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 methods 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 methods 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 methods in this chapter find a local minimum of a function f, that is a point x* s.t. for all x near x*fx≥fx*.

The E05 class contains methods to find the global minima of a function f. At a global minimum x* fx≥fx* for all x.

The H (not in this release) contains methods 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 [Types of Optimization Problems], [Geometric Representation and Terminology], [Scaling], [Analysis of Computed Results] and [Recommendations on Choice and Use of Available Methods].

Syntax

C#
public static class E04

Visual Basic (Declaration)
Public NotInheritable Class E04

Visual C++
public ref class E04 abstract sealed

F#
[<AbstractClassAttribute>] [<SealedAttribute>] type E04 = class end

Background to the Problems

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 methods 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 methods to solve problems in all such categories.

Unconstrained minimization

In unconstrained minimization problems there are no constraints on the variables. The problem can be stated mathematically as follows:

minimizexFx

where x∈Rn, that is, x = x1,x2,…,xnT .

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 , x∈Rn

where the ith element of the m-vector f is the function fix.

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., x5≤10, but no constraints of a more general form are present.

The problem can be stated mathematically as follows:

minimizex Fx, x∈Rn

subject to li≤xi≤ui, 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.

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+2x2≥4. The various types of linear constraint are reflected in the following mathematical statement of the problem:

minimizex Fx, x∈Rn

subject to the

equality constraints:	aiT x=bi	i=1,2,…,m1;
inequality constraints:	aiT x≥bi	i=m1+1,m1+2,…,m2;
	aiT x≤bi	i=m2+1,m2+2,…,m3;
range constraints:	sj ≤ aiT x≤tj	i=m3+1,m3+2,…,m4;
		j=1,2,…,m4-m3;
bounds constraints:	li≤xi≤ui	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).

Minimization subject to nonlinear constraints

A problem is included in this category if at least one constraint function is nonlinear, e.g., x12+x3+x4-2≥0. 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:	cix≥0	i=m5+1,m5+2,…,m6;
range constraints:	vj≤cix≤wj	i=m6+1,m6+2,…,m7,
		j=1,2,…,m7-m6

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 cix≤0, since this is equivalent to -cix≥0.

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).

Minimization subject to bounds on the objective function

In all of the above problem categories it is assumed that

a≤Fx≤b

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 [Function Evaluations at Infeasible Points].

Multi-objective optimization

You 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 E05 class may be employed to address such problems.

Geometric Representation and Terminology

To illustrate the nature of optimization problems it is useful to consider the following example in two dimensions:

Fx=ex14x12+2x22+4x1x2+2x2+1.

(This function is used as the example function in the documentation for the unconstrained methods.)

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 x- 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 x1≥0 (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-x1x2-32≥0, 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 x^≃1.1825,-1.7397T, the feasible point with the smallest function value (where Fx^≃3.0607).

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.

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/∂xi∂xj. If Fx has continuous second derivatives, then Gx must be positive semi-definite at any unconstrained minimum of F.

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

Note that if cix is a linear constraint involving aiT x , then its vector of first partial derivatives is simply the vector ai.

Sufficient Conditions for a Solution

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

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.

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.

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

A^Tx*=b^.

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^.

Define

gZx=ZTgx, the projected gradient vector of Fx;
GZx=ZTGxZ, the projected Hessian matrix of Fx.

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:

λ*=A^TA^-1A^Tgx*.

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.

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:

Lx,λ=Fx-λTc^x.

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^xλ; and
(iii)	G^Lx* is positive-definite; and
(iv)	λi>0 if λi corresponds to a constraint of the form c^i≥0. 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*.

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 [Analysis of Computed Results].

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 [One-dimensional optimization].

One-dimensional optimization

The Library contains two special methods for minimizing a function of a single variable. Both methods are based on safeguarded polynomial approximation. One method 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).

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 methods in the Library either require a method 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).

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.

Since

Fx=∑i=1mfi2x

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).

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 methods 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.

Methods for handling multi-objective optimization

Suppose we have objective functions fxx, 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 fxx should be of comparable size at a solution.

(b)

Order the objectives in order of importance to you. Suppose fi are ordered s.t. fxx 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 minfx,x subject to rk≤fkx≤rk, for k=1,2,…,i-1 plus the other constraints.

Clearly the bounds on fk might be relaxed at your discretion.

In general, ifnAG methods from the E04 class 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 methods from E05 class might be used or, alternatively, a pragmatic attempt to derive a global minimum might be tried. 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.

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 methods in the Library, scaling is automatically performed by default in the methods which solve sparse LP, QP or NLP problems and in some newer dense solver methods. The following sections present some general comments on problem scaling.

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

xnew=Dxold,

where D is a diagonal matrix with constant coefficients. Our experience suggests that more use should be made of the transformation

xnew=Dxold+v,

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 150–170. One possible scaling would be to define a new variable x-3, given by

x-3=x3170.

A better transformation, however, is given by defining x-3 as

x-3=x3-16010.

Frequently, an improvement in the accuracy of evaluation of Fx can result if the variables are scaled before the methods 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.

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 method. 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.

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 method, 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).

Analysis of Computed Results

Convergence criteria

The convergence criteria inevitably vary from method to method, 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 methods 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.

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 method reports success. Frequently a ‘solution’ may have been found even when the method does not report a success. The reason for this apparent contradiction is that the method 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 method has returned a nonzero value of ifail 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.

Monitoring progress

Many of the methods 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 method 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 nonzero value of ifail.

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 [Methods for nonlinear least-squares problems]). The Jacobian, J, or a factorization of J is returned by all the comprehensive least-squares methods and, in addition, a method is available in the Library to estimate variances of the parameters following the use of most of the nonlinear least-squares methods, 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

var⁡x-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-var⁡x-i. t1-β/2,m-n<xi*<x-i+var⁡x-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

var⁡ϕ=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

ϕz-,x--var⁡ϕ.tβ/2,m-n<ϕz-,x*<ϕz-,x-+var⁡ϕ.tβ/2,m-n.

For further details on the analysis of least-squares solutions see Bard (1974) and Wolberg (1967).

Recommendations on Choice and Use of Available Methods

The choice of method 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 methods); whether or not storage is a problem; whether or not the method is to be used in a multithreaded environment; and whether computational time has a high priority. Not all choices are catered for in the current version of the Library.

Easy-to-use and Comprehensive Methods

Many methods appear in the Library in two forms: a comprehensive form and an easy-to-use form. The objective in the easy-to-use forms is to make the method simple to use by including in the calling sequence only those parameters absolutely essential to the definition of the problem, as opposed to parameters relevant to the solution method. If you are an experienced user the comprehensive methods have additional parameters which enable you to improve their efficiency by ‘tuning’ the method to a particular problem. If you are a casual or inexperienced user, this feature is of little value and may in some cases cause a failure because of a poor choice of some parameters.

In the easy-to-use methods, these extra parameters are determined either by fixing them at a known safe and reasonably efficient value, or by an auxiliary method which generates a ‘good’ value automatically.

For methods introduced since Mark 12 of the Library a different approach has been adopted towards the choice of easy-to-use and comprehensive methods. The optimization method has an easy-to-use parameter list, but additional parameters may be changed from their default values by calling an ‘option’ setting method before the call to the main optimization method. This approach has the advantages of allowing the options to be given in the form of keywords and requiring only those options that are to be different from their default values to be set.

Reverse Communication Methods

Most of the methods in this chapter are called just once in order to compute the minimum of a given objective function subject to a set of constraints on the variables. The objective function and nonlinear constraints (if any) are specified by you and written as methods to a very rigid format described in the relevant method document. Such methods usually appear in the argument list of the minimization method.

For the majority of applications this is the simplest and most convenient usage. Sometimes however this approach can be restrictive:

(i)	when the required format of the method does not allow useful information to be passed conveniently to and from your calling program;
(ii)	when the minimization method is being called from another computer language, such as Visual Basic, which does not fully support procedure arguments in a way that is compatible with the Library.

A way around these problems is to utilize reverse communication methods. Instead of performing complete optimizations, these methods perform one step in the solution process before returning to the calling program with an appropriate flag (irevcm) set. The value of irevcm determines whether the minimization process has finished or whether fresh information is required. In the latter case you calculate this information (in the form of a vector or as a scalar, as appropriate) and re-enter the reverse communication method with the information contained in appropriate arguments. Thus you have the responsibility for providing the iterative loop in the minimization process, but as compensation, you have an extremely flexible and basic user-interface to the reverse communication method.

The only reverse communication methods in this chapter are e04uf, which solve dense NLP problems using a sequential quadratic programming method.

Service Methods

One of the most common errors in the use of optimization methods is that user-supplied delegates 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 methods are provided to perform an elementary check on the gradients you supplied. These methods are inexpensive to use in terms of the number of calls they require to user-supplied delegates.

The appropriate checking methods are as follows:

Minimization method	Checking method(s)
e04kd	e04hc
e04lb	e04hc and e04hd
(e04gb not in this release)	e04ya
e04gd	e04ya
e04he	e04ya and e04yb

It should be noted that methods e04uf, e04us, e04vh and e04wd 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 method computes a set of finite-differences to be used when approximating first derivatives. Such differences are required as input parameters by some methods that use only function evaluations.

(e04yc not in this release) estimates selected elements of the variance-covariance matrix for the computed regression parameters following the use of a nonlinear least-squares method.

e04xa estimates the gradient and Hessian of a function at a point, given a method to calculate function values only, or estimates the Hessian of a function at a point, given a method to calculate function and gradient values.

e04zc checks that user-supplied delegates for evaluating an objective function, constraint functions and their first derivatives produce derivative values which are consistent with the function and constraint values calculated.

Function Evaluations at Infeasible Points

All the methods 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 methods 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 methods allow you to return the parameter (iflag or mode) with a negative value to force an immediate clean exit from the minimization method when the objective function (or nonlinear constraints where appropriate) cannot be evaluated.

Choosing Between Variant Methods for Some Problems

As evidenced by the wide variety of methods available in E04 class, 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 method, and that is what the decision trees in [Decision Trees] help to do.

Sometimes in E04 class more than one method is available to solve precisely the same minimization problem. Thus, for example, the general nonlinear programming methods e04uc and e04wd have very similar argument lists and are based on similar methods. Experience shows that although both methods can usually solve the same problem and get similar results, sometimes one method will be faster, sometimes one might find a different local minimum to the other, or, in difficult cases, one method may obtain a solution when the other one fails.

After using one of these methods, if the results obtained are unacceptable for some reason, it may be worthwhile trying the other method instead. In the absence of any other information, in the first instance you are recommended to try using e04uc, and if that proves unsatisfactory, try using e04wd. Although the algorithms used are very similar, the two methods each have slightly different optional arguments which may allow the course of the computation to be altered in different ways.

Other pairs of methods which solve the same kind of problem are e04nk or e04nq, for sparse quadratic or linear programming problems, and e04ug or e04vh, for sparse nonlinear programming. In these cases the argument lists are not so similar as e04uc or e04wd, but the same considerations apply.

Decision Trees

Tree 1: Selection chart for unconstrained problems

Only one variable?	_ yes	Are first derivatives available?	_ yes	e04bb
\|		no \|
\|		e04ab
no \|
Does the function have many discontinuities?	_ yes	e04cb
no \|
Is store size a problem?	_ yes	e04dg
no \|
Is the function a sum of squares?	_ yes	Are you an experienced user?	_ yes	Are first derivatives available?	_ yes	Are second derivatives available?	_ yes	e04he
\|		\|		\|		no \|
\|		\|		\|		Are there more than ten variables?	_ yes	(e04gb not in this release)
\|		\|		\|		no \|
\|		\|		\|		e04gd
\|		\|		no \|
\|		\|		e04fc
\|		no \|
\|		Are first derivatives available?	_ yes	Are second derivatives available?	_ yes	e04hy
\|		\|		no \|
\|		\|		Are there more than ten variables?	_ yes	e04gy
\|		\|		no \|
\|		\|		e04gz
\|		no \|
\|		e04fy
no \|
Are you an experienced user?	_ yes	Are first derivatives available?	_ yes	Are second derivatives available?	_ yes	e04lb
\|		\|		no \|
\|		\|		Is computational cost critical?	_ yes	e04uc, e04uf, e04ug, e04vh or e04wd
\|		\|		no \|
\|		\|		e04kd
\|		no \|
\|		e04uc, e04uf, e04ug, e04vh or e04wd
no \|
Are first derivatives available?	_ yes	Are second derivatives available?	_ yes	e04ly
\|		no \|
\|		Is computational cost critical?	_ yes	e04ky
\|		no \|
\|		e04kz
no \|
e04jy

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

Are there any nonlinear constraints?	_ yes	Is the objective function a sum of squares? (A least-squares problem)	_ yes	e04us
\|		no \|
\|		Are the constraints sparse?	_ yes	e04ug or e04vh
\|		no \|
\|		e04uc, e04uf or e04wd
no \|
Is the objective function linear? (An LP problem)	_ yes	See Tree 3
no \|
Is the objective function quadratic? (A QP or least-squares problem)	_ yes	Is the problem a least-squares problem?	_ yes	e04nc
\|		no \|
\|		See Tree 4
no \|
Is the objective function a sum of squares? (A least-squares problem)	_ yes	e04us
no \|
Are the constraints simple bounds?	_ yes	Are you an experienced user?	_ yes	Are first derivatives available?	_ yes	Are second derivatives available?	_ yes	e04lb
\|		\|		\|		no \|
\|		\|		\|		e04kd, e04uc, e04uf, e04ug or e04vh
\|		\|		no \|
\|		\|		e04uc, e04uf, e04ug, e04vh or e04wd
\|		no \|
\|		Are first derivatives available?	_ yes	Are second derivatives available?	_ yes	e04ly
\|		\|		no \|
\|		\|		Is computational cost critical?	_ yes	e04ky
\|		\|		no \|
\|		\|		e04kz
\|		no \|
\|		e04jy
no \|
e04uc, e04uf, e04ug or e04vh

Tree 3: Linear programming

Is the objective function linear (an LP problem) and is the linear constraint matrix sparse?	_ yes	e04nk, e04nq, e04ug or e04vh
no \|
e04mf

Tree 4: Quadratic programming

Is the linear constraint matrix sparse?	_ yes	e04nk, e04nq, e04ug or e04vh
no \|
Is the problem a convex QP problem?	_ yes	e04nc
no \|
(e04nf not in this release)

References

Inheritance Hierarchy

System..::.Object
NagLibrary..::.E04