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Optimization using the NAG Library

The Numerical Algorithms Group, Ltd.

Wilkinson House, Jordan Hill Road

Oxford OX2 8DR, United Kingdom

1. Introduction

This note describes some aspects of optimization, with an emphasis on the way in

which the NAG Library [1] can be used in the solution of problems in this area. The

following section contains a brief discussion (§2.1) of mathematical optimization,

which then focusses attention (§2.2) on the way problems in this area can be

categorized. This is important because it is necessary to use a solver which is

appropriate for the type of problem at hand; we illustrate this point in §2.3 with a

simple example. Section 3 describes some of the functionality of the NAG Library

optimization solvers; we draw a distinction between so-called local solvers (§3.1) and

global solvers (§3.2), before highlighting, in §3.3, the routines which have been added

to the Library in its latest release. We conclude §3 by looking at some related

problems which can be solved using routines from other chapters of the Library. Our

attention turns to applications of optimization in Section 4, which involves taking a

detailed look at a specific example from financial analysis in §4.1, before briefly

surveying (§4.2) a few other applications in finance, manufacturing and business

analytics. A final section (§5) contains the conclusions of this note.

2. Background

2.1 What is optimization?

Optimization is a wide-ranging and technically detailed subject, to which several

authoritative texts – for example, [2] [3] – have been devoted. It can be defined as

“the selection of a best element (with regard to some criteria) from some set of

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available alternatives” [4]. Mathematically, it consists of locating the maximum or

minimum of a function of one or more variables (called the objective function):

( )

( )

( )

(1)

and, since maximizing ( ) is equivalent to minimizing the negative of ( ), we can

without loss of generality consider minimization only from hereon. A distinction may

be drawn between finding a local minimum – where, for all near , ( )

( ) – and a global minimum, where this condition is fulfilled for all . While many

algorithms have been developed for local optimization, the development of reliable

methods for global optimization has been harder to do – we return to this point below

in §3.2.

Methods for optimization often require information related to the derivatives of the

objective function, in order to assist with the search for the minimum. More

specifically, the elements of the gradient vector and the Hessian matrix are

( )

( )

(2)

A simple illustration of this point can be found below, in the description of the so-

called gradient descent method, which uses explicitly – see equation (10).

2.2 Constraints and categorization

The optimization problem may be supplemented by restrictions on the values of the

solution, which are usually specified by a set of constraint functions, for example:

(3)

which are bound constraints, or

(4)

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which is an example of a linear constraint, or

(5)

which is a non-linear constraint. The three types of constraint can be expressed

generally as

{

( )

} (6)

in which and are vectors of lower and upper bound values, is a constant matrix,

and is a vector of nonlinear constraint functions.

The optimization problem is categorized by the form of the constraints, and also by

the properties of the objective function. More specifically, a distinction is made

between objective functions that are linear, quadratic and nonlinear. Another

categorization may be made if (1) can be written as

{ ∑

( ) (7)

in which case the optimization is referred to as a least squares problem (which may be

further classified as nonlinear or linear, depending on the form of the functions ).

Some specific combinations of the properties of the objective function and the

constraints receive special designation; thus, linear constraints together with a linear

is referred to as a linear programming (LP) problem; linear constraints plus a

quadratic makes a quadratic programming (QP) problem, while nonlinear

constraints with a nonlinear is called a nonlinear programming (NLP) problem.

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These designations are important because a single, all-purpose optimization method

does not currently exist; instead, different methods have been developed which are

suitable for particular forms of the problem. Using a method that is inappropriate for

the problem at hand can be cumbersome and/or inefficient. We illustrate this point in

the following subsection using a concrete example.

2.3 An example problem

Figure 1 shows two attempts to find the minimum of the so-called Rosenbrock

function [5]:

Figure 1. Finding the minimum of Rosenbrock's function in

MATLAB® using the method of steepest descents (top) and a method

from the NAG Library (bottom)

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( ) ( ) (

) (8)

This is a function which was designed as a performance test for optimization

algorithms. It has a global minimum at

( ) (9)

which is within a long, narrow, parabolic-shaped flat valley. The two methods used in

Figure 1 to minimize (8) are called gradient (or steepest) descent [6], and a sequential

quadratic programming (SQP) method [7], as implemented in the NAG Library [8].

The former is easier to understand (and program) than the latter: specifically, at each

point, the method takes a step in the direction of the negative gradient at that point:

( ) ( ) ( ) ( ( )) (10)

where ( ), the step size at the -th step, is proportional to | ( ( ))|, the modulus of

the gradient at that step. Figure 1 shows that, whilst this method is able to find the

location of the valley in a few big steps, its progress along the valley takes many more

small steps, because of the link between the size of the step and the gradient. The

method terminates after 200 steps in this example, with the current step still being a

long way away from the minimum. By contrast, the SQP method converges to the

minimum in 55 steps. (The number of steps taken by an optimization method should

be as small as possible for efficiency’s sake, because it is proportional to the number

of times the objective function is evaluated, which can be computationally expensive.)

We recall that many optimization methods require information about the objective

function’s derivatives. The NAG implementation of the SQP method has an option

for these to be specified explicitly (in the same way that the form of the objective

function is described); otherwise, it approximates the derivatives using finite

difference methods. This clearly requires many more evaluations of the objective

function – to be specific, in this example, it increases the number of steps required for

convergence to 280.

Finally in this section, we note that the Rosenbrock function (8) can be written in the

form of (7), which turns its minimization into a non-linear least squares problem:

( ) ( )

( ) (11)

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( ) ( )

( ) ( )

Switching to a quasi-Newton method [9] which is specially designed for this type of

problem – implemented in the NAG Library at [10] – results in a convergence to the

minimum in 19 steps (which, since this method requires specification of derivatives,

is to be compared with the 55 steps required by the more general SQP method when

derivatives are provided).

3. Optimization and the NAG Library

In this section we look at the range of functionality and applicability that the NAG

Library offers for optimization problems. We consider both local (§3.1) and global

(§3.2) optimization and, whilst we do not for the most part discuss individual routines

in detail, describe in §3.3 the enhancements in optimization functionality which have

been included [11] in the most recent release (Mark 23) of the NAG Library. Finally

in this section, we mention (§3.4) a few problems which are related to optimization

that can be solved using functions in other chapters of the Library.

We note in passing that the Library is a collection of numerical routines which can be

accessed from a variety of computing environments and languages [12] including

MATLAB and Microsoft Excel (see figures 1-3 for examples). In addition, some

work has been recently done [13] to build interfaces between NAG solvers and

AMPL, a specialized language which is used for describing optimization problems.

The interface to two solvers [14] [15] is described in detail (and is available for

download from the NAG website), which provides a starting point for the

development of AMPL interfaces to other routines, and the construction of

mechanisms for calling NAG routines from other optimization environments.

3.1 Local optimization

The NAG Library contains a number of routines for the local optimization of linear,

quadratic or nonlinear objective functions; of objective functions that are sums of

squares of linear or nonlinear functions; subject to bounded, linear, sparse linear,

nonlinear or no constraints. The documentation of the local optimization chapter [16]

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describes how to choose the most appropriate routine for the problem at hand with the

help of a decision tree which incorporates questions such as:

What kind of constraints does the problem have?

o None

o Bound

o Linear

If so, is the linear constraints matrix sparse?

o Nonlinear

What kind of form does the objective function have?

o Linear

o Quadratic

o Sum of squares

o Nonlinear

Does it have one variable?

o If not, does it have more than ten?

Are first derivatives available?

o If so, are second derivatives available?

Is computational cost critical?

Is storage critical?

Are you an experienced user?

The final question is related to the fact that many of the optimization methods are

implemented in two routines: namely, a comprehensive form and an easy-to-use form.

The latter are more appropriate for less experienced users, since they include only

those parameters which are essential to the definition of the problem (as opposed to

parameters relevant to the solution method). The comprehensive routines use

additional parameters which allow the user to improve the method’s efficiency by

‘tuning’ it to a particular problem.

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Figure 2. Globally optimizing a function with more than one minimum (displayed as

a surface at top) using a method from the NAG Library (bottom)

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3.2 Global optimization

The Library contains implementations of three methods for the solution of global

optimization problems. As previously (§3.1), the documentation for this chapter [17]

provides the user with suggestions about the choice of the most appropriate routine for

the problem at hand.

nag_glopt_bnd_mcs_solve [18] uses a so-called multilevel coordinate

search [19] which recursively splits the search space (i.e. that within which the

minimum is being sought) into smaller subspaces in a non-uniform fashion.

The problem may be unconstrained, or subject to bound constraints. The

routine does not require derivative information, but the objective function

must be continuous in the neighbourhood of the minimum, otherwise the

method is not guaranteed to converge. The use of this routine is illustrated in

Figure 2, the upper part of which shows a demo function having more than one

minimum1. The workings of the method are shown in the bottom part of the

figure, which displays the function as a contour plot, along with the search

subspaces and objective function evaluation points chosen by the method; it

can be seen that – for this example – the method spends most of its time in and

around the two minima, before correctly identifying the global minimum as

the lower of the two.

nag_glopt_nlp_pso [20] uses a stochastic method based on particle

swarm optimization [21] to search for the global minimum of the objective

function. The problem may be unconstrained, or subject to bound, or linear, or

nonlinear constraints (there also exists a simpler variant [22] of the routine

which only handles bound – or no – constraints). The routine does not require

derivative information, although this could be used by the optional

accompanying local optimizers. A particular feature of this routine is that it

can exploit parallel hardware, since it allows multiple threads to advance

subiterations of the algorithm in an asynchronous fashion.

nag_glopt_nlp_multistart_sqp [23] finds the global minimum of an

arbitrary smooth function subject to constraints (which may include simple

bounds on the variables, linear constraints and smooth nonlinear constraints)

1 More specifically, this is MATLAB’s peaks function.

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by generating a number of different starting points and performing a local

search from each using SQP. This method can also exploit parallel hardware,

in the same fashion as the particle swarm method.

3.3 Recent additions to NAG Library optimization functionality

Two of the global optimization routines described in §3.2 (specifically,

nag_glopt_nlp_pso and nag_glopt_nlp_multistart_sqp) are new in

the most recent release of the NAG Library. In addition, the following local

optimization routines were also added to the Library at that release:

nag_opt_bounds_qa_no_deriv [24] is an easy-to-use routine that uses

methods of quadratic approximation to find a local minimum of an objective

function ( ), subject to bound constraints on the . Derivatives of are not

required. The routine is intended for functions that are continuous and that

have continuous first and second derivatives (although it will usually work

even if the derivatives have occasional discontinuities). The routine uses the

BOBYQA (Bound Optimization BY Quadratic Approximation) algorithm [25]

whose efficiency is preserved for large problem sizes2. For example, it solves

the problem of distributing 50 points on a sphere to have maximal pairwise

separation (starting from equally spaced points on the equator) using 4633

evaluations of . This is to be compared with, for example, the NAG routine

[26] which uses the so-called Nelder-Mead simplex solver [27] and takes

16757 evaluations.

nag_opt_nlp_revcomm [28] is designed to locally minimize an arbitrary

smooth function subject to constraints (which may include simple bounds,

linear constraints and smooth nonlinear constraints) using an SQP method.

The user should supply as many first derivatives as possible; any unspecified

derivatives are approximated by finite differences. It may also be used for

unconstrained, bound-constrained and linearly constrained optimization. It

2 i.e., large in equation (1).

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uses a similar algorithm to nag_opt_nlp [15], but has an alternative

interface3.

3.4 Solving optimization-related problems using other NAG routines

In addition to the standard types of optimization problem which the routines from the

two optimization chapters [16] [17] can be used to solve, there are a few related

problems that can be solved using functions in other chapters of the NAG Library:

nag_ip_bb [29] solves various types of integer programming problem.

Briefly, this is an optimization problem of the form of (1) – possibly

incorporating constraints (6) – with the restriction that some (or all) of the

elements of the solution vector take integer values. The routine uses a so-

called branch and bound method, which divides the problem up into sub-

problems, each of which is solved internally using a QP solver from the local

optimization chapter [30].

There are several routines elsewhere in the Library [31] which solve linear

least squares problems – i.e., the minimization of

( ) ∑ ( )

(12)

where the residual is defined as

( ) ∑

(13)

In a similar fashion, the minimization of other metrics – for example the so-

called norm [32],

( ) ∑| ( )|

(14)

or the norm,

3 Specifically, it uses so-called reverse communication [51], as opposed to direct communication which

requires that user-supplied procedures – such as that which calculates ( ) – be included in the

function argument list.

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( )

| ( )| (15)

can be solved4 using routines from the curve and surface fitting chapter of the

Library.

4. Applications

A complete review of applications for optimization [4] is beyond the scope of this

note; instead we first present (§4.1) a brief description of a specific, simple,

application taken from finance – namely, portfolio optimization – before discussing

(§4.2) a few other example applications of interest.

We note in passing that use of the NAG Library for portfolio optimization has

previously been discussed elsewhere [33]. Our account in §4.1 is less comprehensive,

but takes more recent developments – including the Library interfaces with MATLAB

and Microsoft Excel, and the treatment of the so-called nearest correlation matrix [34]

– into account. The interested reader is referred to [33] for a more detailed and

authoritative account of this subject.

4.1 Portfolio optimization

Portfolio optimization addresses the question of diversification – that is, how to

combine different assets in a portfolio which achieves maximum return with minimal

risk. This problem was initially addressed by Markowitz [35]; we outline its

treatment here although more refined models are often used by professional investors

as the Markowitz model has certain practical drawbacks.. The basic theory is

however still relevant and is taught in postgraduate finance courses at universities.

We assume that the portfolio contains assets, each of which has a return . Then,

the return from the portfolio is

( ) ∑

(16)

where is the proportion of the portfolio invested in asset . We further assume that

each is normally distributed with mean and variance ; the vector of returns

4 nag_lone_fit [53] for the norm, or nag_linf_fit [52] for the norm

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then has ( ) distribution, with the vector of means and the covariance

matrix. These can be, for example, calculated from historical data:

∑ ( )

∑ ( ) ( )

(17)

for a collection of observations of at times (we note in passing that the NAG

Library contains a routine [36] for calculating and , as well as other statistical

data).

The expectation and variance of the portfolio return are

( ) ∑

( ) ∑∑

(18)

The portfolio optimization problem is then to determine the which results in a

specific return , with minimum risk. We define the latter as the variance of the

return – see (18). The objective function is then ( ) – i.e., the optimization

problem can then be stated as

(19)

with the constraint

∑

(20)

This problem can be solved using a QP method [37], as implemented in the NAG

Library at [38]. The problem may incorporate constraints – for example, that the

whole portfolio be fully invested:

∑

(21)

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In addition, the composition of the portfolio may be constrained across financial

sectors. For example, the requirement that up to 20% be invested in commodities, and

at least 10% in emerging markets may be specified as

∑

∑

(22)

where each element of the coefficient vectors and is either 0 or 1, depending on

the classification of asset . Other types of linear constraints might capture the

selection of assets based on geography or stock momentum, while simple bound

constraints of the form of (3) can be used to allow or prohibit shorting5.

Figure 3 is a screenshot from an application [39] implemented in Excel which allows

the user to enter or edit the historical returns for a portfolio of eleven stocks, before

calculating and from (17). These are then used in (19) and (20), which are solved

using a NAG method6 [38]. The application calculates the portfolio which provides

a return (entered by the user), at minimum risk. It also calculates the so-called

efficient frontier [40], which shows the highest expected return for each level of risk.

5 A company with a licence to short a position sells stock it doesn’t own in the expectation that when

the time comes to deliver on the sale, they will be able to buy the stock at a lower price than it received

initially, thereby making a profit on the transaction.

6 More specifically, the method takes the form of a routine within the NAG Library (implemented as a

Dynamic Link Library, or DLL), which we call from within a Visual Basic for Applications (VBA)

function attached to the spreadsheet. More information about the technical details of how this works

can be found at [50].

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As is well-known [35], this simplified portfolio model suffers from a number of

drawbacks that impede its application to real-world problems. For example, the

assumption that the returns are normally distributed is not borne out in practice, nor is

it the case that correlations between assets do not change in time7. It also assumes

that over-performance (a return greater than ) is to be eschewed in the same way as

under-performance; such an equal weighting clearly does not correspond to the

desires of real investors. In addition, the costs of each transaction – and other

refinements such as taxes and broker fees – are ignored in this model, which also

assumes that all assets can be divided into parcels of any size (which, in reality, is not

7 For example, during a financial crisis, all assets tend to become positively correlated, because they all

move in the same direction (i.e. down).

Figure 3. Portfolio optimization using a NAG method in a Microsoft Excel

spreadsheet, showing the so-called efficient frontier (see text).

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the case). We must emphasise that in practice quantitative analysts use models that are

more sophisticated than that described here – see, for example, the work described in

[41] – which in turn require the use of a more general optimizer from the NAG

Library.

Before leaving this subject, we mention a related area where NAG methods are

proving helpful. The treatment above – and also that of more sophisticated models –

assumes that the covariance matrix is positive semi-definite – i.e., that the quantity

is non-negative for all (this is part of the definition of a covariance matrix,

and is a requirement for the optimizer to work). In practice, this is often not the case,

as is calculated from historical data [see (16)], which might be incomplete, thus

rendering inappropriate for use here.

In other models is a correlation matrix (a covariance matrix with unit diagonal). A

method which determines , the nearest correlation matrix to , by minimizing their

Frobenius norm distance has been recently developed [42] [43], and has been

implemented in the NAG Library [34].

4.2 Other applications

Other examples of applications which use NAG routines for optimization include

index tracking [44]. This problem – which is related to portfolio optimization (§4.1) –

aims to replicate the movements of the index of a particular financial market,

regardless of changes in market conditions, and can also be tackled using optimization

methods. In fact, the same NAG method [38] that was used for the portfolio

optimization problem can also be used here.

Optimization is also heavily used in the calibration of derivative pricing models [45],

in which values for the parameters of the model are calculated by determining the best

match between the results of the model and historical prices (known as the calibration

instruments). This can be expressed as the minimization of, for example, the chi-

squared metric:

∑

( ) (23)

i.e., the weighted sum, over the instruments in the calibration set, of the difference

between the market price of instrument and its price ( ) as predicted by the

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model for some set of parameters . The weight is used to reflect the confidence

with which the market price for is known. As noted above (§3.4), it is also possible

to use other metrics – such as the (weighted) and norms – here. The

optimization problem is then how to determine the which minimizes the metric.

Other users [46] have optimized the performance of power stations by developing

mathematical models of engineering processes, and using NAG optimization routines

to solve them. The same routines have also found application in the solution of shape

optimization problems in manufacturing and engineering [47].

In the field of business analytics, NAG optimization routines have been used [48] in

the development of applications to help customers with – for example – fitting to a

pricing model used in the planning of promotions, and maximizing product income by

combining its specification, distribution channel and promotional tactics.

Other applications include the use of optimization for parameter estimation in the

fitting of statistical models to observed data [49].

5. Conclusions

In this note, we have examined some characteristics of optimization problems,

including their categorization, and the importance of using an appropriate solver for

the problem at hand. In this context, we have described some of the solvers which

can be found in the optimization chapters of the NAG Library, together with other

NAG routines that can be applied to the solution of optimization problems. We have

also briefly reviewed a few example applications which have incorporated the use of

NAG solvers.

Finally, it should perhaps be mentioned that, along with the routines that have been

mentioned in this note, the NAG Library [1] contains user-callable routines for

treatment of a wide variety of numerical and statistical problems, including – for

example – linear algebra, statistical analysis, quadrature, correlation and regression

analysis and random number generation.

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6. Acknowledgements

We thank our colleagues David Sayers, Martyn Byng, Jan Fiala, Lawrence

Mulholland and Mick Pont for helpful technical discussions whilst preparing this

note.

7. Bibliography

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http://www.nag.com/numeric/numerical_libraries.asp.

2. Gill, P.E. and Murray, W., [ed.]. Numerical Methods for Constrained

Optimization. London : Academic Press, 1974.

3. Fletcher, R. Practical Methods of Optimization. 2nd edn. Chichester : Wiley, 1987.

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http://en.wikipedia.org/wiki/Rosenbrock_function.

6. —. Gradient descent. [Online] http://en.wikipedia.org/wiki/Gradient_descent.

7. SNOPT: An SQP Algorithm for Large-scale Constrained Optimization. Gill, P.E.,

Murray, W. and Saunders, M.A. 2002, SIAM J. Optim., Vol. 12, pp. 979-1006.

8. Numerical Algorithms Group. nag_opt_nlp_solve (e04wd) routine document.

[Online] http://www.nag.co.uk/numeric/CL/nagdoc_cl23/pdf/E04/e04wdc.pdf.

9. Algorithms for the solution of the nonlinear least-squares problem. Gill, P.E. and

Murray, W. 1978, SIAM J. Numer. Anal., Vol. 15, pp. 977-992.

10. Numerical Algorithms Group. nag_opt_lsq_deriv (e04gb) routine document.

[Online] http://www.nag.co.uk/numeric/CL/nagdoc_cl23/pdf/E04/e04gbc.pdf.

11. —. NAG C Library News, Mark 23. [Online]

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12. —. Languages and Environments. [Online] http://www.nag.co.uk/languages-and-

environments.

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13. Fiala, Jan. Nonlinear Optimization Made Easier with the AMPL Modelling

Language and NAG Solvers. [Online]

http://www.nag.co.uk/NAGNews/NAGNews_Issue95.asp#Article1.

14. Numerical Algorithms Group. nag_opt_nlp_sparse (e04ugc) routine document.

[Online] http://www.nag.co.uk/numeric/CL/nagdoc_cl23/pdf/E04/e04ugc.pdf.

15. —. nag_opt_nlp (e04ucc) function document. [Online]

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16. —. NAG Chapter Introduction: e04 - Minimizing or Maximizing a Function.

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http://www.nag.co.uk/numeric/CL/nagdoc_cl23/html/E05/e05intro.html.

18. —. nag_glopt_bnd_mcs_solve (e05jb) routine document. [Online]

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19. Global optimization by multi-level coordinate search. Huyer, W. and Neumaier,

A. 1999, Journal of Global Optimization, Vol. 14, pp. 331-355.

20. Numerical Algorithms Group. nag_glopt_nlp_pso (e05sb) routine document.

[Online] http://www.nag.co.uk/numeric/CL/nagdoc_cl23/pdf/E05/e05sbc.pdf.

21. Vaz, A.I. and Vicente, L.N. A Particle Swarm Pattern Search Method for Bound

Constrained Global Optimization. Journal of Global Optimization. Vol. 39, 2, pp.

197-219.

22. Numerical Algorithms Group. nag_glopt_bnd_pso (e05sa) routine document.

[Online] http://www.nag.co.uk/numeric/CL/nagdoc_cl23/pdf/E05/e05sac.pdf.

23. —. nag_glopt_nlp_multistart_sqp (e05ucc) routine document. [Online]

http://www.nag.co.uk/numeric/CL/nagdoc_cl23/html/E05/e05ucc.html.

24. —. nag_opt_bounds_qa_no_deriv (e04jcc) function document. [Online]

http://www.nag.co.uk/numeric/CL/nagdoc_cl23/pdf/E04/e04jcc.pdf.

25. Powell, M.J.D. The BOBYQA algorithm for bound constrained optimization

without derivatives. University of Cambridge. 2009. Report DAMTP 2009/NA06.

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26. Numerical Algorithms Group. nag_opt_simplex_easy (e04cbc) function

document. [Online]

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27. A simplex method for function minimization. Nelder, J.A. and Mead, R. 1965,

Comput. J., Vol. 7, pp. 308–313.

28. Numerical Algorithms Group. nag_opt_nlp_revcomm (e04ufc) function

document. [Online]

http://www.nag.co.uk/numeric/CL/nagdoc_cl23/pdf/E04/e04ufc.pdf.

29. —. nag_ip_bb (ho2bbc) routine document. [Online]

http://www.nag.co.uk/numeric/CL/nagdoc_cl23/pdf/H/h02bbc.pdf.

30. —. nag_opt_qp (e04nfc) routine document. [Online]

http://www.nag.co.uk/numeric/CL/nagdoc_cl23/pdf/E04/e04nfc.pdf.

31. —. NAG chapter introduction - f08: Least Squares and Eigenvalue Problems.

[Online] http://www.nag.co.uk/numeric/CL/nagdoc_cl23/pdf/F08/f08intro.pdf.

32. Wikipedia. Norm (mathematics). [Online]

http://en.wikipedia.org/wiki/Norm_(mathematics).

33. Fernando, K.V. Practical Portfolio Optimization. [Online]

http://www.nag.co.uk/doc/techrep/index.html#np3484.

34. Numerical Algorithms Group. nag_nearest_correlation (g02aa) routine

document. [Online]

http://www.nag.co.uk/numeric/CL/nagdoc_cl23/pdf/G02/g02aac.pdf.

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37. Gill, P.E., et al. Users’ guide for LSSOL (Version 1.0) Report SOL 86-1.

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38. Numerical Algorithms Group. nag_opt_lin_lsq (e04nc) routine document.

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39. —. Demo Excel spreadsheet using NAG method to solve portfolio optimization

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