Open access peer-reviewed chapter

Some Unconstrained Optimization Methods

By Snezana S. Djordjevic

Submitted: September 12th 2018Reviewed: December 20th 2018Published: February 20th 2019

DOI: 10.5772/intechopen.83679

Downloaded: 196

Abstract

Although it is a very old theme, unconstrained optimization is an area which is always actual for many scientists. Today, the results of unconstrained optimization are applied in different branches of science, as well as generally in practice. Here, we present the line search techniques. Further, in this chapter we consider some unconstrained optimization methods. We try to present these methods but also to present some contemporary results in this area.

Keywords

  • unconstrained optimization
  • line search
  • steepest descent method
  • Barzilai-Borwein method
  • Newton method
  • modified Newton method
  • inexact Newton method
  • quasi-Newton method

1. Introduction

Optimization is a very old subject of a great interest; we can search deep into a human history to find important examples of applying optimization in the usual life of a human being, for example, the need of finding the best way to produce food yielded finding the best piece of land for producing, as well as (later on, how the time was going) the best ways of treatment of the chosen land and the chosen seedlings to get the best results.

From the very beginning of manufacturing, the manufacturers were trying to find the ways to get maximum income with minimum expenses.

There are plenty of examples of optimization processes in pharmacology (for determination of the geometry of a molecule), in meteorology, in optimization of a trajectory of a deep-water vehicle, in optimization of power management (optimization of the production of electrical power plants), etc.

Optimization presents an important tool in decision theory and analysis of physical systems.

Optimization theory is a very developed area with its wide application in science, engineering, business management, military, and space technology.

Optimization can be defined as the process of finding the best solution to a problem in a certain sense and under certain conditions.

Along with the passage of time, optimization was evolving. Optimization became an independent area of mathematics in 1940, when Dantzig presented the so-called simplex algorithm for linear programming.

The development of nonlinear programming became great after presentation of conjugate gradient methods and quasi-Newton methods in the 1950s.

Today, there exist many modern optimization methods which are made to solve a variety of optimization problems. Now, they present the necessary tool for solving problems in diverse fields.

At the beginning, it is necessary to define an objective function, which, for example, could be a technical expense, profit or purity of materials, time, potential energy, etc.

The object function depends on certain characteristics of the system, which are known as variables. The goal is to find the values of those variables, for which the object function reaches its best value, which we call an extremum or an optimum.

It can happen that those variables are chosen in such a way that they satisfy certain conditions, i.e., restrictions.

The process of identifying the object function, variables, and restrictions for the given problem is called modeling.

The first and the most important step in an optimization process is the construction of the appropriate model, and this step can be the problem by itself. Namely, in the case that the model is too much simplified, it cannot be a faithful reflection of the practical problem. By the other side, if the constructed model is too complicated, then solving the problem is also too complicated.

After the construction of the appropriate model, it is necessary to apply the appropriate algorithm to solve the problem. It is no need to emphasize that there does not exist a universal algorithm for solving the set problem.

Sometimes, in the applications, the set of input parameters is bounded, i.e., the input parameters have values within the allowed space of input parameters Dx; we can write

xDx.E1

Except (1), the next conditions can also be imposed:

φlx1xn=φ0l,l=1,,m1<n,E2
ψjx1xnψ0j,j=1,,m2.E3

Optimization task is to find the minimum (maximum) of the objective function fx=fx1xn, under the conditions (1), (2), and (3).

If the object function is linear, and the functions φlx1xnl=1,,m1and ψjx1xnj=1,,m2are linear, then it is about the linear programming problem, but if at least one of the mentioned functions is nonlinear, it is about the nonlinear programming problem.

Unconstrained optimization problem can be presented as

minxRnfx,E4

where fRnis a smooth function.

Problem (4) is, in fact, the unconstrained minimization problem. But, it is well known that the unconstrained minimization problem is equivalent to an unconstrained maximization problem, i.e.

minfx=maxfx,E5

as well as

maxfx=minfx.E6

Definition 1.1.1 xis called a global minimizer of fif fxfxfor all xRn.

The ideal situation is finding a global minimizer of f. Because of the fact that our knowledge of the function fis usually only local, the global minimizer can be very difficult to find. We usually do not have the total knowledge about f. In fact, most algorithms are able to find only a local minimizer, i.e., a point that achieves the smallest value of fin its neighborhood.

So, we could be satisfied by finding the local minimizer of the function f. We distinguish weak and strict (or strong) local minimizer.

Formal definitions of local weak and strict minimizer of the function fare the next two definitions, respectively.

Definition 1.1.2 xis called a weak local minimizer of fif there exists a neighborhood Nof x, such that fxfxfor all xN.

Definition 1.1.3 xis called a strict (strong) local minimizer of fif there exists a neighborhood Nof x, such that fx<fxfor all xN.

Considering backward definitions 1.1.2 and 1.1.3, the procedure of finding local minimizer (weak or strict) does not seem such easy; it seems that we should examine all points from the neighborhood of x, and it looks like a very difficult task.

Fortunately, if the object function fsatisfies some special conditions, we can solve this task in a much easier way.

For example, we can assume that the object function fis smooth or, furthermore, twice continuously differentiable. Then, we concentrate to the gradient fxas well as to the Hessian 2fx.

All algorithms for unconstrained minimization require the user to start from a certain point, so-called the starting point, which we usually denote by x0. It is good to choose x0such that it is a reasonable estimation of the solution. But, to find such estimation, a little more knowledge about the considered set of data is needed, and the systematic investigation is needed also. So, it seems much simpler to use one of the algorithms to find x0or to take it arbitrarily.

There exist two important classes of iterative methods—line search methods and trust-region methods—made in the aim to solve the unconstrained optimization problem (4).

In this chapter, at first, we discuss different kinds of line search. Then, we consider some line search optimization methods in details, i.e., we study steepest descent method, Barzilai-Borwein gradient method, Newton method, and quasi-Newton method.

Also, we try to give some of the most recent results in these areas.

2. Line search

Now, let us consider the problem

minxRnfx,E7

where f:RnRis a continuously differentiable function, bounded from below.

There exists a great number of methods made in the aim to solve the problem (7).

The optimization methods based on line search utilize the next iterative scheme:

xk+1=xk+tkdk,E8

where xkis the current iterative point, xk+1is the next iterative point, dkis the search direction, and tkis the step size in the direction dk.

At first, we consider the monotone line search.

Now, we give the iterative scheme of this kind of search.

Algorithm 1.2.1. (Monotone line search).

Assumptions: ϵ>0, x0, k0.

Step 1. If gkϵ, then STOP.

Step 2. Find the descent direction dk.

Step 3. Find the step size tk, such that fxk+tkdk<fxk.

Step 4. Set xk+1=xk+tkdk.

Step 5. Take kk+1and go to Step 1.

Denote

Φt=fxk+tdk.

Trying to solve the minimization problem, we are going to search for the step size t=tk, in the direction dk, such that the next relation holds:

Φtk<Φ0.

That procedure is called the monotone line search.

We can search for the step size tkin such a way that the next relation holds:

fxk+tkdk=mint0fxk+tkdk,E9

i.e.

Φtk=mint0Φt,E10

or we can use the next formula:

tk=mintgxk+tdkTdk=0t0.E11

In this case we are talking about the exact or the optimal line search, where the parameter tk, which is received as the solution of the one-dimensional problem (10), is the optimal step size.

By the other side, instead of using the relation (9), or the relation (11), we can be satisfied by searching for such tk, which is acceptable if the next relation suits us:

fxkfxk+tkdk>δk>0.

Then, we are talking about the inexact or the approximate or the acceptable line search, which is very much utilized in the practice.

There are several reasons to use the inexact instead of the exact line search. One of them is that the exact line search is expensive. Further, in the cases when the iteration is far from the solution, the exact line search is not efficient. Next, in the practice, the convergence rate of many optimization methods (such as Newton or quasi-Newton) does not depend on the exact line search.

First, we are going to mention so-called basic and, by the way, very well-known inexact line searches.

Algorithm 1.2.2. (Backtracking).

Assumptions: xk, the descent direction dk, 0<δ<12, η01.

Step 1. t1.

Step 2. While fxk+tdk>fxk+δtgkTdk, ttη.

Step 3. Set tk=t.

Now, we describe the Armijo rule.

Theorem 1.2.1. [1] Let fC1Rnand let dkbe the descent direction. Then, there exists the nonnegative number mk, such that

fxk+ηmkdkfxk+c1ηmkgkTdk,

where c101and η01.

Next, we describe the Goldstein rule [2].

The step size tkis chosen in such a way that

fxk+tdkfxk+δtgkTdk,fxk+tdk>fxk+1δtgkTdk,

where 0<δ<12.

Now, Wolfe line search rules follow [3], [4].

Standard Wolfe line search conditions are

fxk+tkdkfxkδtkgkTdk,E12
gk+1TdkσgkTdk,E13

where dkis a descent direction and 0<δσ<1.

This efficient strategy means that we should accept a positive step length tk, if conditions (12)(13) are satisfied.

Strong Wolfe line search conditions consist of (12) and the next, stronger version of (13):

gk+1TdkσgkTdk.E14

In the generalized Wolfe line search conditions, the absolute value in (14) is replaced by the inequalities:

σ1gkTdkgk+1Tdkσ2gkTdk,0<δσ1<1,σ20.E15

By the other side, in the approximate Wolfe line search conditions, the inequalities (15) are changed into the next ones:

σgkTdkgk+1Tdk2δ1gkTdk,0<δ<12,δ<σ<1.E16

The next lemma is very important.

Lemma 1.2.1. [5] Let fCRn. Let dkbe a descent direction at the point xk, and assume that the function fis bounded from below along the direction xk+tdkt>0. Then, if 0<δ<σ<1, there exist the intervals inside which the step length satisfies standard Wolfe conditions and strong Wolfe conditions.

By the other side, the introduction of the non-monotone line search is motivated by the existence of the problems where the search direction does not have to be a descent direction. This can happen, for example, in stochastic optimization [6].

Next, some efficient quasi-Newton methods, for example, SR1update, do not produce the descent direction in every iteration [5].

Further, some efficient methods like spectral are not monotone at all.

Some numerical results given in [7, 8, 9, 10, 11] show that non-monotone techniques are better than the monotone ones if the problem is to find the global optimal values of the object function.

Algorithms of the non-monotone line search do not insist on a descent of the object function in every step. But, even these algorithms require the reduction of the object function after a predetermined number of iterations.

The first non-monotone line search technique is presented in [12]. Namely, in [12], the problem is to find the step size which satisfies

fxk+tkdkmax0jmkfxkj+δtkgkTdk,

where m0=0, 0mkminmk1+1M, for k1, δ01, where Mis a nonnegative integer.

This strategy is in fact the generalization of Armijo line search. In the same work, the authors suppose that the search directions satisfy the next conditions for some positive constants b1and b2:

gkTdkb1gk2,dkb2gk.

The next non-monotone line search is described in [11].

Let x0be the starting point, and let

0ηminηmax1,0<δ<σ<1<ρ,μ>0.

Let C0=fx0, Q0=1.

The step size has to satisfy the next conditions:

fxk+tkdkCk+δtkgkTdk,E17
gxk+tkdkσgkTdk.E18

The value ηkis chosen from the interval ηminηmaxand then

Qk+1=ηkQk+1,Ck+1=ηkQkCk+fxk+1Qk+1.

Non-monotone rules which contain the sequence of nonnegative parameters ϵkare used firstly in [13], and they are successfully used in many other algorithms, for example, in [14]. The next property of the parameters ϵkis assumed:

ϵk>0,kϵk=ϵ<,

and the corresponding rule is

fxk+tkdkfxk+c1tkgkTdk+ϵk.

Now, we give the non-monotone line search algorithm, shortly NLSA, presented in [11].

Algorithm 1.2.3. (NLSA).

Assumptions: x0, 0ηminηmax1, 0<δ<σ<1<ρ, μ>0.

Set C0=fx0, Q0=1, k=0.

Step 1. If fxkis sufficiently small, then STOP.

Step 2. Set xk+1=xk+tkdk, where tksatisfies either the (non-monotone) Wolfe conditions (17) and (18) or the (non-monotone) Armijo conditions: tk=t¯kρhk, where t¯k>0is the trial step and hkis the largest integer such that (17) holds and tkμ.

Step 3. Choose ηkηminηmax, and set

Qk+1=ηkQk+1,Ck+1=ηkQkCk+fxk+1/Qk+1.

Step 4. Set kk+1and go to Step 1.

We can notice [11] that Ck+1is a convex combination of fx0,fx1,,fxk. The parameter ηkcontrols the degree of non-monotonicity.

If ηk=0for all k, then this non-monotone line search becomes monotone Wolfe or Armijo line search.

If ηk=1for all k, then Ck=Ak, where

Ak=1k+1i=0kfxi.

Lemma 1.2.2. [11] If fxkTdk0for each k, then for the iterates generated by the non-monotone line search algorithm, we have fkCkAkfor each k. Moreover, if fxkTdk<0and fxare bounded from below, then there exists tksatisfying either Wolfe or Armijo conditions of the line search update.

This study would be very incomplete unless we mention that there are many modifications of the abovementioned line searches. All these modifications are made to improve the previous results.

For example, in [15], the new inexact line search is described by the next way.

Let β01, σ012; let Bkbe a symmetric positive definite matrix which approximates 2fxkand sk=gkTdkdkTBkdk. The step size tkis the largest one in skskβskβ2such that

fxk+tdkfxkσtgkTdk+12tdkTBkdk.

Further, in [16], a new inexact line search rule is presented. This rule is a modified version of the classical Armijo line search rule. We describe it now.

Let g=fxbe a Lipschitz continuous function and Lthe Lipschitz constant. Let Lkbe an approximation of L. Set

βk=gkTdkLkdk2.

Find a step size tkas the largest component in the set βkβkρβkρ2such that the inequality

fxk+tkdkfxk+σtkgkTdk12tkμLkdk2

holds, where σ01, μ0, and ρ01are given constants.

Next, in [17], a new, modified Wolfe line search is given in the next way.

Find tk>0such that

fxk+tkdkfxkminδtkgkTdkγtk2dk2,gxk+tkdkTdkσgkTdk,

where δ01, σδ1, and γ>0.

More recent results on this topic can be found, for example, in [18, 19, 20, 21, 22, 23].

2.1 Steepest descent (SD)

The classical steepest descent method which is designed by Cauchy [24] can be considered as one among the most important procedures for minimization of real-valued function defined on Rn.

Steepest descent is one of the simplest minimization methods for unconstrained optimization. Since it uses the negative gradient as its search direction, it is known also as the gradient method.

It has low computational cost and low matrix storage requirement, because it does not need the computations of the second derivatives to be solved to calculate the search direction [25].

Suppose that fxis continuously differentiable in a certain neighborhood of a point xkand also suppose that gkfxk0.

Using Taylor expansion of the function fnear xkas well as Cauchy-Schwartz inequality, one can easily prove that the greatest fall of fexists if and only if dk=gk, i.e., gkis the steepest descent direction.

The iterative scheme of the SDmethod is

xk+1=xktkgk.E19

The classical steepest descent method uses the exact line search.

Now, we give the algorithm of the steepest descent method which refers to the exact as well as to the inexact line search.

Algorithm 1.2.4. (Steepest descent method, i.e., SDmethod).

Assumptions: 0<ϵ1, x0Rn. Let k=0.

Step 1. If gkε, then STOP, else set dk=gk.

Step 2. Find the step size tk, which is the solution of the problem

mint0fxk+tdk,E20

else find the step size tkby any of the inexact line search methods.

Step 3. Set xk+1=xk+tkdk.

Step 4. Set kk+1and go to Step 1.

The classical and the oldest steepest descent step size tk, which was designed by Cauchy (in the case of the exact line search), is computed as [26]

tk=gkTgkgkTGgk,

where gk=fxkand G=2fxk.

Theorem 1.2.2. [27] (Global convergence theorem of the SDmethod) Let fC1. Then, each accumulation point of the iterative sequence xk, generated by Algorithm 1.2.4, is a stationary point.

Remark 1.2.1. The steepest descent method has at least the linear convergence rate.

More information about the convergence of the SD method can be found in [5, 27].

Although known as the first unconstrained optimization method, this method is still a theme considered by scientists.

Different modifications of this method are made, for example, see [25, 28, 29, 30, 31, 32].

In [28], the authors presented a new search direction from Cauchy’s method in the form of two parameters known as Zubai’ah-Mustafa-Rivaie-Ismail method, shortly, ZMRImethod:

dk=gkgkgk1.E21

So, in [28], a new modification of SDmethod is suggested using a new search direction, dk, given by (21). The numerical results are presented based on the number of iterations and CPU time. It is shown that this new method is efficient when it is compared to the classical SD.

In [25], a new scaled search direction of SDmethod is presented. The inspiration for this new method is the work of Andrei [33], in which the author presents and analyzes a new scaled conjugate gradient algorithm, based on an interpretation of the secant equation and on the inexact Wolfe line search conditions.

The method proposed in [25] is known as Rashidah-Rivaie-Mamat (RRM) method, and it suggests the direction dkgiven by the next relation:

dk=gk,ifk=0,θkgkgkgk1,E22

where θkis a scaling parameter, θk=dk1Tyk1gk12, yk1=gkgk1.

Further, in [25], a comparison among RRM, ZMRI, and SDmethods is made; it is shown that RRMmethod is better than ZMRIand SDmethods.

It is interesting that the exact line search is used in [25].

In [34], the properties of steepest descent method from the literature are reviewed together with advantages and disadvantages of each step size procedure.

Namely, the step size procedures, which are compared in this paper, are:

1. tk=gkTgkgkTHkgk: Step size method by Cauchy [24], computed by exact line search (Cstep size).

2. Given s>0,β,σ01,tk=maxssβ2such that

fxk+tkdkfxk+σtkgkTdkArmijosline searchAstep size.

3. Given β,σ01,t˜0=1, and tk=βt˜ksuch that

fxk+tkdkfxk+σtkgkTdkBacktracking line searchBstep size.

4. tk=sk1Tyk1yk12, (BB1), tk=sk12sk1Tyk1, (BB2), sk1=xkxk1yk1=gkgk1,: Barzilai and Borwein’s formula. The convergence is R-superlinear.

5. tk=tk12gkTgk2(fxk+tkdkfxk+tk1gkTgk,: Elimination line search (ELstep size), which estimates the step size without computation of the Hessian.

The comparison is based on time execution, number of total iteration, total percentage of function, gradient and Hessian evaluation, and the most decreased value of objective function obtained.

From the numerical results, the authors conclude that the Amethod and BB1method are the best methods among others.

Further, in [34], the general conclusions about the steepest descent method are given:

  1. This method is sensitive to the initial point.

  2. This method has a descent property, and it is a logical starting procedure for all gradient based methods.

  3. xkapproaches the minimizer slowly, in fact in a zigzag way.

In [35], in the aim to achieve fast convergence and the monotone property, a new step size for the steepest descent method is suggested.

In [36], for quadratic positive definite problems, an over-relaxation has been considered. Namely, Raydan and Svaiter [36] proved that the poor behavior of the steepest descent method is due to the optimal Cauchy choice of step size and not to the choice of the search direction. These results are extended in [29] to convex, well-conditioned functions. Further, in [29], it is shown that a simple modification of the step length by means of a random variable uniformly distributed in 01, for the strongly convex functions, represents an improvement of the classical gradient descent algorithm. Namely, in this paper, the idea is to modify the gradient descent method by introducing a relaxation of the following form:

xk+1=xk+θktkdk,E23

where θkis the relaxation parameter, a random variable uniformly distributed between 0and 1.

In the recent years, the steepest descent method has been applied in many branches of science; one can be inspired, for example, by [37, 38, 39, 40, 41, 42, 43].

2.2 Barzilai and Borwein gradient method

Remind to the fact that SDmethod performs poorly, converges linearly, and is badly affected by the ill-conditioning.

Also, remind to the fact that this poor behavior of SDmethod is due to the optimal choice of the step size and not to the choice of the steepest descent direction gk.

Barzilai and Borwein presented [44] a two-point step size gradient method, which is well known as BBmethod.

The step size is derived from a two-point approximation to the secant equation.

Consider the gradient iteration form:

xk+1=xktkgk.

It can be rewritten as xk+1=xkDkgk,where Dk=tkI.

To make the matrix Dkhaving quasi-Newton property, the step size tkis computed in such a way that we get

minsk1Dkyk1.

This yields that

tkBB1=sk1Tyk1yk1Tyk1,sk1=xkxk1,yk1=gkgk1.E24

But, using symmetry, we may minimize Dk1sk1yk1, with respect to tk, and we get:

tkBB2=sk12sk1Tyk1,sk1=xkxk1,yk1=gkgk1.E25

Now, we give the algorithm of BBmethod.

Algorithm 1.2.5. (Barzilai-Borwein gradient method, i.e., BBmethod).

Assumptions: 0<ϵ1, x0Rn. Let k=0.

Step 1. If gkϵ, then STOP, else set dk=gk.

Step 2. If k=0, then find the step size t0by the line search, else compute tkusing the formula (24) or (25).

Step 3. Set xk+1=xk+tkdk.

Step 4. Set kk+1and go to Step 1.

Considering Algorithm 1.2.5, we can conclude that this method does not require any matrix computation or any line search.

The Barzilai-Borwein method is in fact the gradient method, which requires less computational work than SDmethod, and it speeds up the convergence of the gradient method. Barzilai and Borwein proved that BBalgorithm is Rsuperlinearly convergent for the quadratic case.

In the general non-quadratic case, a globalization strategy based on non-monotone line search is applied in this method.

In this general case, tk, computed by (24) or (25), may be unacceptably large or small. That is the reason why we assume that there exist the numbers tland tr, such that

0<tltktr,forallk.

Using the iteration

xk+1=xk1tkgk=xkλkgk,E26

with

tk=sk1Tyk1sk1Tsk1,λk=1tk,
sk=1tkgk=λkgk,

we get

tk+1=skTykskTsk=λkgkTykλk2gkTgk=gkTykλkgkTgk.

Now, we give the algorithm of the Barzilai-Borwein method with non-monotone line search.

Algorithm 1.2.6. (BBmethod with non-monotone line search).

Assumptions: 0<ϵ1, x0Rn, M0is an integer, ρ01, δ>0, 0<σ1<σ2<1, tl, tr. Let k=0.

Step 1. If gkϵ, then STOP.

Step 2. If tktl, or tktr, then set tk=δ.

Step 3. Set λ=1tk.

Step 4. (non-monotone line search) If

fxkλgkmax0jminkMfxkjρλgkTgk,

then set

λk=λ,xk+1=xkλkgk,

and go to Step 6.

Step 5. Choose σσ1σ2, set λ=σλ, and go to Step 4.

Step 6. Set tk+1=gkTykλkgkTgkand kk+1, and return to Step 1.

Obviously, the above algorithm is globally convergent.

Several authors paid attention to the Barzilai-Borwein method, and they proposed some variants of this method.

In [8], the globally convergent Barzilai-Borwein method is proposed by using non-monotone line search by Grippo et al. [12]. In the same paper, Raydan proves the global convergence of the non-monotone Barzilai-Borwein method.

Further, Grippo and Sciandrone [45] propose another type of the non-monotone Barzilai-Borwein method.

Dai [7] gives the basic analysis of the non-monotone line search strategy.

Moreover, in [46] numerical results are presented, using

tk=sνkTyνksνkTsνk.E27

and

νk=Mck1Mc,

where for rR, rdenotes the largest integer jsuch that jrand Mc is a positive integer. The gradient method with (27) is called the cyclic Barzilai-Borwein method. Numerical results in [46] prove that their method performs better than the Barzilai-Borwein method.

Many researchers study the gradient method for minimizing a strictly convex quadratic function, namely,

minfx=12xTAxbTx,E28

where ARn×nis a symmetric positive definite matrix and bRnis a given vector. For an application of the Barzilai-Borwein method to the problem (28), Raydan [47] establishes global convergence, and Dai and Liao [48] prove R-linear rate of convergence. Friedlander, Martinez, Molina, and Raydan [49] propose a new gradient method with retards, in which tkis defined by

tk=gνkTAρk+1gνkgνkTAρkgνk,νkkk1max0kmE29

and ρkq1qm, where mis a positive integer and q1,,qm2are integers. In the same paper, they establish its global convergence for problem (28) and prove the Q-superlinear rate of convergence in the special case.

In [50], the authors extend the Barzilai-Borwein method, and they give extended Barzilai-Borwein method, which they denote EBB. They also establish global and Qsuperlinear convergence properties of the proposed method for minimizing a strictly convex quadratic function. Furthermore, they discuss an application of their method to general objective functions. In [50], a new step size is proposed by extending (29). Namely, in this paper, following Friedlander et al. [49], a new step size is proposed as follows:

tk=i=1lϕigνikTAρik+1gνikgνikTAρikgνik,ϕi0,i=1nϕi=1,νikkk1max0km

and

ϕikq1qm,

where land mare positive integers and q1,,qmare integers.

Also, an application of algorithm EBBto general unconstrained minimization problems (4) is considered.

Following Raydan [8], the authors [50] further combine the non-monotone line search and algorithm EBBto get the algorithm called NEBB. They also prove the global convergence of the algorithm NEBB, under some classical assumptions.

The Barzilai-Borwein method and its related methods are reviewed by Dai and Yuan [51] and Fletcher [52].

In [53], a new concept of the approximate optimal step size for gradient method is introduced and used to interpret the BBmethod; an efficient gradient method with the approximate optimal step size for unconstrained optimization is presented. The next definition is introduced in [53].

Definition 1.2.1. Let Φtbe an approximation model of fxktgk. A positive constant tis called approximate optimal step size associated to Φtfor gradient method, if tsatisfies

t=argmint>0Φt.

The approximate optimal step size is different from the steepest descent step size, which will lead to the expensive computational cost. The approximate optimal step size is generally calculated easily, and it can be applied to unconstrained optimization.

Due to the effectiveness of tkBB1and the fact that tkBB1=argmint>0Φt,we can naturally ask if more suitable approximation models can be constructed to generate more efficient approximate optimal step-sizes.

This is the purpose of work [53]. Further, if the objective function fxis not close to a quadratic function on the line segment between xk1and xk, in this paper a conic model is developed to generate the approximate optimal step size if the conic model is suitable to be used. Otherwise, the authors consider two cases:

  1. If sk1Tyk1>0, the authors construct a new quadratic model, to derive the approximate optimal step size.

  2. If sk1Tyk10, they construct a new quadratic model or two other new approximation models to generate the approximate optimal step size for gradient method. They also analyze the convergence of the proposed method under some suitable conditions. Numerical results show the proposed method is better than the BB method.

In [54], derivative-free iterative scheme that uses the residual vector as search direction for solving large-scale systems of nonlinear monotone equations is presented.

The Barzilai-Borwein method is widely used; some interesting results can be found in [55, 56, 57].

2.3 Newton method

The basic idea of Newton method for unconstrained optimization is the iterative usage of the quadratic approximation qkto the objective function fat the current iterate xkand then minimization of such approximation qk.

Let f:RnRbe twice continuously differentiable, xkRn, and let the Hessian 2fxkbe positive definite.

We model fat the current point xkby the quadratic approximation qk:

fxk+sqks=fxk+fxkTs+12sT2fxks,s=xxk.

Minimization of qksgives the next iterative scheme:

xk+1=xk2fxk1fxk,

which is known as Newton formula.

Denote Gk=2fxk, gk=fxk.

Then, we have a simpler form:

xk+1=xkGk1gk.E30

A Newton direction is

sk=xk+1xk=Gk1gk.E31

We have supposed that Gkis positive definite. So, the Newton direction is a descent direction. This we can conclude from

gkTsk=gkTGk1gk<0.

Now, we give the algorithm of the Newton method.

Algorithm 1.2.7. (Newton method).

Assumptions: ϵ>0, x0Rn. Let k=0.

Step 1. If gkϵ, then STOP.

Step 2. Solve Gks=gkfor sk.

Step 3. Set xk+1=xk+sk.

Step 4. kk+1, return to Step 1.

The next theorem shows the local convergence and the quadratic convergence rate of Newton method.

Theorem 1.2.3. [27] (Convergence theorem of Newton method) Let fC2and xkbe close enough to the solution xof the minimization problem with gx=0. If the Hessian Gxis positively definite and Gxsatisfies Lipschitz condition

GijxGijyβxy,for someβ,foralli,j,

where Gijxis the ijelement of Gxand then for all k, Newton direction (31) is well-defined; the generated sequence xkconverges to xwith a quadratic rate.

But, in spite of this quadratic rate, the Newton method is a local method: when the starting point is far away from the solution, there is a possibility that Gkis not positive definite, as well as Newton direction is not a descent direction.

So, to guarantee the global convergence, we can use Newton method with line search. We can remind to the fact that only when the step size sequence tktends to 1, Newton method is convergent with the quadratic rate.

Newton iteration with line search is as follows:

dk=Gk1gk,E32
xk+1=xk+tkdk.E33

Now, we give the algorithm.

Algorithm 1.2.8. (Newton method with line search).

Assumptions: ϵ>0, x0Rn. Let k=0.

Step 1. If gkϵ, then STOP.

Step 2. Solve Gkd=gkfor dk.

Step 3. Line search step: find tksuch that

fxk+tkdk=mint0fxk+tdk,

or find tksuch that (inexact) Wolfe line search rules hold.

Step 4. Set xk+1=xk+tkdkand k=k+1, and go to Step 1.

The next theorems claim that Algorithm 1.2.8 with the exact line search, as well as Algorithm 1.2.8 with the inexact line search, are globally convergent.

Theorem 1.2.4. [27] Let f:RnRbe twice continuously differentiable on open convex set DRn. Assume that for any x0Dthere exists a constant m>0, such that fxsatisfies

uT2fxumu2,foralluRn,xLx0,E34

where Lx0=xfxfx0is the corresponding level set. Then, the sequence xk, generated by Algorithm 1.2.8, with the exact line search, satisfies:

  1. When xkis a finite sequence, gk=0for some k.

  2. When xkis an infinite sequence, xkconverges to the unique minimizer xof f.

Note that the next relation holds from the standard Wolfe line search:

fxkfxk+tkdkη¯gk2cos2dkgk,E35

where the constant η¯does not depend on k.

Theorem 1.2.5. [27] Let f:RnRbe twice continuously differentiable on open convex set DRn. Assume that for any x0Dthere exists a constant m>0, such that fxsatisfies the relation (34) on the level set Lx0. If the line search satisfies the relation (35), then the sequence xk, generated by Algorithm 1.2.8, with the inexact Wolfe line search, satisfies

limkgk=0

and xkconverges to the unique minimizer of fx.

2.4 Modified Newton method

The main problem in Newton method could be the fact that the Hessian Gkmay be not positive definite. In that case, we are not sure that the objective function fhas its minimizers; furthermore, when Gkis indefinite, the objective function fis unbounded.

So, many modified schemes are made. Now, we describe the next two methods shortly.

In [58], Goldstein and Price use the steepest descent method when Gkis not positive definite. Denoting the angle between dkand gkby θ, as well as having in view the angle rule, θπ2μ, where μ>0, they determine the direction dkas

dk=Gk1gk,ifcosθη,gk,otherwise,

where η>0is a given constant.

In [59], the authors present another modified Newton method. When Gkis not positive definite, Hessian Gkis changed into Gk+νkI, where νk>0is chosen in such a way that Gk+νkIis positive definite and well-conditioned. Otherwise, when Gkis positive definite, νk=0.

To consider the other modified Newton methods, such as finite difference Newton method, negative curvature direction method, Gill-Murray stable Newton method, etc., one can see [27], for example.

2.5 Inexact Newton method

By the other side, because of the high cost of the exact Newton method, especially when the dimension nis large, the inexact Newton method might be a good solution. This type of method means that we only approximately solve the Newton equation.

Consider solving the nonlinear equations:

Fx=0,E36

where F:RnRnis assumed to have the next properties:

A1 There exists xsuch that Fx=0.

A2 Fis continuously differentiable in the neighborhood of x.

A3 Fxis nonsingular.

Remind that the basic Newton step is obtained by solving

Fxksk=Fxk

and setting

xk+1=xk+sk.

The inexact Newton method means that we solve

Fxksk=Fxk+rk,E37

where

rkηkFxk.E38

Set

xk+1=xk+sk.E39

Here, rkdenotes the residual, and the sequence ηk, where 0<ηk<1, is the sequence which controls the inexactness.

Now, we give two theorems; the first of them claims the linear convergence, and the second claims the superlinear convergence of the inexact Newton method.

Theorem 1.2.6. [27] Let F:RnRnsatisfy the assumptions A1A3. Let the sequence ηksatisfies 0ηkη<t<1. Then, for some ϵ>0, if the starting point is sufficiently near x, the sequence xkgenerated by inexact Newton’s method (37)(39) converges to x, and the convergence rate is linear, i.e.

xk+1xtxkx,

where y=Fxy.

Theorem 1.2.7. [27] Let all assumptions of Theorem 1.2.6 hold. Assume that the sequence xk, generated by the inexact Newton method, converges to x. then

rk=oFxk,k,

if and only if xkconverges to xsuperlinearly.

The relation

xk+1=xkfxkfxkfxk1xkxk1,E40

presents the secant method.

In [60], a modification of the classical secant method for solving nonlinear, univariate, and unconstrained optimization problems based on the development of the cubic approximation is presented. The iteration formula including an approximation of the third derivative of fxby using the Taylor series expansion is derived. The basic assumption on the objective function fxis that fxis a real-valued function of a single, real variable xand that fxhas a minimum at x. Furthermore, in this chapter it is noted that the secant method is the simplification of Newton method. But, the order of the secant method is lower than one of the Newton methods; it is Q-superlinearly convergent, and its order is p=5+121,618.

This modified secant method is constructed in [60], having in view, as it is emphasized, that it is possible to construct a cubic function which agrees with fxup to the third derivatives. The third derivative of the objective function fis approximated as

fx=32fxkfxkfxk1xkxk1xkxk1fxkxk1xk.

In [61], the authors propose an inexact Newton-like conditional gradient method for solving constrained systems of nonlinear equations. The local convergence of the new method as well as results on its rate is established by using a general majorant condition.

2.6 Quasi-Newton method

Consider the Newton method.

For various practical problems, the computation of Hessian may be very expensive, or difficult, or Hessian can be unavailable analytically. So, the class of so-called quasi-Newton methods is formed, such that it uses only the objective function values and the gradients of the objective function and it is close to Newton method. Quasi-Newton method is such a class of methods which does not compute Hessian, but it generates a sequence of Hessian approximations and maintains a fast rate of convergence.

So, we would like to construct Hessian approximation Bkin quasi-Newton method. Naturally, it is desirable that the sequence Bkpossesses positive definiteness, as well as its direction dk=Bk1gkshould be a descent one.

Now, let f:RnRbe twice continuously differentiable function on an open set DRn. Consider the quadratic approximation of fat xk+1:

fxfxk+1+gk+1Txxk+1+12xxk+1TGk+1xxk+1.

Finding the derivatives, we get

gxgk+1+Gk+1xxk+1.

Setting x=xkand using the standard notation: sk=xk+1xk, yk=gk+1gk, from the last relation, we get

Gk+11yksk.E41

Relation (41) transforms into the next one if fis the quadratic function:

Gk+11yk=sk.E42

Let Hkbe the approximation of the inverse of Hessian. Then, we want Hkto satisfy the relation (42). In this way, we come to the quasi-Newton condition or quasi-Newton equation:

Hk+1yk=sk.E43

Let Bk+1=Hk+11be the approximation of Hessian Gk+1. Then

Bk+1sk=ykE44

is also the quasi-Newton equation.

If

skTyk>0,E45

then the matrix Bk+1is positive definite. The condition (45) is known as the curvature condition.

Algorithm 1.2.9. (A general quasi-Newton method).

Assumptions: 0ϵ<1, x0Rn, H0Rn×n. Let k=0.

Step 1. If gkϵ, then STOP.

Step 2. Compute dk=Hkgk.

Step 3. Find tkby line search and set xk+1=xk+tkdk.

Step 4. Update Hkinto Hk+1such that quasi-Newton equation (43) holds.

Step 5. Set k=k+1and go to Step 1.

In Algorithm 1.2.9, usually we take H0=I, where Iis an identity matrix.

Sometimes, instead of Hk, we use Bkin Algorithm 1.2.9.

Then, Step 2 becomes

Step 2. Solve

Bkd=gk,fordk.

By the other side, Step 4 becomes

Step 4. Update Bkinto Bk+1in such a way that quasi-Newton equation (44) holds.

2.7 Symmetric rank-one (SR1) update

Let Hkbe the inverse Hessian approximation of the kth iteration. We are trying to update Hkinto Hk+1, i.e.

Hk+1=Hk+Ek,

where Ekis a matrix with a lower rank. If it is about a rank-one update, we get

Hk+1=Hk+uvT,E46

where u,vRn. Using quasi-Newton equation (43), we can get

Hk+1yk=Hk+uvTyk=sk,

wherefrom

vTyku=skHkyk.E47

Further, from (46) and (47), we have

Hk+1=Hk+1vTykskHkykvT.

Having in view that the inverse Hessian approximation Hkhas to be the symmetric one, we use v=skHkyk, so we get the symmetric rank-one update (i.e., SR1update):

Hk+1=Hk+skHkykskHkykTskHkykTyk.E48

Theorem 1.2.8. [27] (Property theorem of SR1update) Let s0, s1, and sn1be linearly independent. Then, for quadratic function with a positive definite Hessian, SR1method terminates at n+1steps, i.e., Hn=G1.

More information about SR1 update can be found.

2.8 Davidon-Fletcher-Powell (DFP) update

There exists another type of update, which is a rank-two update. In fact, we get Hk+1using two symmetric, rank-one matrices:

Hk+1=Hk+auuT+bvvT,E49

where u,vRnand a,bare scalars which have to be determined.

Using quasi-Newton equation (43), we can get

Hkyk+auuTyk+bvvTyk=sk.E50

The values of u,vare not determined in a unique way, but the good choice is

u=sk,v=Hkyk.

Now, from (50), we get:

a=1skTyk,b=1ykTHkyk.

Hence, we get the formula

Hk+1=Hk+skskTskTykHkykykTHkykTHkyk,E51

which is DFPupdate.

Theorem 1.2.9. [27] (Positive definiteness of DFPupdate) DFPupdate (51) retains positive definiteness if and only if skTyk>0.

Theorem 1.2.10. [27] (Quadratic termination theorem of DFPmethod) Let fxbe a quadratic function with positive definite Hessian G. Then, if the exact line search is used, the sequence sj, generated from DFPmethod, satisfies, for i=0,1,,m, where mn1:

  1. Hi+1yj=sj,j=0,1,,ihereditary property.

  2. siTGsj=0,j=0,1,,i1conjugate direction property.

  3. The method terminatesatm+1nsteps.Ifm=n1,thenHn=G1.

2.9 Broyden-Fletcher-Goldfarb-Shanno (BFGS) update

BFGSupdate is given by the formula

Bk+1BFGS=Bk+ykykTykTskBkskskTBkskTBksk.E52

The BFGSupdate is also said to be a complement to DFPupdate.

In [62], an adaptive scaled BFGSmethod for unconstrained optimization is presented. In this paper, the author emphasizes that the BFGSmethod is one of the most efficient quasi-Newton methods for solving small-size and medium-size unconstrained optimization problems. The third term in the standard BFGSupdate formula is scaled in order to reduce the large eigenvalues of the approximation to the Hessian of the minimizing function. In fact, in [62], the general scaling BFGSupdating formula is considered:

Bk+1=BkBkskskTBkskTBksk+γkykykTykTsk,E53

where γkis a positive parameter. Obviously, using γk=1for all k=0,1,, we get the standard BFGSformula. By the way, there exist several procedures created to select the scaling parameter γk, for example, see [62, 63, 64, 65, 66, 67, 68, 69]. The approach for determining the scaling parameters of the terms of the BFGSupdate in [62] is to minimize the Byrd and Nocedal measure function.

Namely, in [70], the next function was introduced:

φA=trAlndetA,E54

which is defined on positive definite matrices.

This function is a measure of matrices involving all the eigenvalues of A, not only the smallest one and the largest one, as it is traditionally used in the analysis of the quasi-Newton method based on the condition number of matrices.

Observe that function φworks simultaneously with the trace and the determinant, thus simplifying the analysis of the quasi-Newton methods. Fletcher [71] proves that this function is strictly convex on the set of symmetric and positive definite matrices, and it is minimized by A=I. Besides, this function becomes unbounded when Abecomes singular or infinite, and therefore it works as a barrier function that keeps Apositive definite. It is worth saying that the BFGSupdate tends to generate updates with large eigenvalues.

Further, in [62], a double-parameter scaling BFGSupdate is considered, in which the first two terms on the right-hand side of the BFGSupdate (52) are scaled with a positive parameter, while the third one is scaled with another positive parameter:

Bk+1=δkBkBkskskTBkskTBksk+γkykykTykTsk,E55

where δkand γkare the two positive parameters that have to be determined.

In [62], the next proposition is proved.

Proposition 1.2.1. If the step size tkis determined by the standard Wolfe line search (12) and (13), Bkis positive definite and γk>0, and then Bk+1, given by (55), is also positive definite.

From (55), it can be seen that φBk+1depends on the scaling parameters δkand γk. In [62], these scaling parameters are determined as solution of the minimizing problem:

minδk>0,γk>0φBk+1.E56

Further, the next values of the scaling parameters δkand γkare reached:

δk=n1trBkBksk2skTBkskE57
γk=ykTskyk2.E58

Consider the relation

xk+1=xk+tkdk,E59

where dkis the BFGSsearch direction obtained as solution of the linear algebraic system

Bkdk=gk,

where the matrix Bkis the BFGSapproximation to the Hessian 2fxk, being updated by the classical formula (52).

The next theorems are also given in [62].

Theorem 1.2.11. If the step size in (59) is determined by the Wolfe search conditions (12)(13), then the scaling parameters given by (57) and (58) are the unique global solutions of the problem (56).

Theorem 1.2.12. Let δkbe computed by (57). Then, for any k=0,1,, δkis positive and close to 1.

Next, in [72], using chain rule, a modified secant equation is given, to get a more accurate approximation of the second curvature of the objective function. Then, based on this modified secant equation, a new BFGSmethod is presented. The proposed method makes use of both gradient and function values, and it utilizes information from two most recent steps, while the usual secant relation uses only the latest step information. Under appropriate conditions, it is shown that the proposed method is globally convergent without convexity assumption on the objective function.

Some interesting applications of Newton, modified Newton, inexact Newton, and quasi-Newton methods can be found, for example, in [73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83], etc.

A very interesting paper is [84].

An interesting application of BFGSmethod can be found in [85].

3. Conclusion

Today, the modifications of the line search techniques are very actual and all in the aim to create new, better optimization methods.

Further, following recent trends in unconstrained optimization, we can notice that almost all optimization methods, which are considered in this chapter, are still actual.

They are applied in the other areas of Mathematics, as well as in practice. Also, different modifications of these methods are made, in the aim to improve them.

Let us emphasize that BFGSupdate is very popular now.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Snezana S. Djordjevic (February 20th 2019). Some Unconstrained Optimization Methods, Applied Mathematics, Bruno Carpentieri, IntechOpen, DOI: 10.5772/intechopen.83679. Available from:

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