Numerical Nonlinear Global Optimization
Introduction
Numerical algorithms for constrained nonlinear optimization can be broadly categorized into
gradient-based methods and
direct search methods. Gradient-based methods use first derivatives (gradients) or second derivatives (Hessians). Examples are the sequential quadratic programming (SQP) method, the augmented Lagrangian method, and the (nonlinear) interior point method. Direct search methods do not use derivative information. Examples are Nelder-Mead, genetic algorithm and differential evolution, and simulated annealing. Direct search methods tend to converge more slowly, but can be more tolerant to the presence of noise in the function and constraints.
Typically, algorithms only build up a local model of the problems. Furthermore, many such algorithms insist on certain decrease of the objective function, or decrease of a merit function that is a combination of the objective and constraints, to ensure convergence of the iterative process. Such algorithms will, if convergent, only find local optima, and are called
local optimization algorithms. In
Mathematica local optimization problems can be solved using
FindMinimum.
Global optimization algorithms, on the other hand, attempt to find the global optimum, typically by allowing decrease as well as increase of the objective/merit function. Such algorithms are usually computationally more expensive. Global optimization problems can be solved exactly using
Minimize or numerically using
NMinimize.
This solves a nonlinear programming problem,
using
Minimize,which gives an exact solution
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This solves the same problem numerically.
NMinimize returns a machine-number solution.
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FindMinimum numerically finds a local minimum. In this example the local minimum found is also a global minimum.
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The NMinimize Function
NMinimize and
NMaximize implement several algorithms for finding constrained global optima. The methods are flexible enough to cope with functions that are not differentiable or continuous and are not easily trapped by local optima.
Finding a global optimum can be arbitrarily difficult, even without constraints, and so the methods used may fail. It may frequently be useful to optimize the function several times with different starting conditions and take the best of the results.
This finds the maximum of
.
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This finds the minimum of
subject to the constraints
and
.
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The constraints to
NMinimize and
NMaximize may be either a list or a logical combination of equalities, inequalities, and domain specifications. Equalities and inequalities may be nonlinear. Any strong inequalities will be converted to weak inequalities due to the limits of working with approximate numbers. Specify a domain for a variable using
Element, for example,
Element[x, Integers] or
Integers. Variables must be either integers or real numbers, and will be assumed to be real numbers unless specified otherwise. Constraints are generally enforced by adding penalties when points leave the feasible region.
Constraints can contain logical operators like
And,
Or, and so on.
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Here
is restricted to being an integer.
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In order for
NMinimize to work, it needs a rectangular initial region in which to start. This is similar to giving other numerical methods a starting point or starting points. The initial region is specified by giving each variable a finite upper and lower bound. This is done by including
in the constraints, or
in the variables. If both are given, the bounds in the variables are used for the initial region, and the constraints are just used as constraints. If no initial region is specified for a variable
x, the default initial region of
is used. Different variables can have initial regions defined in different ways.
Here the initial region is taken from the variables. The problem is unconstrained.
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Here the initial region is taken from the constraints.
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Here the initial region for
is taken from the constraints, the initial region for
is taken from the variables, and the initial region for
is taken to be the default. The problem is unconstrained in
and
, but not
.
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The polynomial
has global minima at
.
finds one of the minima.
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The other minimum can be found by using a different
.
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NMinimize and
NMaximize have several optimization methods available:
Automatic,
,
,
, and
. The optimization method is controlled by the
Method option, which either takes the method as a string, or takes a list whose first element is the method as a string and whose remaining elements are method-specific options. All method-specific option, left-hand sides should also be given as strings.
The following function has a large number of local minima.
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Use
to find a minimum.
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Use
with more starting points to find the global minimum.
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With the default method,
NMinimize picks which method to use based on the type of problem. If the objective function and constraints are linear,
LinearProgramming is used. If there are integer variables, or if the head of the objective function is not a numeric function, differential evolution is used. For everything else, it uses Nelder-Mead, but if Nelder-Mead does poorly, it switches to differential evolution.
Because the methods used by
NMinimize may not improve every iteration, convergence is only checked after several iterations have occurred.
Numerical Algorithms for Constrained Global Optimization
Nelder-Mead
The Nelder-Mead method is a direct search method. For a function of
variables, the algorithm maintains a set of
points forming the vertices of a polytope in
-dimensional space. This method is often termed the "simplex" method, which should not be confused with the well-known simplex method for linear programming.
At each iteration,
points
form a polytope. The points are ordered so that
A new point is then generated to replace the worst point
Let
be the centroid of the polytope consisting of the best
points,
. A trial point
is generated by reflecting the worst point through the centroid,
, where
is a parameter.
If the new point
is neither a new worst point nor a new best point,
,
replaces
.
If the new point
is better than the best point,
, the reflection is very successful and can be carried out further to
, where
is a parameter to expand the polytope. If the expansion is successful,
,
replaces
; otherwise the expansion failed, and
replaces
.
If the new point
is worse than the second worst point,
, the polytope is assumed to be too large and needs to be contracted. A new trial point is defined as
where
is a parameter. If
, the contraction is successful, and
replaces
. Otherwise a further contraction is carried out.
The process is assumed to have converged if the difference between the best function values in the new and old polytope, as well as the distance between the new best point and the old best point, are less than the tolerances provided by
AccuracyGoal and
PrecisionGoal.
Strictly speaking, Nelder-Mead is not a true global optimization algorithm; however, in practice it tends to work reasonably well for problems that do not have many local minima.
| | |
| "ContractRatio" | 0.5 | ratio used for contraction |
| "ExpandRatio" | 2.0 | ratio used for expansion |
| "InitialPoints" | Automatic | set of initial points |
| "PenaltyFunction" | Automatic | function applied to constraints to penalize invalid points |
| "PostProcess" | Automatic | whether to post-process using local search methods |
| "RandomSeed" | 0 | starting value for the random number generator |
| "ReflectRatio" | 1.0 | ratio used for reflection |
| "ShrinkRatio" | 0.5 | ratio used for shrinking |
| "Tolerance" | 0.001 | tolerance for accepting constraint violations |
specific options.
Here the function inside the unit disk is minimized using
.
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Here is a function with several local minima that are all different depths.
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With the default parameters,
is too easily trapped in a local minimum.
By using settings that are more aggressive and less likely to make the simplex smaller, the results are better.
Differential Evolution
Differential evolution is a simple stochastic function minimizer.
The algorithm maintains a population of
points,
, where typically
, with
being the number of variables.
During each iteration of the algorithm, a new population of
points is generated. The
new point is generated by picking three random points,
,
, and
, from the old population, and forming
, where
is a real scaling factor. Then a new point
is constructed from
and
by taking the
coordinate from
with probability
and otherwise taking the coordinate from
. If
, then
replaces
in the population. The probability
is controlled by the
option.
The process is assumed to have converged if the difference between the best function values in the new and old populations, as well as the distance between the new best point and the old best point, are less than the tolerances provided by
AccuracyGoal and
PrecisionGoal.
The differential evolution method is computationally expensive, but is relatively robust and tends to work well for problems that have more local minima.
| | |
| "CrossProbability" | 0.5 | probability that a gene is taken from  |
| "InitialPoints" | Automatic | set of initial points |
| "PenaltyFunction" | Automatic | function applied to constraints to penalize invalid points |
| "PostProcess" | Automatic | whether to post-process using local search methods |
| "RandomSeed" | 0 | starting value for the random number generator |
| "ScalingFactor" | 0.6 | scale applied to the difference vector in creating a mate |
| "SearchPoints" | Automatic | size of the population used for evolution |
| "Tolerance" | 0.001 | tolerance for accepting constraint violations |
specific options.
Here the function inside the unit disk is minimized using
.
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The following constrained optimization problem has a global minimum of
.
With the default settings for
, an unsatisfactory solution results.
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By adjusting
, the results are much better. In this case, the increased
gives
better mobility with respect to the integer variables.
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Simulated Annealing
Simulated annealing is a simple stochastic function minimizer. It is motivated from the physical process of annealing, where a metal object is heated to a high temperature and allowed to cool slowly. The process allows the atomic structure of the metal to settle to a lower energy state, thus becoming a tougher metal. Using optimization terminology, annealing allows the structure to escape from a local minimum, and to explore and settle on a better, hopefully global, minimum.
At each iteration, a new point,
, is generated in the neighborhood of the current point,
. The radius of the neighborhood decreases with each iteration. The best point found so far,
, is also tracked.
If
,
replaces
and
. Otherwise,
replaces
with a probability
. Here
is the function defined by
,
is the current iteration,
is the change in the objective function value, and
is the value of the objective function from the previous iteration. The default function for
is
.
Like the
method,
uses multiple starting points, and finds an optimum starting from each of them.
The default number of starting points, given by the option
, is
, where
is the number of variables.
For each starting point, this is repeated until the maximum number of iterations is reached, the method converges to a point, or the method stays at the same point consecutively for the number of iterations given by
.
| | |
| "BoltzmannExponent" | Automatic | exponent of the probability function |
| "InitialPoints" | Automatic | set of initial points |
| "LevelIterations" | 50 | maximum number of iterations to stay at a given point |
| "PenaltyFunction" | Automatic | function applied to constraints to penalize invalid points |
| "PerturbationScale" | 1.0 | scale for the random jump |
| "PostProcess" | Automatic | whether to post-process using local search methods |
| "RandomSeed" | 0 | starting value for the random number generator |
| "SearchPoints" | Automatic | number of initial points |
| "Tolerance" | 0.001 | tolerance for accepting constraint violations |
specific options.
Here a function in two variables is minimized using
.
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Here is a function with many local minima.
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By default, the step size for
is not large enough to escape from the local minima.
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By increasing
, larger step sizes are taken to produce a much better solution.
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Here
is set to use an exponential cooling function that gives faster convergence. (Note that the modified
is still being used as well.)
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Random Search
The random search algorithm works by generating a population of random starting points and uses a local optimization method from each of the starting points to converge to a local minimum. The best local minimum is chosen to be the solution.
The possible local search methods are
Automatic and
. The default method is
Automatic, which uses
FindMinimum with unconstrained methods applied to a system with penalty terms added for the constraints. When
Method is set to
, a nonlinear interior-point method is used.
The default number of starting points, given by the option
, is
, where
is the number of variables.
Convergence for
is determined by convergence of the local method for each starting point.
is fast, but does not scale very well with the dimension of the search space. It also suffers from many of the same limitations as
FindMinimum. It is not well suited for discrete problems and others where derivatives or secants give little useful information about the problem.
| | |
| "InitialPoints" | Automatic | set of initial points |
| "Method" | Automatic | which method to use for minimization |
| "PenaltyFunction" | Automatic | function applied to constraints to penalize invalid points |
| "PostProcess" | Automatic | whether to post-process using local search methods |
| "RandomSeed" | 0 | starting value for the random number generator |
| "SearchPoints" | Automatic | number of points to use for starting local searches |
| "Tolerance" | 0.001 | tolerance for accepting constraint violations |
specific options.
Here the function inside the unit disk is minimized using
.
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Here is a function with several local minima that are all different depths and are generally difficult to optimize.
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With the default number of
, sometimes the minimum is not found.
Using many more
produces better answers.
Here points are generated on a grid for use as initial points.
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This uses nonlinear interior point methods to find the minimum of a sum of squares.
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For some classes of problems, limiting the number of
can be much faster without affecting the quality of the solution.
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