E05 (Glopt)
Chapter Introduction
FL CL
NAG CL Interface
E05 (Glopt)
Global Optimization of a Function
E05 (Glopt)
Chapter Introduction
FL CL
1
Scope of the Chapter
Global optimization involves finding the absolute maximum or minimum value of a function (the
objective function) of several variables, possibly subject to restrictions (defined by a set of bounds or
constraint functions) on the values of the variables. Where
Chapter E04 treats local nonlinear optimization problems as well as convex optimization problems (for which local optima are by definition also global), this chapter tries to handle finding global optima of non-convex nonlinear optimization problems. Such problems can be much harder to solve than local optimization problems because it is difficult to determine whether a potential optimum found is global, and because of the nonlocal methods required to avoid becoming trapped near local optima.
This introduction is a brief guide to the subject of global optimization, designed for the casual user. For further details you may find it beneficial to consult a more detailed text, see
Neumaier (2004). Furthermore, much of the material in the
E04 Chapter Introduction is also relevant in this context and it is strongly recommended that you read
Section 2.6 in the
E04 Chapter Introduction.
Some solvers in this chapter have been integrated in to the
NAG optimization modelling suite, which provides an easy-to-use and flexible way of defining optimization problems among numerous solvers of the
NAG library. All utility routines from the suite can be used to define a compatible problem to be solved by the integrated solvers; see
Section 4.1 in the
E04 Chapter Introduction for more details.
2
Background to the Problems
2.1
Problem Formulation
For the purposes of this Library, the global optimization problem is
where
(the
objective function) is a real function; the vectors
and
are elements of
, where
denotes the extended reals
; and where
is a vector of
constraint functions
, with
and
defining the constraints on
. If
the problem is said to be
bound constrained. Relational operators between vectors are interpreted elementwise. The
feasible region is the set of all points (
feasible points) that satisfy all of the constraints. A
solution of
(1) is a feasible point
satisfying
A local minimum minimizes only on some neighbourhood of . If a local minimum has the smallest objective value over all the local minima, then it is a global minimum.
2.2
Terminology
2.2.1
Complete Methods
A global optimization algorithm is called
asymptotically complete if
-
(i)assuming indefinitely long run-time and exact computations, a global minimum will be found with certainty (probability one), but
-
(ii)the algorithm has no way of knowing when a global minimum has been found.
In comparison, a
complete method satisfies
(i) as well as the algorithm being able to recognize a global minimum (to prescribed tolerances) within a finite amount of time.
It is important to appreciate that, for finding a solution exactly, bounds on the amount of work may be very pessimistic. What complete methods guarantee is the absence of any deficiency that would prevent a global minimum from
eventually being found. To achieve termination with certainty in a finite amount of time, the algorithm requires access to global information about the problem. In the case where only function values are available, as in
e05kbc, stopping criteria based on heuristics are present. This is because such a class of method can only terminate with certainty by performing an expensive dense search.
In contrast,
incomplete methods have intuitive heuristics for searching but no guarantee of not getting stuck near nonglobal, local, minima. Often, to make incomplete methods efficient, expert knowledge on the particular problem class to be solved is required. Examples of incomplete methods include Particle Swarm Optimization (PSO), Genetic Algorithms (GA), Simulated Annealing (SA), Ant Colony Optimization (ACO) and Covariance Matrix Adaptation Evolutionary Strategies (CMA-ES). PSO has been implemented in the functions
e05sac and
e05sbc. Such functions must also use heuristics to stop the algorithm as again an expensive, dense search would be required to guarantee that no superior optima are present.
2.2.2
Branching
Most complete methods recursively split the original problem into smaller, more manageable subproblems. This technique is called branching. Branching is usually accompanied by a selection process that splits favourable branches more frequently than others. For example, with branch and bound methods, bounds on the objective function for each subproblem are computed in an attempt to eliminate those subregions where no improvement will occur.
Branching methods use a
branching scheme to generate sequences of sub-boxes that eventually cover the feasible region. At least one function evaluation is made for every sub-box, and new sub-boxes are generated by splitting existing ones. Using appropriate
splitting rules, convergence to zero of the diameters of sub-boxes is assured. For example, always splitting the oldest box along the oldest side, provided the children do not have too small a volume compared with the parent, guarantees convergence of the method, in the sense described in
Neumaier (2004).
Efficiency can be enhanced by carefully balancing global and local searches. While the global part of the search splits sub-boxes with large unexplored territory, the local part usually entails splitting boxes with good function values. For example, the sub-box with the best function value should always be split. A method may also be improved by launching local searches from appropriate candidate local minima.
2.3
Methods of Global Optimization
2.3.1
Multi-level Coordinate Search (MCS)
The function
e05kbc searches for a global minimizer using branching to recursively split the search space in a nonuniform manner. It divides, or
splits, the
root box of the search into smaller sub-boxes. Each sub-box contains a distinguished
basepoint at which the objective function is sampled. We shall sometimes say ‘the function value of the (sub)box’ as shorthand for ‘the function value of the basepoint of the (sub)box’. The splitting procedure biases the search in favour of those sub-boxes where low function values are expected.
The global part of the algorithm entails splitting sub-boxes that enclose large unexplored territory, while the local part of the algorithm entails splitting sub-boxes that have good function values. A balance between the global and local part is achieved using a
multi-level approach, where every sub-box is assigned a
level . You can control the value of
using the optional parameter
. Whenever a sub-box of intermediate level
is split each descendant will be given a new level, and the original sub-box's level is set to
: a sub-box with level
has already been split; a sub-box with level
will be split no further. This entire process is described in more detail in
Section 11.1 in
e05kbc, where the
initialization procedure used to produce an initial set of sub-boxes is outlined, and the method by which the algorithm
sweeps through levels is discussed. Each sweep starts with the sub-boxes at the lowest level, a process thus forming the global part of the algorithm. At each level the sub-box with the best function value is selected for splitting; this forms the local part of the algorithm.
The process by which sub-boxes are split is explained in
Section 11.2 in
e05kbc. It is a variant of the standard coordinate search method: the solver splits along a single coordinate at a time, at adaptively chosen points. In most cases one new function evaluation is needed to split a sub-box into two or three children. Each child is given a basepoint chosen to differ from the basepoint of the parent in at most one coordinate, and safeguards are present to ensure a degree of symmetry in the splits.
If you set the optional parameter
to be
, then the basepoints and function values of sub-boxes of maximum level
are put into a ‘shopping basket’ of candidate minima. Turning
(the default setting) will enable local searches to be started from these basepoints before they go into the shopping basket. The local search will go ahead providing the basepoint is not likely to be in the basin of attraction of a previously-found local minimum. The search itself uses a trust region approach, and is explained in
Section 11.3 in
e05kbc: local quadratic models are built by a
triple search, then a linesearch is made along the direction obtained by minimizing the quadratic on a region where it is a good approximation to the objective function. The quadratic need not be positive definite, so a general nonlinear optimizer is used to minimize the model.
2.3.2
Particle Swarm Optimization
The functions
e05sac and
e05sbc search for a global optimum using a variant of the Particle Swarm Optimization (PSO) algorithm. PSO is an heuristic algorithm similar in its behaviour to GA, ACO, SA and others. A set of particles (the swarm) is generated in the search space, and advances at each iteration following an heuristic velocity based upon the best candidate found by an individual particle (cognitive memory), the best candidate found by all the particles (global memory) and inertia. The inertia is provided by a decreasingly weighted contribution from a particle's current velocity. This mix allows for a global search of the domain in question.
The rate at which the swarm contracts and expands about potential optima is user controllable, allowing expert knowledge to be used when available. Furthermore, the algorithm may be coupled with a selection of local optimizers. These may be called during the iterations of the heuristic algorithm (the interior phase) to hasten the discovery of locally optimal points. They may also be called following the heuristic iterations (the exterior phase) to attempt to refine the final solution. Different options may be set for the local optimizer in each phase. For further details see Section 11 in
e05sac and
e05sbc.
2.3.3
Multiple Start
Function
e05ucc attempts to find the global minimum of an arbitrary smooth function subject to constraints (which may include simple bounds on the variables, linear constraints and smooth nonlinear constraints) by generating a number of different starting points and using
a sequential quadratic programming local minimizer.
Function
e05usc takes the same approach in attempting to find the global minimum of an arbitrary smooth sum of squares function using
a sequential quadratic programming local minimizer.
The more starting points chosen, the greater the degree of confidence that you might have in the returned results. Facilities are provided to allow you to specify the starting points and to provide for subsequent runs with different starting points as an additional means of gaining confidence in the results.
You may also request that a number of solutions be provided, ordered in increasing value of the local optima. This may be useful if a local solution has a desirable property not exhibited by the best local optimum computed, the putative global optimum.
3
Recommendations on Choice and Use of Available Functions
e05kbc is based on the
multi-level coordinate search method of
Huyer and Neumaier (1999). It is an asymptotically complete method for bound constrained problems based on local information (function values) only, employing branching and local searches to accelerate convergence. This algorithm has been integrated to the
NAG optimization modelling suite, see
Section 4.1 in the
E04 Chapter Introduction for more details.
If the problem has nonlinear constraints and is sufficiently smooth then you are advised to consider a multiple start technique.
e05ucc and
e05usc are provided for this purpose. Both
e05ucc and
e05usc use the functions
e05zkc and
e05zlc for initialization and option setting.
The suite of particle swarm optimization (PSO) functions are to be considered as experimental and are not recommended for production or mission-critical applications. They are only recommended as a last resort (should other methods fail) or for comparitive purposes.
The suite consists of the solver functions:
Both
e05sac and
e05sbc use the functions
e05zkc and
e05zlc for initialization and option setting. These functions predominantly use function values only, although derivatives can be provided for coupled local minimization functions.
e05sac is a simplified version of
e05sbc with less functionality. In particular,
e05sac does not support general constraint handling whereas
e05sbc does support general nonlinear, non-equality constraints.
If the objective function is smooth and the problem has only simple bound constraints then
all
algorithms are applicable. For low dimensional problems (up to
)
e05kbc is preferred. With increasing dimension the multi-start methods may be
better.
The particle swarm methods are potentially useful when there is no smoothness in the objective function (e.g., due to noise) and, for the simple-bound constrained problem,
e05sac may be appropriate.
Currently there is no routine in this chapter using a complete method that can handle constraints that are not bound constraints.
4
Functionality Index
Nonlinear programming (NLP) – global
optimization,
|
|
|
branching algorithm, multi-level coordinate search
|
|
e05kbc
|
heuristic algorithm, particle swarm optimization (PSO)
|
|
e05sac
|
generic, including nonlinearly constrained,
|
|
|
heuristic algorithm, particle swarm optimization (PSO)
|
|
e05sbc
|
Nonlinear least squares, data fitting – global optimization,
|
|
|
generic, including nonlinearly constrained,
|
|
|
option setting functions,
|
|
|
supply optional parameter values from a character string
|
|
e05zkc
|
retrieve optional parameter values
|
|
e05zlc
|
5
Auxiliary Functions Associated with Library Function Arguments
None.
6
Withdrawn or Deprecated Functions
None.
7
References
Gill P E, Murray W and Wright M H (1981) Practical Optimization Academic Press
Huyer W and Neumaier A (1999) Global optimization by multi-level coordinate search Journal of Global Optimization 14 331–355
Kennedy J and Eberhart R C (1995) Particle Swarm Optimization Proceedings of the 1995 IEEE International Conference on Neural Networks 1942–1948
Koh B, George A D, Haftka R T and Fregly B J (2006) Parallel Asynchronous Particle Swarm Optimization International Journal for Numerical Methods in Engineering 67(4) 578–595
Neumaier A (2004) Complete search in constrained global optimization Acta Numerica 13 271–369
Vaz A I and Vicente L N (2007) A Particle Swarm Pattern Search Method for Bound Constrained Global Optimization Journal of Global Optimization 39(2) 197–219 Kluwer Academic Publishers
E05 (Glopt)
Chapter Introduction
FL CL