e04 Chapter Contents
e04 Chapter Introduction (PDF version)
NAG Library Manual

NAG Library Chapter Introduction

e04 – Minimizing or Maximizing a Function


2  Background to the Problems

1  Scope of the Chapter

An optimization problem involves minimizing a function (called the objective function) of several variables, possibly subject to restrictions on the values of the variables defined by a set of constraint functions. Most functions in the Library are concerned with function minimization only, since the problem of maximizing a given objective function F(x) is equivalent to minimizing -Fx. Some functions allow you to specify whether you are solving a minimization or maximization problem, carrying out the required transformation of the objective function in the latter case.
In general functions in this chapter find a local minimum of a function f, that is a point x* s.t. for all x near x* fxfx*.
The Chapter e05 contains functions to find the global minimum of a function f. At a global minimum x* fxfx* for all x.
The Chapter h contains functions typically regarded as belonging to the field of operations research.
This introduction is only a brief guide to the subject of optimization designed for the casual user. Anyone with a difficult or protracted problem to solve will find it beneficial to consult a more detailed text, such as Gill et al. (1981) or Fletcher (1987).
If you are unfamiliar with the mathematics of the subject you may find some sections difficult at first reading; if so, you should concentrate on Sections 2.1, 2.2, 2.5, 2.6 and 4.

2  Background to the Problems

2.1  Types of Optimization Problems

The solution of optimization problems by a single, all-purpose, method is cumbersome and inefficient. Optimization problems are therefore classified into particular categories, where each category is defined by the properties of the objective and constraint functions, as illustrated by some examples below.
Properties of Objective Function Properties of Constraints
Nonlinear Nonlinear
Sums of squares of nonlinear functions Sparse linear
Quadratic Linear
Sums of squares of linear functions Bounds
Linear None
For instance, a specific problem category involves the minimization of a nonlinear objective function subject to bounds on the variables. In the following sections we define the particular categories of problems that can be solved by functions contained in this chapter. Not every category is given special treatment in the current version of the Library; however, the long-term objective is to provide a comprehensive set of functions to solve problems in all such categories.

2.1.1  Unconstrained minimization

In unconstrained minimization problems there are no constraints on the variables. The problem can be stated mathematically as follows:
where xRn, that is, x = x1,x2,,xnT .

2.1.2  Nonlinear least squares problems

Special consideration is given to the problem for which the function to be minimized can be expressed as a sum of squared functions. The least squares problem can be stated mathematically as follows:
minimizex fTf=i=1mfi2x ,  xRn  
where the ith element of the m-vector f is the function fix.

2.1.3  Minimization subject to bounds on the variables

These problems differ from the unconstrained problem in that at least one of the variables is subject to a simple bound (or restriction) on its value, e.g., x510, but no constraints of a more general form are present.
The problem can be stated mathematically as follows:
minimizex Fx,  xRn  
subject to lixiui, for i=1,2,,n.
This format assumes that upper and lower bounds exist on all the variables. By conceptually allowing ui=+ and li=- all the variables need not be restricted.

2.1.4  Minimization subject to linear constraints

A general linear constraint is defined as a constraint function that is linear in more than one of the variables, e.g., 3x1+2x24. The various types of linear constraint are reflected in the following mathematical statement of the problem:
minimizex Fx,  xRn  
subject to the
equality constraints: aiT x=bi i=1,2,,m1;
inequality constraints: aiT xbi i=m1+1,m1+2,,m2;
  aiT xbi i=m2+1,m2+2,,m3;
range constraints: sj aiT xtj i=m3+1,m3+2,,m4;
bounds constraints: lixiui i=1,2,,n
where each ai is a vector of length n; bi, sj and tj are constant scalars; and any of the categories may be empty.
Although the bounds on xi could be included in the definition of general linear constraints, we prefer to distinguish between them for reasons of computational efficiency.
If Fx is a linear function, the linearly-constrained problem is termed a linear programming problem (LP); if Fx is a quadratic function, the problem is termed a quadratic programming problem (QP). For further discussion of LP and QP problems, including the dual formulation of such problems, see Dantzig (1963).

2.1.5  Minimization subject to nonlinear constraints

A problem is included in this category if at least one constraint function is nonlinear, e.g., x12+x3+x4-20. The mathematical statement of the problem is identical to that for the linearly-constrained case, except for the addition of the following constraints:
equality constraints: cix=0 i=1,2,,m5;
inequality constraints: cix0 i=m5+1,m5+2,,m6;
range constraints: vjcixwj i=m6+1,m6+2,,m7,
where each ci is a nonlinear function; vj and wj are constant scalars; and any category may be empty. Note that we do not include a separate category for constraints of the form cix0, since this is equivalent to -cix0.
Although the general linear constraints could be included in the definition of nonlinear constraints, again we prefer to distinguish between them for reasons of computational efficiency.
If Fx is a nonlinear function, the nonlinearly-constrained problem is termed a nonlinear programming problem (NLP). For further discussion of NLP problems, see Gill et al. (1981) or Fletcher (1987).

2.1.6  Minimization subject to bounds on the objective function

In all of the above problem categories it is assumed that
where a=- and b=+. Problems in which a and/or b are finite can be solved by adding an extra constraint of the appropriate type (i.e., linear or nonlinear) depending on the form of Fx. Further advice is given in Section 4.2.

2.1.7  Multi-objective optimization

Sometimes a problem may have two or more objective functions which are to be optimized at the same time. Such problems are called multi-object, multi-criteria or multi-attribute optimization. If the constraints are linear and the objectives are all linear then the terminology ‘goal programming’ is also used.
Techniques used in this chapter and in Chapter e05 may be employed to address such problems.

2.2  Geometric Representation and Terminology

To illustrate the nature of optimization problems it is useful to consider the following example in two dimensions:
(This function is used as the example function in the documentation for the unconstrained functions.)
Figure 1
Figure 1
Figure 1 is a contour diagram of Fx. The contours labelled F0,F1,,F4 are isovalue contours, or lines along which the function Fx takes specific constant values. The point x*=12,-1T is a local unconstrained minimum, that is, the value of Fx* (=0) is less than at all the neighbouring points. A function may have several such minima. The lowest of the local minima is termed a global minimum. In the problem illustrated in Figure 1, x* is the only local minimum. The point xs is said to be a saddle point because it is a minimum along the line AB, but a maximum along CD.
If we add the constraint x10 (a simple bound) to the problem of minimizing Fx, the solution remains unaltered. In Figure 1 this constraint is represented by the straight line passing through x1=0, and the shading on the line indicates the unacceptable region (i.e., x1<0). The region in Rn satisfying the constraints of an optimization problem is termed the feasible region. A point satisfying the constraints is defined as a feasible point.
If we add the nonlinear constraint c1x:x1+x2-x1x2-320, represented by the curved shaded line in Figure 1, then x* is not a feasible point because c1x*<0. The solution of the new constrained problem is xb1.1825,-1.7397T, the feasible point with the smallest function value (where Fxb3.0607).

2.2.1  Gradient vector

The vector of first partial derivatives of Fx is called the gradient vector, and is denoted by gx, i.e.,
gx= Fx x1 , Fx x2 ,, Fx xn T.  
For the function illustrated in Figure 1,
gx= Fx+ex18x1+ 4x2 ex1 4x2+ 4x1+ 2 .  
The gradient vector is of importance in optimization because it must be zero at an unconstrained minimum of any function with continuous first derivatives.

2.2.2  Hessian matrix

The matrix of second partial derivatives of a function is termed its Hessian matrix. The Hessian matrix of Fx is denoted by Gx, and its i,jth element is given by 2Fx/xixj. If Fx has continuous second derivatives, then Gx must be positive definite at any unconstrained minimum of F.

2.2.3  Jacobian matrix; matrix of constraint normals

In nonlinear least squares problems, the matrix of first partial derivatives of the vector-valued function fx is termed the Jacobian matrix of fx and its i,jth component is fi/xj.
The vector of first partial derivatives of the constraint cix is denoted by
aix= cix x1 , cix x2 ,, cix xn T.  
The matrix whose columns are the vectors ai is termed the matrix of constraint normals. At a point x^, the vector aix^ is orthogonal (normal) to the isovalue contour of cix passing through x^; this relationship is illustrated for a two-dimensional function in Figure 2.
Figure 2
Figure 2
Note that if cix is a linear constraint involving aiT x , then its vector of first partial derivatives is simply the vector ai.

2.3  Sufficient Conditions for a Solution

All nonlinear functions will be assumed to have continuous second derivatives in the neighbourhood of the solution.

2.3.1  Unconstrained minimization

The following conditions are sufficient for the point x* to be an unconstrained local minimum of Fx:
(i) gx*=0; and
(ii) Gx* is positive definite,
where g denotes the Euclidean length of g.

2.3.2  Minimization subject to bounds on the variables

At the solution of a bounds-constrained problem, variables which are not on their bounds are termed free variables. If it is known in advance which variables are on their bounds at the solution, the problem can be solved as an unconstrained problem in just the free variables; thus, the sufficient conditions for a solution are similar to those for the unconstrained case, applied only to the free variables.
Sufficient conditions for a feasible point x* to be the solution of a bounds-constrained problem are as follows:
(i) g-x*=0; and
(ii) G-x* is positive definite; and
(iii) gjx*<0,xj=uj; gjx*>0,xj=lj,
where g-x is the gradient of Fx with respect to the free variables, and G-x is the Hessian matrix of Fx with respect to the free variables. The extra condition (iii) ensures that Fx cannot be reduced by moving off one or more of the bounds.

2.3.3  Linearly-constrained minimization

For the sake of simplicity, the following description does not include a specific treatment of bounds or range constraints, since the results for general linear inequality constraints can be applied directly to these cases.
At a solution x*, of a linearly-constrained problem, the constraints which hold as equalities are called the active or binding constraints. Assume that there are t active constraints at the solution x*, and let A^ denote the matrix whose columns are the columns of A corresponding to the active constraints, with b^ the vector similarly obtained from b; then
The matrix Z is defined as an n×n-t matrix satisfying:
A^TZ=0; ZTZ=I.  
The columns of Z form an orthogonal basis for the set of vectors orthogonal to the columns of A^.
At the solution of a linearly-constrained problem, the projected gradient vector must be zero, which implies that the gradient vector gx* can be written as a linear combination of the columns of A^, i.e., gx*=i=1tλi*a^i=A^λ*. The scalar λi* is defined as the Lagrange multiplier corresponding to the ith active constraint. A simple interpretation of the ith Lagrange multiplier is that it gives the gradient of Fx along the ith active constraint normal; a convenient definition of the Lagrange multiplier vector (although not a recommended method for computation) is:
Sufficient conditions for x* to be the solution of a linearly-constrained problem are:
(i) x* is feasible, and A^Tx*=b^; and
(ii) gZx*=0, or equivalently, gx*=A^λ*; and
(iii) GZx* is positive definite; and
(iv) λi*>0 if λi* corresponds to a constraint a^iT x* b^i ;
λi*<0 if λi* corresponds to a constraint a^iT x* b^i .
The sign of λi* is immaterial for equality constraints, which by definition are always active.

2.3.4  Nonlinearly-constrained minimization

For nonlinearly-constrained problems, much of the terminology is defined exactly as in the linearly-constrained case. The set of active constraints at x again means the set of constraints that hold as equalities at x, with corresponding definitions of c^ and A^: the vector c^x contains the active constraint functions, and the columns of A^x are the gradient vectors of the active constraints. As before, Z is defined in terms of A^x as a matrix such that:
A^TZ=0; ZTZ=I  
where the dependence on x has been suppressed for compactness.
The projected gradient vector gZx is the vector ZTgx. At the solution x* of a nonlinearly-constrained problem, the projected gradient must be zero, which implies the existence of Lagrange multipliers corresponding to the active constraints, i.e., gx*=A^x*λ*.
The Lagrangian function is given by:
We define gLx as the gradient of the Lagrangian function; GLx as its Hessian matrix, and G^Lx as its projected Hessian matrix, i.e., G^L=ZTGLZ.
Sufficient conditions for x* to be the solution of a nonlinearly-constrained problem are:
(i) x* is feasible, and c^x*=0; and
(ii) gZx*=0, or, equivalently, gx*=A^x*λ*; and
(iii) G^Lx* is positive definite; and
(iv) λi*>0 if λi* corresponds to a constraint of the form c^i0.
The sign of λi* is immaterial for equality constraints, which by definition are always active.
Note that condition (ii) implies that the projected gradient of the Lagrangian function must also be zero at x*, since the application of ZT annihilates the matrix A^x*.

2.4  Background to Optimization Methods

All the algorithms contained in this chapter generate an iterative sequence x k  that converges to the solution x* in the limit, except for some special problem categories (i.e., linear and quadratic programming). To terminate computation of the sequence, a convergence test is performed to determine whether the current estimate of the solution is an adequate approximation. The convergence tests are discussed in Section 2.6.
Most of the methods construct a sequence x k  satisfying:
x k+1 =x k +α k p k ,  
where the vector p k  is termed the direction of search, and α k  is the steplength. The steplength α k  is chosen so that Fx k+1 <Fx k  and is computed using one of the techniques for one-dimensional optimization referred to in Section 2.4.1.

2.4.1  One-dimensional optimization

The Library contains two special functions for minimizing a function of a single variable. Both functions are based on safeguarded polynomial approximation. One function requires function evaluations only and fits a quadratic polynomial whilst the other requires function and gradient evaluations and fits a cubic polynomial. See Section 4.1 of Gill et al. (1981).

2.4.2  Methods for unconstrained optimization

The distinctions among methods arise primarily from the need to use varying levels of information about derivatives of Fx in defining the search direction. We describe three basic approaches to unconstrained problems, which may be extended to other problem categories. Since a full description of the methods would fill several volumes, the discussion here can do little more than allude to the processes involved, and direct you to other sources for a full explanation.
(a) Newton-type Methods (Modified Newton Methods)
Newton-type methods use the Hessian matrix Gx k , or a finite difference approximation to Gx k , to define the search direction. The functions in the Library either require a function that computes the elements of Gx k  directly, or they approximate Gx k  by finite differences.
Newton-type methods are the most powerful methods available for general problems and will find the minimum of a quadratic function in one iteration. See Sections 4.4 and 4.5.1 of Gill et al. (1981).
(b) Quasi-Newton Methods
Quasi-Newton methods approximate the Hessian Gxk by a matrix Bk which is modified at each iteration to include information obtained about the curvature of F along the current search direction pk. Although not as robust as Newton-type methods, quasi-Newton methods can be more efficient because Gxk is not computed directly, or approximated by finite differences. Quasi-Newton methods minimize a quadratic function in n iterations, where n is the number of variables. See Section 4.5.2 of Gill et al. (1981).
(c) Conjugate-gradient Methods
Unlike Newton-type and quasi-Newton methods, conjugate-gradient methods do not require the storage of an n by n matrix and so are ideally suited to solve large problems. Conjugate-gradient type methods are not usually as reliable or efficient as Newton-type, or quasi-Newton methods. See Section 4.8.3 of Gill et al. (1981).

2.4.3  Methods for nonlinear least squares problems

These methods are similar to those for unconstrained optimization, but exploit the special structure of the Hessian matrix to give improved computational efficiency.
the Hessian matrix Gx is of the form
Gx = 2 JxT Jx + i=1m fix Gix ,  
where Jx is the Jacobian matrix of fx, and Gix is the Hessian matrix of fix.
In the neighbourhood of the solution, fx is often small compared to JxT Jx  (for example, when fx represents the goodness-of-fit of a nonlinear model to observed data). In such cases, 2 JxT Jx  may be an adequate approximation to Gx, thereby avoiding the need to compute or approximate second derivatives of fix. See Section 4.7 of Gill et al. (1981).

2.4.4  Methods for handling constraints

Bounds on the variables are dealt with by fixing some of the variables on their bounds and adjusting the remaining free variables to minimize the function. By examining estimates of the Lagrange multipliers it is possible to adjust the set of variables fixed on their bounds so that eventually the bounds active at the solution should be correctly identified. This type of method is called an active set method. One feature of such methods is that, given an initial feasible point, all approximations x k  are feasible. This approach can be extended to general linear constraints. At a point, x, the set of constraints which hold as equalities being used to predict, or approximate, the set of active constraints is called the working set.
Nonlinear constraints are more difficult to handle. If at all possible, it is usually beneficial to avoid including nonlinear constraints during the formulation of the problem. The methods currently implemented in the Library handle nonlinearly constrained problems by transforming them into a sequence of quadratic programming problems. A feature of such methods is that x k  is not guaranteed to be feasible except in the limit, and this is certainly true of the functions currently in the Library. See Chapter 6, particularly Sections 6.4 and 6.5, of Gill et al. (1981).
Anyone interested in a detailed description of methods for optimization should consult the references.

2.4.5  Methods for handling multi-objective optimization

Suppose we have objective functions fix, i>1, all of which we need to minimize at the same time. There are two main approaches to this problem:
(a) Combine the individual objectives into one composite objective. Typically this might be a weighted sum of the objectives, e.g.,
w1 f1x + w2 f2x + + wn fnx  
Here you choose the weights to express the relative importance of the corresponding objective. Ideally each of the fix should be of comparable size at a solution.
(b) Order the objectives in order of importance. Suppose fi are ordered such that fix is more important than fi+1x, for i=1,2,,n-1. Then in the lexicographical approach to multi-objective optimization a sequence of subproblems are solved. Firstly solve the problem for objective function f1x and denote by r1 the value of this minimum. If i-1 subproblems have been solved with results ri-1 then subproblem i becomes minfix subject to rkfkxrk, for k=1,2,,i-1 plus the other constraints.
Clearly the bounds on fk might be relaxed at your discretion.
In general, if NAG functions from the Chapter e04 are used then only local minima are found. This means that a better solution to an individual objective might be found without worsening the optimal solutions to the other objectives. Ideally you seek a Pareto solution; one in which an improvement in one objective can only be achieved by a worsening of another objective.
To obtain a Pareto solution functions from Chapter e05 might be used or, alternatively, a pragmatic attempt to derive a global minimum might be tried (see nag_glopt_nlp_multistart_sqp (e05ucc)). In this approach a variety of different minima are computed for each subproblem by starting from a range of different starting points. The best solution achieved is taken to be the global minimum. The more starting points chosen the greater confidence you might have in the computed global minimum.

2.5  Scaling

Scaling (in a broadly defined sense) often has a significant influence on the performance of optimization methods. Since convergence tolerances and other criteria are necessarily based on an implicit definition of ‘small’ and ‘large’, problems with unusual or unbalanced scaling may cause difficulties for some algorithms. Although there are currently no user-callable scaling functions in the Library, scaling is automatically performed by default in the functions which solve sparse LP, QP or NLP problems and in some newer dense solver functions. The following sections present some general comments on problem scaling.

2.5.1  Transformation of variables

One method of scaling is to transform the variables from their original representation, which may reflect the physical nature of the problem, to variables that have certain desirable properties in terms of optimization. It is generally helpful for the following conditions to be satisfied:
(i) the variables are all of similar magnitude in the region of interest;
(ii) a fixed change in any of the variables results in similar changes in Fx. Ideally, a unit change in any variable produces a unit change in Fx;
(iii) the variables are transformed so as to avoid cancellation error in the evaluation of Fx.
Normally, you should restrict yourself to linear transformations of variables, although occasionally nonlinear transformations are possible. The most common such transformation (and often the most appropriate) is of the form
where D is a diagonal matrix with constant coefficients. Our experience suggests that more use should be made of the transformation
where v is a constant vector.
Consider, for example, a problem in which the variable x3 represents the position of the peak of a Gaussian curve to be fitted to data for which the extreme values are 150 and 170; therefore x3 is known to lie in the range 150170. One possible scaling would be to define a new variable x-3, given by
A better transformation, however, is given by defining x-3 as
Frequently, an improvement in the accuracy of evaluation of Fx can result if the variables are scaled before the functions to evaluate Fx are coded. For instance, in the above problem just mentioned of Gaussian curve-fitting, x3 may always occur in terms of the form x3-xm, where xm is a constant representing the mean peak position.

2.5.2  Scaling the objective function

The objective function has already been mentioned in the discussion of scaling the variables. The solution of a given problem is unaltered if Fx is multiplied by a positive constant, or if a constant value is added to Fx. It is generally preferable for the objective function to be of the order of unity in the region of interest; thus, if in the original formulation Fx is always of the order of 10+5 (say), then the value of Fx should be multiplied by 10-5 when evaluating the function within an optimization function. If a constant is added or subtracted in the computation of Fx, usually it should be omitted, i.e., it is better to formulate Fx as x12+x22 rather than as x12+x22+1000 or even x12+x22+1. The inclusion of such a constant in the calculation of Fx can result in a loss of significant figures.

2.5.3  Scaling the constraints

A ‘well scaled’ set of constraints has two main properties. Firstly, each constraint should be well-conditioned with respect to perturbations of the variables. Secondly, the constraints should be balanced with respect to each other, i.e., all the constraints should have ‘equal weight’ in the solution process.
The solution of a linearly- or nonlinearly-constrained problem is unaltered if the ith constraint is multiplied by a positive weight wi. At the approximation of the solution determined by a Library function, any active linear constraints will (in general) be satisfied ‘exactly’ (i.e., to within the tolerance defined by machine precision) if they have been properly scaled. This is in contrast to any active nonlinear constraints, which will not (in general) be satisfied ‘exactly’ but will have ‘small’ values (for example, c^1x*=10-8, c^2x*=-10 -6, and so on). In general, this discrepancy will be minimized if the constraints are weighted so that a unit change in x produces a similar change in each constraint.
A second reason for introducing weights is related to the effect of the size of the constraints on the Lagrange multiplier estimates and, consequently, on the active set strategy. This means that different sets of weights may cause an algorithm to produce different sequences of iterates. Additional discussion is given in Gill et al. (1981).

2.6  Analysis of Computed Results

2.6.1  Convergence criteria

The convergence criteria inevitably vary from function to function, since in some cases more information is available to be checked (for example, is the Hessian matrix positive definite?), and different checks need to be made for different problem categories (for example, in constrained minimization it is necessary to verify whether a trial solution is feasible). Nonetheless, the underlying principles of the various criteria are the same; in non-mathematical terms, they are:
(i) is the sequence x k  converging?
(ii) is the sequence F k  converging?
(iii) are the necessary and sufficient conditions for the solution satisfied?
The decision as to whether a sequence is converging is necessarily speculative. The criterion used in the present functions is to assume convergence if the relative change occurring between two successive iterations is less than some prescribed quantity. Criterion (iii) is the most reliable but often the conditions cannot be checked fully because not all the required information may be available.

2.6.2  Checking results

Little a priori guidance can be given as to the quality of the solution found by a nonlinear optimization algorithm, since no guarantees can be given that the methods will not fail. Therefore, you should always check the computed solution even if the function reports success. Frequently a ‘solution’ may have been found even when the function does not report a success. The reason for this apparent contradiction is that the function needs to assess the accuracy of the solution. This assessment is not an exact process and consequently may be unduly pessimistic. Any ‘solution’ is in general only an approximation to the exact solution, and it is possible that the accuracy you have specified is too stringent.
Further confirmation can be sought by trying to check whether or not convergence tests are almost satisfied, or whether or not some of the sufficient conditions are nearly satisfied. When it is thought that a function has returned a value of fail.code other than NE_NOERROR only because the requirements for ‘success’ were too stringent it may be worth restarting with increased convergence tolerances.
For nonlinearly-constrained problems, check whether the solution returned is feasible, or nearly feasible; if not, the solution returned is not an adequate solution.
Confidence in a solution may be increased by resolving the problem with a different initial approximation to the solution. See Section 8.3 of Gill et al. (1981) for further information.

2.6.3  Monitoring progress

Many of the functions in the chapter have facilities to allow you to monitor the progress of the minimization process, and you are encouraged to make use of these facilities. Monitoring information can be a great aid in assessing whether or not a satisfactory solution has been obtained, and in indicating difficulties in the minimization problem or in the ability of the function to cope with the problem.
The behaviour of the function, the estimated solution and first derivatives can help in deciding whether a solution is acceptable and what to do in the event of a return with a fail.code other than NE_NOERROR.

2.6.4  Confidence intervals for least squares solutions

When estimates of the parameters in a nonlinear least squares problem have been found, it may be necessary to estimate the variances of the parameters and the fitted function. These can be calculated from the Hessian of Fx at the solution.
In many least squares problems, the Hessian is adequately approximated at the solution by G=2JTJ (see Section 2.4.3). The Jacobian, J, or a factorization of J is returned by all the comprehensive least squares functions and, in addition, a function is available in the Library to estimate variances of the parameters following the use of most of the nonlinear least squares functions, in the case that G=2JTJ is an adequate approximation.
Let H be the inverse of G, and S be the sum of squares, both calculated at the solution x-; an unbiased estimate of the variance of the ith parameter xi is
varx-i=2S m-n Hii  
and an unbiased estimate of the covariance of x-i and x-j is
covarx-i,x-j=2S m-n Hij.  
If x* is the true solution, then the 1001-β% confidence interval on x- is
x-i-varx-i. t1-β/2,m-n<xi*<x-i+varx-i.t1-β/2,m-n,  i=1,2,,n  
where t1-β/2,m-n is the 1001-β/2 percentage point of the t-distribution with m-n degrees of freedom.
In the majority of problems, the residuals fi, for i=1,2,,m, contain the difference between the values of a model function ϕz,x calculated for m different values of the independent variable z, and the corresponding observed values at these points. The minimization process determines the parameters, or constants x, of the fitted function ϕz,x. For any value, z-, of the independent variable z, an unbiased estimate of the variance of ϕ is
varϕ=2S m-n i=1n j=1n ϕ xi z- ϕ xj z- Hij.  
The 1001-β% confidence interval on F at the point z- is
For further details on the analysis of least squares solutions see Bard (1974) and Wolberg (1967).

3  Optional Facilities

The comments in this section do not apply to functions introduced at Mark 8 and later, viz. nag_opt_sparse_convex_qp_solve (e04nqc), nag_opt_nlp_revcomm (e04ufc), nag_opt_sparse_nlp_solve (e04vhc) and nag_opt_nlp_solve (e04wdc). For details of their optional facilities please refer to their individual documents.
The optimization functions of this chapter provide a range of optional facilities: these offer the possibility of fine control over many of the algorithmic parameters and the means of adjusting the level and nature of the printed results.
Control of these optional facilities is exercised by a structure of type Nag_E04_Opt, the members of the structure being optional input or output arguments to the function. After declaring the structure variable, which is named options in this manual, you must initialize the structure by passing its address in a call to the utility function nag_opt_init (e04xxc). Selected members of the structure may then be set to your required values and the address of the structure passed to the optimization function. Any member which has not been set by you will indicate to the optimization function that the default value should be used for this argument. A more detailed description of this process is given in Section 3.4.
The optimization process may sometimes terminate before a satisfactory answer has been found, for instance when the limit on the number of iterations has been reached. In such cases you may wish to re-enter the function making use of the information already obtained. Functions nag_opt_conj_grad (e04dgc), nag_opt_lsq_no_deriv (e04fcc) and nag_opt_lsq_deriv (e04gbc) can simply be re-entered but the functions nag_opt_bounds_deriv (e04kbc), nag_opt_lp (e04mfc), nag_opt_lin_lsq (e04ncc), nag_opt_qp (e04nfc), nag_opt_sparse_convex_qp (e04nkc), nag_opt_nlp (e04ucc), nag_opt_nlin_lsq (e04unc) and nag_opt_nlp_solve (e04wdc) have a structure member which needs to be set appropriately if the function is to make use of information from the previous call. The member is named start in the functions listed.

3.1  Control of Printed Output

Results from the optimization process are printed by default on the stdout (standard output) stream. These include the results after each iteration and the final results at termination of the search process. The amount of detail printed out may be increased or decreased by setting the optional argument Print Level, i.e., the structure member Print Level. This member is an enum type, Nag_PrintType, and an example value is Nag_Soln which when assigned to Print Level will cause the optimization function to print only the final result; all intermediate results printout is suppressed.
If the results printout is not in the desired form then it may be switched off, by setting Print Level=Nag_NoPrint, or alternatively you can supply your own function to print out or make use of both the intermediate and final results. Such a function would be assigned to the pointer to function member print_fun; the user-defined function would then be called in preference to the NAG print function.
In addition to the results, the values of the arguments to the optimization function are printed out when the function is entered; the Boolean member list may be set to Nag_FALSE if this listing is not required.
Printing may be output to a named file rather than to stdout by providing the name of the file in the options character array member outfile. Error messages will still appear on stderr, if fail.print=Nag_TRUE or the fail argument is not supplied (see the Section 3.6 in the Essential Introduction for details of error handling within the library).

3.2  Memory Management

The options structure contains a number of pointers for the input of data and the output of results. The optimization functions will manage the allocation of memory to these pointers; when all calls to these functions have been completed then a utility function nag_opt_free (e04xzc) can be called by your program to free the NAG allocated memory which is no longer required.
If the calling function is part of a larger program then this utility function allows you to conserve memory by freeing the NAG allocated memory before the options structure goes out of scope. nag_opt_free (e04xzc) can free all NAG allocated memory in a single call, but it may also be used selectively. In this case the memory assigned to certain pointers may be freed leaving the remaining memory still available; pointers to this memory and the results it contains may then be passed to other functions in your program without passing the structure and all its associated memory.
Although the NAG C Library optimization functions will manage all memory allocation and deallocation, it may occasionally be necessary for you to allocate memory to the options structure from within the calling program before entering the optimization function.
An example of this is where you store information in a file from an optimization run and at a later date wish to use that information to solve a similar optimization problem or the same one under slightly changed conditions. The pointer state, for example, would need to be allocated memory by you before the status of the constraints could be assigned from the values in the file. The member Cold Start would need to be appropriately set for functions nag_opt_lp (e04mfc) and nag_opt_qp (e04nfc).
If you assign memory to a pointer within the options structure then the deallocation of this memory must also be performed by you; the utility function nag_opt_free (e04xzc) will only free memory allocated by NAG C Library optimization functions. When your allocated memory is freed using the standard C library function free() then the pointer should be set to NULL immediately afterwards; this will avoid possible confusion in the NAG memory management system if a NAG function is subsequently entered. In general we recommend the use of NAG_ALLOC, NAG_REALLOC and NAG_FREE for allocating and freeing memory used with NAG functions.

3.3  Reading Optional Argument Values From a File

Optional argument values may be placed in a file by you and the function nag_opt_read (e04xyc) used to read the file and assign the values to the options structure. This utility function permits optional argument values to be supplied in any order and altered without recompilation of the program. The values read are also checked before assignment to ensure they are in the correct range for the specified option. Pointers within the options structure cannot be assigned to using nag_opt_read (e04xyc).

3.4  Method of Setting Optional Arguments

The method of using and setting the optional arguments is:
step 1declare a structure of type Nag_E04_Opt.
step 2initialize the structure using nag_opt_init (e04xxc).
step 3assign values to the structure.
step 4pass the address of the structure to the optimization function.
step 5call nag_opt_free (e04xzc) to free any memory allocated by the optimization function.
If after step 4, it is wished to re-enter the optimization function, then step 3 can be returned to directly, i.e., step 5 need only be executed when all calls to the optimization function have been made.
At step 3, values can be assigned directly and/or by means of the option file reading function nag_opt_read (e04xyc). If values are only assigned from the options file then step 2 need not be performed as nag_opt_read (e04xyc) will automatically call nag_opt_init (e04xxc) if the structure has not been initialized.

4  Recommendations on Choice and Use of Available Functions

The choice of function depends on several factors: the type of problem (unconstrained, etc.); the level of derivative information available (function values only, etc.); your experience (there are easy-to-use versions of some functions); whether or not storage is a problem; and whether computational time has a high priority. Not all choices are catered for in the current version of the Library.

4.1  Service Functions

One of the most common errors in the use of optimization functions is that user-supplied functions do not evaluate the relevant partial derivatives correctly. Because exact gradient information normally enhances efficiency in all areas of optimization, you are encouraged to provide analytical derivatives whenever possible. However, mistakes in the computation of derivatives can result in serious and obscure run-time errors. Consequently, service functions are provided to perform an elementary check on the gradients you supplied. These functions are inexpensive to use in terms of the number of calls they require to user-supplied functions.
The appropriate checking function is as follows:
Minimization function Checking function(s)
nag_opt_bounds_2nd_deriv (e04lbc) nag_opt_check_deriv (e04hcc) and nag_opt_check_2nd_deriv (e04hdc)
nag_opt_lsq_deriv (e04gbc) nag_opt_lsq_check_deriv (e04yac)
It should be noted that functions nag_opt_nlp (e04ucc), nag_opt_nlp_revcomm (e04ufc), nag_opt_nlp_sparse (e04ugc), nag_opt_sparse_nlp_solve (e04vhc) and nag_opt_nlp_solve (e04wdc) each incorporate a check on the gradients being supplied. This involves verifying the gradients at the first point that satisfies the linear constraints and bounds. There is also an option to perform a more reliable (but more expensive) check on the individual gradient elements being supplied. Note that the checks are not infallible.
A second type of service function computes a set of finite differences to be used when approximating first derivatives. Such differences are required as input arguments by some functions that use only function evaluations.
nag_opt_lsq_covariance (e04ycc) estimates selected elements of the variance-covariance matrix for the computed regression parameters following the use of a nonlinear least squares function.
nag_opt_estimate_deriv (e04xac) estimates the gradient and Hessian of a function at a point, given a function to calculate function values only, or estimates the Hessian of a function at a point, given a function to calculate function and gradient values.

4.2  Function Evaluations at Infeasible Points

All the functions for constrained problems will ensure that any evaluations of the objective function occur at points which approximately satisfy any simple bounds or linear constraints. Satisfaction of such constraints is only approximate because functions which estimate derivatives by finite differences may require function evaluations at points which just violate such constraints even though the current iteration just satisfies them.
There is no attempt to ensure that the current iteration satisfies any nonlinear constraints. If you wish to prevent your objective function being evaluated outside some known region (where it may be undefined or not practically computable), you may try to confine the iteration within this region by imposing suitable simple bounds or linear constraints (but beware as this may create new local minima where these constraints are active).
Note also that some functions allow you to return the argument (commflag) with a negative value to force an immediate clean exit from the minimization function when the objective function (or nonlinear constraints where appropriate) cannot be evaluated. Please note that nag_opt_sparse_convex_qp_solve (e04nqc), nag_opt_sparse_nlp_solve (e04vhc) and nag_opt_nlp_solve (e04wdc) use the user-supplied function imode instead of commflag.

4.3  Related Problems

Apart from the standard types of optimization problem, there are other related problems which can be solved by functions in this or other chapters of the Library.
nag_ip_bb (h02bbc) solves dense integer LP problems.
Several functions in Chapters f04 and f08 solve linear least squares problems, i.e., minimizei=1mri x 2 where rix=bi-j=1naijxj.
nag_lone_fit (e02gac) solves an overdetermined system of linear equations in the l1 norm, i.e., minimizes i=1mrix, with ri as above.
nag_linf_fit (e02gcc) solves an overdetermined system of linear equations in the l norm, i.e., minimizes maxirix, with ri as above.
Chapter e05 contains functions for global minimization.
Section 2.4.5 describes how a multi-objective optimization problem might be addressed using functions from this chapter and from Chapter e05.

4.4  Choosing Between Variant Functions for Some Problems

As evidenced by the wide variety of functions available in Chapter e04, it is clear that no single algorithm can solve all optimization problems. It is important to try to match the problem to the most suitable function, and that is what the decision trees in Section 5 help to do.
Sometimes in Chapter e04 more than one function is available to solve precisely the same minimization problem. Thus, for example, the general nonlinear programming functions nag_opt_nlp (e04ucc) and nag_opt_nlp_solve (e04wdc) are based on similar methods. Experience shows that although both functions can usually solve the same problem and get similar results, sometimes one function will be faster, sometimes one might find a different local minimum to the other, or, in difficult cases, one function may obtain a solution when the other one fails.
After using one of these functions, if the results obtained are unacceptable for some reason, it may be worthwhile trying the other function instead. In the absence of any other information, in the first instance you are recommended to try using nag_opt_nlp (e04ucc), and if that proves unsatisfactory, try using nag_opt_nlp_solve (e04wdc). Although the algorithms used are very similar, the two functions each have slightly different optional arguments which may allow the course of the computation to be altered in different ways.
Other pairs of functions which solve the same kind of problem are nag_opt_sparse_convex_qp_solve (e04nqc) (recommended first choice) or nag_opt_sparse_convex_qp (e04nkc), for sparse quadratic or linear programming problems, and nag_opt_nlp_sparse (e04ugc) or nag_opt_sparse_nlp_solve (e04vhc), for sparse nonlinear programming. In these cases the argument lists are not so similar as nag_opt_nlp (e04ucc) or nag_opt_nlp_solve (e04wdc), but the same considerations apply.

5  Decision Trees

Tree 1: Selection chart for unconstrained problems

Only one variable?   Are first derivatives available?   e04bbc
  no   no
Does the function have many discontinuities?   e04cbc
Is store size a problem?   e04dgc
Is the function a sum of squares?   Are first derivatives available?   e04gbc
  no   no
Are first derivatives available?   Are second derivatives available?   e04lbc
  no   no
e04ugc, e04vhc or e04wdc
e04ucc, e04ugc, e04vhc or e04wdc

Tree 2: Selection chart for bound-constrained, linearly-constrained and nonlinearly-constrained problems

Are there any nonlinear constraints?   Is the objective function a sum of squares? (A least squares problem)   e04unc
  no   no
Are the constraints sparse?   e04ugc and e04vhc
e04ucc or e04wdc
Is the objective function linear? (An LP problem)   Tree 3
Is the objective function quadratic? (A QP or least squares problem)   Is the problem a least squares problem?   e04ncc
  no   no
Tree 3
Is the objective function a sum of squares? (A least squares problem)   e04unc
Are the constraints simple bounds?   Are the first derivatives available?   Are the second derivatives available?   e04lbc
  no   no   no
e04ucc, e04ugc, e04vhc or e04wdc
e04jcc, e04ucc, e04ugc, e04vhc or e04wdc
e04ucc, e04ugc, e04vhc or e04wdc

Tree 3: Linear and Quadratic Programming (LP and QP)

Is the linear constraint matrix sparse?   e04nqc, e04nkc, e04vhc and e04ugc
Is the objective function linear (an LP problem)?   e04mfc
Is the QP problem convex?   e04ncc

6  Functionality Index

Bound constrained nonlinear minimization, 
    first derivatives required nag_opt_bounds_deriv (e04kbc)
Constrained minimum of a sum of squares, nonlinear constraints, 
    using function values and optionally first derivatives, sequential QP method, 
        (dense) nag_opt_nlin_lsq (e04unc)
Convex QP problem or linearly-constrained linear least squares problem (dense) nag_opt_lin_lsq (e04ncc)
Linear least squares with bounds on the variables nag_opt_bnd_lin_lsq (e04pcc)
Linear programming (LP) problem (dense) nag_opt_lp (e04mfc)
LP or QP problem (sparse) nag_opt_sparse_convex_qp (e04nkc)
LP or QP problem (sparse) (recommended – see Section 4.4) nag_opt_sparse_convex_qp_solve (e04nqc)
Minimum, function of one variable, 
    using first derivative nag_opt_one_var_deriv (e04bbc)
    using function values only nag_opt_one_var_no_deriv (e04abc)
Minimum, function of several variables, nonlinear constraints, 
    using function values and optionally first derivatives, sequential QP method, 
        dense nag_opt_nlp_solve (e04wdc)
        dense (recommended – see Section 4.4) nag_opt_nlp (e04ucc)
        sparse nag_opt_sparse_nlp_solve (e04vhc)
        sparse (recommended – see Section 4.4) nag_opt_nlp_sparse (e04ugc)
Minimum, function of several variables, nonlinear constraints (comprehensive), 
    using function values and optionally first derivatives, sequential QP method, 
        reverse communication (dense) nag_opt_nlp_revcomm (e04ufc)
Minimum, function of several variables, simple bounds, 
    using first and second derivatives, modified Newton algorithm nag_opt_bounds_2nd_deriv (e04lbc)
Minimum, function of several variables, simple bounds (easy-to-use), 
    using function values only, by quadratic approximation nag_opt_bounds_qa_no_deriv (e04jcc)
Quadratic programming (QP) problem (dense) nag_opt_qp (e04nfc)
Service functions, 
    check user's function for calculating, 
        first derivatives of function nag_opt_check_deriv (e04hcc)
        Jacobian of first derivatives nag_opt_lsq_check_deriv (e04yac)
        second derivatives of function nag_opt_check_2nd_deriv (e04hdc)
    convert MPS data file defining LP or QP problem to format required by nag_opt_sparse_convex_qp_solve (e04nqc) (recommended) nag_opt_miqp_mps_read (e04mxc)
    convert MPSX data file defining LP or QP problem to format required by nag_opt_sparse_convex_qp (e04nkc) nag_opt_sparse_mps_free (e04myc)
    convert MPSX data file defining LP or QP problem to format required by nag_opt_sparse_convex_qp_solve (e04nqc) nag_opt_sparse_mps_read (e04mzc)
    covariance matrix for nonlinear least squares problem nag_opt_lsq_covariance (e04ycc)
    determine Jacobian sparsity structure before a call of nag_opt_sparse_nlp_solve (e04vhc) nag_opt_sparse_nlp_jacobian (e04vjc)
    estimate gradient and/or Hessian of a function nag_opt_estimate_deriv (e04xac)
    general option setting, 
        initialization function nag_opt_init (e04xxc)
        memory freeing function nag_opt_free (e04xzc)
        read options from a text file nag_opt_read (e04xyc)
    initialization function for, 
        nag_opt_sparse_convex_qp_solve (e04nqc) nag_opt_sparse_convex_qp_init (e04npc)
        nag_opt_nlp_revcomm (e04ufc) nag_opt_nlp_revcomm_init (e04wbc)
        nag_opt_sparse_nlp_solve (e04vhc) nag_opt_sparse_nlp_init (e04vgc)
        nag_opt_nlp_solve (e04wdc) nag_opt_nlp_init (e04wcc)
    retrieve integer optional argument values used by, 
        nag_opt_sparse_convex_qp_solve (e04nqc) nag_opt_sparse_convex_qp_option_get_integer (e04nxc)
        nag_opt_sparse_nlp_solve (e04vhc) nag_opt_sparse_nlp_option_get_integer (e04vrc)
        nag_opt_nlp_solve (e04wdc) nag_opt_nlp_option_get_integer (e04wkc)
    retrieve real optional argument values used by, 
        nag_opt_sparse_convex_qp_solve (e04nqc) nag_opt_sparse_convex_qp_option_get_double (e04nyc)
        nag_opt_sparse_nlp_solve (e04vhc) nag_opt_sparse_nlp_option_get_double (e04vsc)
        nag_opt_nlp_solve (e04wdc) nag_opt_nlp_option_get_double (e04wlc)
    supply integer optional argument values to, 
        nag_opt_sparse_convex_qp_solve (e04nqc) nag_opt_sparse_convex_qp_option_set_integer (e04ntc)
        nag_opt_sparse_nlp_solve (e04vhc) nag_opt_sparse_nlp_option_set_integer (e04vmc)
        nag_opt_nlp_solve (e04wdc) nag_opt_nlp_option_set_integer (e04wgc)
    supply optional argument values from external file for, 
        nag_opt_sparse_convex_qp_solve (e04nqc) nag_opt_sparse_convex_qp_option_set_file (e04nrc)
        nag_opt_nlp (e04ucc) nag_opt_nlp_revcomm_option_set_file (e04udc)
        nag_opt_sparse_nlp_solve (e04vhc) nag_opt_sparse_nlp_option_set_file (e04vkc)
        nag_opt_nlp_solve (e04wdc) nag_opt_nlp_option_set_file (e04wec)
    supply optional argument values to, 
        nag_opt_sparse_convex_qp_solve (e04nqc) nag_opt_sparse_convex_qp_option_set_string (e04nsc)
        nag_opt_nlp (e04ucc) nag_opt_nlp_revcomm_option_set_string (e04uec)
        nag_opt_sparse_nlp_solve (e04vhc) nag_opt_sparse_nlp_option_set_string (e04vlc)
        nag_opt_nlp_solve (e04wdc) nag_opt_nlp_option_set_string (e04wfc)
    supply real optional argument values to, 
        nag_opt_sparse_convex_qp_solve (e04nqc) nag_opt_sparse_convex_qp_option_set_double (e04nuc)
        nag_opt_sparse_nlp_solve (e04vhc) nag_opt_sparse_nlp_option_set_double (e04vnc)
        nag_opt_nlp_solve (e04wdc) nag_opt_nlp_option_set_double (e04whc)
Unconstrained minimum, function of several variables, 
    using first derivatives, pre-conditioned conjugate gradient algorithm nag_opt_conj_grad (e04dgc)
    using function values only, simplex algorithm nag_opt_simplex_easy (e04cbc)
Unconstrained minimum of a sum of squares (comprehensive): 
    using first derivatives, 
        combined Gauss–Newton and quasi-Newton algorithm nag_opt_lsq_deriv (e04gbc)
    using function values only, 
        combined Gauss–Newton and modified Newton algorithm nag_opt_lsq_no_deriv (e04fcc)

7  Auxiliary Functions Associated with Library Function Arguments


8  Functions Withdrawn or Scheduled for Withdrawal

The following lists all those functions that have been withdrawn since Mark 23 of the Library or are scheduled for withdrawal at one of the next two marks.
Mark of

Replacement Function(s)
nag_opt_simplex (e04ccc)24nag_opt_simplex_easy (e04cbc)
nag_opt_bounds_no_deriv (e04jbc)26nag_opt_nlp (e04ucc)

9  References

Bard Y (1974) Nonlinear Parameter Estimation Academic Press
Dantzig G B (1963) Linear Programming and Extensions Princeton University Press
Fletcher R (1987) Practical Methods of Optimization (2nd Edition) Wiley
Gill P E and Murray W (ed.) (1974) Numerical Methods for Constrained Optimization Academic Press
Gill P E, Murray W and Wright M H (1981) Practical Optimization Academic Press
Murray W (ed.) (1972) Numerical Methods for Unconstrained Optimization Academic Press
Wolberg J R (1967) Prediction Analysis Van Nostrand

e04 Chapter Contents
e04 Chapter Introduction (PDF version)
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015