, Hasnaa Kasbioui

Laboratory of Study and Research in Applied Mathematics (LRMA), Mohammadia School of Engineering, Mohammed V University (Adgal), Av Ibn Sina, Rabat, Morocco

Received June 21, 2012; revised August 28, 2012; accepted September 4, 2012

Keywords: Global Optimization; Piyavskii’s Algorithm

ABSTRACT

Piyavskii’s algorithm maximizes a univariate function satisfying a Lipschitz condition. We propose a modified Piyavskii’s sequential algorithm which maximizes a univariate differentiable function f by iteratively constructing an upper bounding piece-wise concave function Φ of f and evaluating f at a point where Φ reaches its maximum. We compare the numbers of iterations needed by the modified Piyavskii’s algorithm (nC) to obtain a bounding piece-wise concave function Φ whose maximum is within ε of the globally optimal value fopt with that required by the reference sequential algorithm (nref). The main result is that nC ≤ 2nref + 1 and this bound is sharp. We also show that the number of iterations needed by modified Piyavskii’s algorithm to obtain a globally ε-optimal value together with a corresponding point (nB) satisfies nB ≤ 4nref + 1 Lower and upper bounds for nref are obtained as functions of f(x), ε, M1 and M0 where M0 is a constant defined by and M1 ≥ M0 is an evaluation of M0.

1. Introduction

We consider the following general global optimization problems for a function defined on a compact set

(P) Find

such that

(P1) Find such that fopt = f(xopt) ≥ f* − ε, where ε is a small positive constant.

Many recent papers and books propose several approaches for the numerical resolution of the problem (P) and give a classification of the problems and their methods of resolution. For instance, the book of Horst and Tuy [1] provides a general discussion concerning deterministic algorithms. Piyavskii [2,3] proposes a deterministic sequential method which solves (P) by iteratively constructing an upper bounding function F of f and evaluating f at a point where F reaches its maximum, Shubert [4], Basso [5,6], Schoen [7], Shen and Zhu [8] and Horst and Tuy [9] give a special aspect of its application by examples involving functions satisfying a Lipschitz condition and propose other formulations of the Piyavskii’s algorithm, Sergeyev [10] use a smooth auxiliary functions for an upper bounding function, Jones et al. [11] consider a global optimization without the Lipschitz constant. Multidimensional extensions are proposed by Danilin and Piyavskii [12], Mayne and Polak [13], Mladineo [14] and Meewella and Mayne [15], Evtushenko [16], Galperin [17] and Hansen, Jaumard and Lu [18,19] propose other algorithms for the problem (P) or its multidimensional case extension. Hansen and Jaumard [20] summarize and discuss the algorithms proposed in the literature and present them in a simplified and uniform way in a high-level computer language. Another aspect of the application of Piyavskii’s algorithm has been developed by Brent [21], the requirement is that the function is defined on a compact interval, with a bounded second derivative. Jacobsen and Torabi [22] assume that the differentiable function is the sum of a convex and concave functions. Recently, a multivariate extension is proposed by Breiman and Cutler [23] which use the Taylor development of f to build an upper bounding function of f. Baritompa and Cutler [24] give a variation and an acceleration of the Breiman and Cutler’s method. In this paper, we suggest a modified Piyavskii’s sequential algorithm which maximizes a univariate differentiable function f. The theoretical study of the number of iterations of Piyavskii’s algorithm was initiated by Danilin [25]. Danilin’s result was improved by Hansen, Jaumard and Lu [26]. In the same way, we develop a reference algorithm in order to study the relationship between nB and nref where nB denotes the number of iterations used by the modified Piyavskii’s algorithm to obtain an ε-optimal value, and nref denotes the number of iterations used by a reference algorithm to find an upper bounding function, whose maximum is within ε of the maximum of f. Our main result is that nB ≤ 4nref + 1. The last purpose of this paper is to derive bounds for nref. Lower and upper bounds for nref are obtained as functions of f(x), ε, M1 and M0 where M0 is a constant defined by

and M1 ≥ M0 is an evaluation of M0.

2. Upper Bounding Piecewise Concave Function

Theorem 1. Let

and suppose that there is a positive constant M such that

(1)

Let

and

Then

Proof. Let

we have

Then Ψ is a concave function, the minimum of Ψ over [a, b] occurs at a or b. Then

for all.

This proves the theorem.

Remark 1. Let

Then the maximum Φ* of Φ(x) is:

If the function f is not differentiable, we generalize the above result by the following one:

Theorem 2. Let f be a continuous on [a, b] and suppose that there is a positive constant M such that for every h > 0

(2)

Then

Proof. without loss of generality, we assume that f(a) = f(b) = 0 and M = 0.

Let us consider the function f(x) − Φ(x) instead of the function f(x). It suffice to prove that

Suppose by contradiction that f* > 0, and let

The function f is continuous, f(u) = f* > 0, thus a < u < b, consequently for h > 0 small and we have f(u − h) ≤ f(u) and f(u + h) < f(u).

Since M = 0, this contradicts the hypothesis (3). Hence f* ≤ 0.

Remark 2. Since the above algorithm is based entirely on the result of Theorem 1, it is clear that the same algorithm will be adopted for functions satisfying condition (2).

If f is twice continuously differentiable, the conditions (1) and (2) are equivalents.

3. Modified Piyavskii’s Algorithm

We call subproblem the set of information

describing the bounding concave function

The algorithm that we propose memorizes all the subproblems and, in particular, stores the maximum of each bounding concave function Φi over [ai, bi] in a structure called Heap. A Heap is a data structure that allows an access to the maximum of the k values that it stores in constant time and is updated in O(Log2k) time. In the following discussion, we denote the Heap by H.

Modified Piyavskii’s algorithm can be described as follows:

3.1. Step 1 (Initialization)

• k ← 0

• [a0, b0] ←[a, b]

• Compute f(a0) and f(b0)

• Let be an upper bounding function of f over [a0, b0]

• Compute

• if Stop, is an ε-optimal value

3.2. Current Step (k = 1, 2, ···)

While H is no empty do

• k ← k + 1

• Extract from H the subproblem

with the largest upper bound

3.2.1 Update of the Best Current Solution

for j = l, k do

3.2.1.1. Lower Bound

If then

Else

End if

3.2.1.2. Upper Bound

• Build an upper bounding function Φj on [aj, bj]

• Compute

End for

• Delete from H all subproblems

with

3.2.2. Optimality Test

• Let be the maximum of all

If then Stop, is an ε-optimal value

End While

Let [an, bn] denote the smallest subinterval of [a, b] containing xn, whose end points are evaluation points. Then the partial upper bounding function Φn1(x) spanning [an, bn] is deleted and replaced by two partial upper bounding functions, the first one spanning [an, xn] denoted by Φnl(x) and the second one spanning [xn, bn] denoted by Φnr(x) (Figure 1).

Proposition 1. For n ≥ 2 the upper bounding function Φn(x) is easily deduced from as follows:

and

Proof. In the case where

Figure 1. Upper bounding piece-wise concave function.

is in [an, xn], then from remark 1, the maximum of is given as follows

(3)

The maximum of spanning [an, bn] is reached at the point

We deduce that

then

By substitution in expression (3), we have the result. We show in the same way that the maximum of defined in [xn, bn] is given by:

Remark 3. Modified Piyavskii’s algorithm obtains an upper bound on f within ε of the maximum value of f(x) when the gap is not larger than ε, where

But f* and

are unknown. So modified Piyavskii’s algorithm stops only after a solution of value within ε of the upper bound has been found, i.e., when the error is not larger than ε. The number of iterations needed to satisfy each of these conditions is studied below.

4. Convergence of the Algorithm

We now study the error and the gap of Φn in modified Piyavskii’s algorithm as a function of the number n of iterations. The following proposition shows how they decrease when the number of iterations increases and provides also a relationship between them.

Proposition 2. Let and denote respectively the error and the gap after n iterations of modified Piyavskii's algorithm. Then, for 1) 2) 3)

Proof. First, notice that and are non increasing sequences and is nondecreasing. After n iterations of the modified Piyavskii's algorithm, the upper bounding piece-wise concave function contains partial upper bounding concave functions, which we call first generation function. We call the partial upper bounding concave functions obtained by splitting these second generation function. must belong to the second generation for the first time after no more than another iterations of the modified Piyavskii’s algorithm, at iteration m (m ≤ 2n − 1).

Let k be the iteration after which the maximum of has been obtained, by splitting the maximum of Then, in the case where

is in and from proposition 1, we get:

where

and is one of the endpoints of the interval over which the function is defined.

1) We have

where

If therefore, we have

If we will consider two cases:

• case 1

If then and if then and the algorithm stops.

• case 2

then we have

hence

(proposition 1).

Moreover

We deduce from these two inequalities that

Therefore

2) We have

We follow the same steps to prove the second point of 1) and show that

3)

• case 1

If or then and the algorithm stops.

• case 2 First, we have

and

therefore Hence

This proves the proposition.

Proposition 3. Modified Piyavskii’s algorithm (with) is convergent, i.e. either it terminates in a finite number of iterations or

Proof. This is an immediate consequence of the definition of and i) of proposition 2.

5. Reference Algorithm

Since the number of necessary function evaluations for solving problem (P) measures the efficiency of a method, Danilin [25] and Hansen, Jaumard and Lu [26] suggested studying the minimum number of evaluation points required to obtain a guaranteed solution of problem (P) where the functions involved are Lipschitz-Continuous on

In the same way, we propose to study the minimum number of evaluation points required to obtain a guaranteed solution of problem (P) where the functions involved satisfy the condition (1).

For a given tolerance and constant M this can be done by constructing a reference bounding piece-wise concave function such that

with a smallest possible number of evaluations points in

Such a reference bounding piece-wise concave function is constructed with which is assumed to be known. It is of course, designed not to solve problem (P) from the outset, but rather to give a reference number of necessary evaluation points in order to study the efficiency of other algorithms. It is easy to see that a reference bounding function can be obtained in the following way:

Description of the Reference Algorithm

1). Initialization

root of the equation

2). Reference bounding function

While do

• compute

root of the equation

End While.

3). Minimum number of evaluations points

A reference bounding function is then defined as follows (see Figure 2)

6. Estimation of the Efficiency of the Modified Piyavskii’s Algorithm

Proposition 4. For a given a tolerance let be a reference bounding function with evaluations points and Let be a bounding function obtained by the modified Piyavskii’s algorithm with evaluation points where nC is the smallest number of iterations such that

Then we have

(4)

and the bound is sharp for all values of

Proof. The proof is by induction on We initialize the induction at

• Case 1 (see Figure 3)

We have to show that We have

Figure 2. Reference bounding function.

Figure 3. Case y1 = b.

and

But

hence

Therefore

hence the result holds.

• case 2 y1 < b. (see Figure 4)

If nc = 2, (4) holds. In this case, we have

where

Assuming that and ( the proof is similar for). We denote by and the partial bounding functions defined respectively on and and by a maximum value of We show that

and that Indeed, we have

and

thus

If

is in then

Figure 4. Case y1 < b.

As

(see the proof of proposition 2), we have

If then

If ≤ x3 then and (4) holds.

Assume now that (4) holds for and consider nref = k + 1. The relation (4) holds for nc = 2, therefore we may assume that

If we let

the modified Piyavskii’s algorithm has to be implemented on two subintervals [x1, x3] and [x3, x2] There are two cases which need to be discussed separately:

• case 1 (see Figure 5)

There is a subinterval containing all the nref evaluation points of the best bounding function Without loss of generality, we may assume that contains all nref evaluation points of. Let and be the solution of By symmetry, we have

Let be the abscissa of the maximum of the last partial bounding function of Due to our assumption, we have

since otherwise there is an another evaluation point of f on the interval

Let be the abscissa of the maximum of the partial bounding function of preceding In this case, we have

But implies that nref = 1, which is not the case according to the assumption (nref ≥ 2). Therefore, this case will never exist.

Figure 5. The subinterval contains all the nref evaluation points.

• case 2 None of the subintervals contains all the nref evaluation points of Φref(x). We can then apply induction, reasoning on the subintervals [x1, x3] and [x3, x2] obtained by modified Piyavskii’s algorithm. If they contain and evaluation points respectively, [x1, x2] contains evaluation points as x3 belongs to both subintervals. Then (4) follows by induction on nref.

Consider the function f(x) = x defined on [0, 1], the constant M is equal to 4. For ε = 0 we have nC = nref + 1 (see Figure 6), and the bound (4) is sharp.

As noticed in remark 3, the modified Piyavskii’s algorithm does not necessarily stop as soon as a cover is found as described in proposition 4, but only when the error does not exceed ε. We now study the number of evaluation points necessary for this to happen.

Proposition 5. For a given tolerance ε, let nref be the number of evaluation points in a reference bounding function of f and nB the number of iterations after which the modified Piyavskii’s algorithm stops. We then have

(5)

Proof. Let Due to proposition 4, after iterations, we have Due to 3) of proposition 2, after iterations, we have

which shows that the termination of modified Piyavskii’s algorithm is satisfied. This proves (5).

7. Bounds on the Number of Evaluation Points

In the previous section, we compared the number of evaluation functions in the modified Piyavskii’s algorithm and in the reference algorithm. We now evaluate the number nB of evaluation functions of the modified Piyavskii’s algorithm itself. To achieve this, we derive bounds on nref, from which bounds on nB are readily obtained using the relation nref ≤ nB ≤ 4nref + 1.

Proposition 6. Let f be a C2 function defined on the interval [a, b]. Set

and let

Then the number nref of evaluation points in a reference cover Φref using the constant M1 ≥ M2 is bounded by the following inequalities:

(6)

Figure 6. None of the subintervals contains all the nref evaluation points.

(7)

Proof. We suppose that the reference cover Φref has nref − 1 partial upper bounding functions defined by the evaluation points

We consider an arbitrary partial upper bounding function and the corresponding subinterval [yi, yi + 1] fori i = 1, ···, nref − 1. To simplify, we move the origin to the point (yi, f(yi)). Let d = yi + 1 − yi , z = f(yi + 1) − f(yi) and h = f* + ε. We assume z ≥ 0 (See Figure 7).

Let be the partial upper bounding function defined on [yi, yi + 1] and the point where

is reached, then

We deduce that

Let g1 be the function defined on [yi, yi + 1] by:

We have

Thus we have

Consider the function

Figure 7. The reference cover Φref has nref − 1 partial upper bounding functions.

Let x1 and x2 be the roots of the equation F1(x) = 0; they are given by

Then F1 is written in the following way

Let

We have

Since θ ≥ 0 and g1 reaches its minimum at the point θ, then

and

Since

and

then

and

This proves (6).

Now let us consider the function G defined and continuous on [yi, yi + 1] such that

Two cases are considered case 1: If

(8)

then, the function G is given by

Hence

We consider

its derivative is

Thus F2 is expressed as follows

hence

Since

and

then

and

Moreover, the inequality

implies (8), this proves the first inequality of (7).

Case 2: Suppose that:

(9)

Let X1 et X2 be the roots of the equation

X1 and X2 are given by

In this case, G is given by

We have

Moreover, M0 = M1 implies the condition (9) and we have

Therefore

and

Hence

Table 1. Computational results with .

Table 2. The number of function evaluations of modified Piyavskii's algorithm for different values of ε and M.

8. Computational Experiences

In this section, we report the results of computational experiences performed on fourteen test functions (see Tables 1 and 2). Most of these functions are test functions drawn from the Lipschitz optimization literature (see Hansen and Jaumard [20]).

The performance of the Modified Piyavskii’s algorithm is measured in terms of NC, the number of function evaluations. The number of function evaluations NC is compared with (nref), the number of function evaluations required by the reference sequential algorithm. We observe that NC is on the average only 1.35 larger than (nref). More precisely, we have the following estimation

.

For the first three test functions, we observe that the influence of the parameter M is not very important, since the number of function evaluations increase appreciably for a same precision ε.

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NOTES

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