Provided by: libnlopt-dev_2.4.1+dfsg-1ubuntu1_amd64 
NAME
nlopt - Nonlinear optimization library
SYNOPSIS
#include <nlopt.h>
nlopt_opt opt = nlopt_create(algorithm, n);
nlopt_set_min_objective(opt, f, f_data);
nlopt_set_ftol_rel(opt, tol);
...
nlopt_optimize(opt, x , &opt_f);
nlopt_destroy(opt);
The "..." indicates any number of calls to NLopt functions, below, to
set parameters of the optimization, constraints, and stopping
criteria. Here, nlopt_set_ftol_rel is merely an example of a
possible stopping criterion. You should link the resulting program
with the linker flags -lnlopt -lm on Unix.
DESCRIPTION
NLopt is a library for nonlinear optimization. It attempts to minimize (or maximize) a
given nonlinear objective function f of n design variables, using the specified algorithm,
possibly subject to linear or nonlinear constraints. The optimum function value found is
returned in opt_f (type double) with the corresponding design variable values returned in
the (double) array x of length n. The input values in x should be a starting guess for
the optimum.
The parameters of the optimization are controlled via the object opt of type nlopt_opt,
which is created by the function nlopt_create and disposed of by nlopt_destroy. By
calling various functions in the NLopt library, one can specify stopping criteria (e.g., a
relative tolerance on the objective function value is specified by nlopt_set_ftol_rel),
upper and/or lower bounds on the design parameters x, and even arbitrary nonlinear
inequality and equality constraints.
By changing the parameter algorithm among several predefined constants described below,
one can switch easily between a variety of minimization algorithms. Some of these
algorithms require the gradient (derivatives) of the function to be supplied via f, and
other algorithms do not require derivatives. Some of the algorithms attempt to find a
global optimum within the given bounds, and others find only a local optimum. Most of the
algorithms only handle the case where there are no nonlinear constraints. The NLopt
library is a wrapper around several free/open-source minimization packages, as well as
some new implementations of published optimization algorithms. You could, of course,
compile and call these packages separately, and in some cases this will provide greater
flexibility than is available via NLopt. However, depending upon the specific function
being optimized, the different algorithms will vary in effectiveness. The intent of NLopt
is to allow you to quickly switch between algorithms in order to experiment with them for
your problem, by providing a simple unified interface to these subroutines.
OBJECTIVE FUNCTION
The objective function is specified by calling one of:
nlopt_result nlopt_set_min_objective(nlopt_opt opt,
nlopt_func f,
void* f_data);
nlopt_result nlopt_set_max_objective(nlopt_opt opt,
nlopt_func f,
void* f_data);
depending on whether one wishes to minimize or maximize the objective function f,
respectively. The function f should be of the form:
double f(unsigned n,
const double* x,
double* grad,
void* f_data);
The return value should be the value of the function at the point x, where x points to an
array of length n of the design variables. The dimension n is identical to the one passed
to nlopt_create.
In addition, if the argument grad is not NULL, then grad points to an array of length n
which should (upon return) be set to the gradient of the function with respect to the
design variables at x. That is, grad[i] should upon return contain the partial derivative
df/dx[i], for 0 <= i < n, if grad is non-NULL. Not all of the optimization algorithms
(below) use the gradient information: for algorithms listed as "derivative-free," the grad
argument will always be NULL and need never be computed. (For algorithms that do use
gradient information, however, grad may still be NULL for some calls.)
The f_data argument is the same as the one passed to nlopt_set_min_objective or
nlopt_set_max_objective, and may be used to pass any additional data through to the
function. (That is, it may be a pointer to some caller-defined data structure/type
containing information your function needs, which you convert from void* by a typecast.)
BOUND CONSTRAINTS
Most of the algorithms in NLopt are designed for minimization of functions with simple
bound constraints on the inputs. That is, the input vectors x[i] are constrainted to lie
in a hyperrectangle lb[i] <= x[i] <= ub[i] for 0 <= i < n. These bounds are specified by
passing arrays lb and ub of length n to one or both of the functions:
nlopt_result nlopt_set_lower_bounds(nlopt_opt opt,
const double* lb);
nlopt_result nlopt_set_upper_bounds(nlopt_opt opt,
const double* ub);
If a lower/upper bound is not set, the default is no bound (unconstrained, i.e. a bound of
infinity); it is possible to have lower bounds but not upper bounds or vice versa.
Alternatively, the user can call one of the above functions and explicitly pass a lower
bound of -HUGE_VAL and/or an upper bound of +HUGE_VAL for some design variables to make
them have no lower/upper bound, respectively. (HUGE_VAL is the standard C constant for a
floating-point infinity, found in the math.h header file.)
Note, however, that some of the algorithms in NLopt, in particular most of the global-
optimization algorithms, do not support unconstrained optimization and will return an
error if you do not supply finite lower and upper bounds.
For convenience, the following two functions are supplied in order to set the lower/upper
bounds for all design variables to a single constant (so that you don't have to fill an
array with a constant value):
nlopt_result nlopt_set_lower_bounds1(nlopt_opt opt,
double lb);
nlopt_result nlopt_set_upper_bounds1(nlopt_opt opt,
double ub);
NONLINEAR CONSTRAINTS
Several of the algorithms in NLopt (MMA and ORIG_DIRECT) also support arbitrary nonlinear
inequality constraints, and some also allow nonlinear equality constraints (COBYLA, SLSQP,
ISRES, and AUGLAG). For these algorithms, you can specify as many nonlinear constraints
as you wish by calling the following functions multiple times.
In particular, a nonlinear inequality constraint of the form fc(x) <= 0, where the
function fc is of the same form as the objective function described above, can be
specified by calling:
nlopt_result nlopt_add_inequality_constraint(nlopt_opt opt,
nlopt_func fc,
void* fc_data,
double tol);
Just as for the objective function, fc_data is a pointer to arbitrary user data that will
be passed through to the fc function whenever it is called. The parameter tol is a
tolerance that is used for the purpose of stopping criteria only: a point x is considered
feasible for judging whether to stop the optimization if fc(x) <= tol. A tolerance of
zero means that NLopt will try not to consider any x to be converged unless fc is strictly
non-positive; generally, at least a small positive tolerance is advisable to reduce
sensitivity to rounding errors.
A nonlinear equality constraint of the form h(x) = 0, where the function h is of the same
form as the objective function described above, can be specified by calling:
nlopt_result nlopt_add_equality_constraint(nlopt_opt opt,
nlopt_func h,
void* h_data,
double tol);
Just as for the objective function, h_data is a pointer to arbitrary user data that will
be passed through to the h function whenever it is called. The parameter tol is a
tolerance that is used for the purpose of stopping criteria only: a point x is considered
feasible for judging whether to stop the optimization if |h(x)| <= tol. For equality
constraints, a small positive tolerance is strongly advised in order to allow NLopt to
converge even if the equality constraint is slightly nonzero.
(For any algorithm listed as "derivative-free" below, the grad argument to fc or h will
always be NULL and need never be computed.)
To remove all of the inequality and/or equality constraints from a given problem opt, you
can call the following functions:
nlopt_result nlopt_remove_inequality_constraints(nlopt_opt opt);
nlopt_result nlopt_remove_equality_constraints(nlopt_opt opt);
ALGORITHMS
The algorithm parameter specifies the optimization algorithm (for more detail on these,
see the README files in the source-code subdirectories), and can take on any of the
following constant values.
Constants with _G{N,D}_ in their names refer to global optimization methods, whereas
_L{N,D}_ refers to local optimization methods (that try to find a local optimum starting
from the starting guess x). Constants with _{G,L}N_ refer to non-gradient (derivative-
free) algorithms that do not require the objective function to supply a gradient, whereas
_{G,L}D_ refers to derivative-based algorithms that require the objective function to
supply a gradient. (Especially for local optimization, derivative-based algorithms are
generally superior to derivative-free ones: the gradient is good to have if you can
compute it cheaply, e.g. via an adjoint method.)
The algorithm specified for a given problem opt is returned by the function:
nlopt_algorithm nlopt_get_algorithm(nlopt_opt opt);
The available algorithms are:
NLOPT_GN_DIRECT_L
Perform a global (G) derivative-free (N) optimization using the DIRECT-L search
algorithm by Jones et al. as modified by Gablonsky et al. to be more weighted
towards local search. Does not support unconstrainted optimization. There are
also several other variants of the DIRECT algorithm that are supported:
NLOPT_GLOBAL_DIRECT, which is the original DIRECT algorithm;
NLOPT_GLOBAL_DIRECT_L_RAND, a slightly randomized version of DIRECT-L that may be
better in high-dimensional search spaces; NLOPT_GLOBAL_DIRECT_NOSCAL,
NLOPT_GLOBAL_DIRECT_L_NOSCAL, and NLOPT_GLOBAL_DIRECT_L_RAND_NOSCAL, which are
versions of DIRECT where the dimensions are not rescaled to a unit hypercube (which
means that dimensions with larger bounds are given more weight).
NLOPT_GN_ORIG_DIRECT_L
A global (G) derivative-free optimization using the DIRECT-L algorithm as above,
along with NLOPT_GN_ORIG_DIRECT which is the original DIRECT algorithm. Unlike
NLOPT_GN_DIRECT_L above, these two algorithms refer to code based on the original
Fortran code of Gablonsky et al., which has some hard-coded limitations on the
number of subdivisions etc. and does not support all of the NLopt stopping
criteria, but on the other hand it supports arbitrary nonlinear inequality
constraints.
NLOPT_GD_STOGO
Global (G) optimization using the StoGO algorithm by Madsen et al. StoGO exploits
gradient information (D) (which must be supplied by the objective) for its local
searches, and performs the global search by a branch-and-bound technique. Only
bound-constrained optimization is supported. There is also another variant of this
algorithm, NLOPT_GD_STOGO_RAND, which is a randomized version of the StoGO search
scheme. The StoGO algorithms are only available if NLopt is compiled with C++ code
enabled, and should be linked via -lnlopt_cxx instead of -lnlopt (via a C++
compiler, in order to link the C++ standard libraries).
NLOPT_LN_NELDERMEAD
Perform a local (L) derivative-free (N) optimization, starting at x, using the
Nelder-Mead simplex algorithm, modified to support bound constraints. Nelder-Mead,
while popular, is known to occasionally fail to converge for some objective
functions, so it should be used with caution. Anecdotal evidence, on the other
hand, suggests that it works fairly well for some cases that are hard to handle
otherwise, e.g. noisy/discontinuous objectives. See also NLOPT_LN_SBPLX below.
NLOPT_LN_SBPLX
Perform a local (L) derivative-free (N) optimization, starting at x, using an
algorithm based on the Subplex algorithm of Rowan et al., which is an improved
variant of Nelder-Mead (above). Our implementation does not use Rowan's original
code, and has some minor modifications such as explicit support for bound
constraints. (Like Nelder-Mead, Subplex often works well in practice, even for
noisy/discontinuous objectives, but there is no rigorous guarantee that it will
converge.)
NLOPT_LN_PRAXIS
Local (L) derivative-free (N) optimization using the principal-axis method, based
on code by Richard Brent. Designed for unconstrained optimization, although bound
constraints are supported too (via the inefficient method of returning +Inf when
the constraints are violated).
NLOPT_LD_LBFGS
Local (L) gradient-based (D) optimization using the limited-memory BFGS (L-BFGS)
algorithm. (The objective function must supply the gradient.) Unconstrained
optimization is supported in addition to simple bound constraints (see above).
Based on an implementation by Luksan et al.
NLOPT_LD_VAR2
Local (L) gradient-based (D) optimization using a shifted limited-memory variable-
metric method based on code by Luksan et al., supporting both unconstrained and
bound-constrained optimization. NLOPT_LD_VAR2 uses a rank-2 method, while .B
NLOPT_LD_VAR1 is another variant using a rank-1 method.
NLOPT_LD_TNEWTON_PRECOND_RESTART
Local (L) gradient-based (D) optimization using an LBFGS-preconditioned truncated
Newton method with steepest-descent restarting, based on code by Luksan et al.,
supporting both unconstrained and bound-constrained optimization. There are
several other variants of this algorithm: NLOPT_LD_TNEWTON_PRECOND (same without
restarting), NLOPT_LD_TNEWTON_RESTART (same without preconditioning), and
NLOPT_LD_TNEWTON (same without restarting or preconditioning).
NLOPT_GN_CRS2_LM
Global (G) derivative-free (N) optimization using the controlled random search
(CRS2) algorithm of Price, with the "local mutation" (LM) modification suggested by
Kaelo and Ali.
NLOPT_GN_ISRES
Global (G) derivative-free (N) optimization using a genetic algorithm (mutation and
differential evolution), using a stochastic ranking to handle nonlinear inequality
and equality constraints as suggested by Runarsson and Yao.
NLOPT_G_MLSL_LDS, NLOPT_G_MLSL
Global (G) optimization using the multi-level single-linkage (MLSL) algorithm with
a low-discrepancy sequence (LDS) or pseudorandom numbers, respectively. This
algorithm executes a low-discrepancy or pseudorandom sequence of local searches,
with a clustering heuristic to avoid multiple local searches for the same local
optimum. The local search algorithm must be specified, along with termination
criteria/tolerances for the local searches, by nlopt_set_local_optimizer. (This
subsidiary algorithm can be with or without derivatives, and determines whether the
objective function needs gradients.)
NLOPT_LD_MMA, NLOPT_LD_CCSAQ
Local (L) gradient-based (D) optimization using the method of moving asymptotes
(MMA), or rather a refined version of the algorithm as published by Svanberg
(2002). (NLopt uses an independent free-software/open-source implementation of
Svanberg's algorithm.) CCSAQ is a related algorithm from Svanberg's paper which
uses a local quadratic approximation rather than the more-complicated MMA model;
the two usually have similar convergence rates. The NLOPT_LD_MMA algorithm
supports both bound-constrained and unconstrained optimization, and also supports
an arbitrary number (m) of nonlinear inequality (not equality) constraints as
described above.
NLOPT_LD_SLSQP
Local (L) gradient-based (D) optimization using sequential quadratic programming
and BFGS updates, supporting arbitrary nonlinear inequality and equality
constraints, based on the code by Dieter Kraft (1988) adapted for use by the SciPy
project. Note that this algorithm uses dense-matrix methods requiring O(n^2)
storage and O(n^3) time, making it less practical for problems involving more than
a few thousand parameters.
NLOPT_LN_COBYLA
Local (L) derivative-free (N) optimization using the COBYLA algorithm of Powell
(Constrained Optimization BY Linear Approximations). The NLOPT_LN_COBYLA algorithm
supports both bound-constrained and unconstrained optimization, and also supports
an arbitrary number (m) of nonlinear inequality/equality constraints as described
above.
NLOPT_LN_NEWUOA
Local (L) derivative-free (N) optimization using a variant of the NEWUOA algorithm
of Powell, based on successive quadratic approximations of the objective function.
We have modified the algorithm to support bound constraints. The original NEWUOA
algorithm is also available, as NLOPT_LN_NEWUOA, but this algorithm ignores the
bound constraints lb and ub, and so it should only be used for unconstrained
problems. Mostly superseded by BOBYQA.
NLOPT_LN_BOBYQA
Local (L) derivative-free (N) optimization using the BOBYQA algorithm of Powell,
based on successive quadratic approximations of the objective function, supporting
bound constraints.
NLOPT_AUGLAG
Optimize an objective with nonlinear inequality/equality constraints via an
unconstrained (or bound-constrained) optimization algorithm, using a gradually
increasing "augmented Lagrangian" penalty for violated constraints. Requires you
to specify another optimization algorithm for optimizing the objective+penalty
function, using nlopt_set_local_optimizer. (This subsidiary algorithm can be
global or local and with or without derivatives, but you must specify its own
termination criteria.) A variant, NLOPT_AUGLAG_EQ, only uses the penalty approach
for equality constraints, while inequality constraints are handled directly by the
subsidiary algorithm (restricting the choice of subsidiary algorithms to those that
can handle inequality constraints).
STOPPING CRITERIA
Multiple stopping criteria for the optimization are supported, as specified by the
functions to modify a given optimization problem opt. The optimization halts whenever any
one of these criteria is satisfied. In some cases, the precise interpretation of the
stopping criterion depends on the optimization algorithm above (although we have tried to
make them as consistent as reasonably possible), and some algorithms do not support all of
the stopping criteria.
Important: you do not need to use all of the stopping criteria! In most cases, you only
need one or two, and can omit the remainder (all criteria are disabled by default).
nlopt_result nlopt_set_stopval(nlopt_opt opt,
double stopval);
Stop when an objective value of at least stopval is found: stop minimizing when a
value <= stopval is found, or stop maximizing when a value >= stopval is found.
(Setting stopval to -HUGE_VAL for minimizing or +HUGE_VAL for maximizing disables
this stopping criterion.)
nlopt_result nlopt_set_ftol_rel(nlopt_opt opt,
double tol);
Set relative tolerance on function value: stop when an optimization step (or an
estimate of the optimum) changes the function value by less than tol multiplied by
the absolute value of the function value. (If there is any chance that your
optimum function value is close to zero, you might want to set an absolute
tolerance with nlopt_set_ftol_abs as well.) Criterion is disabled if tol is non-
positive.
nlopt_result nlopt_set_ftol_abs(nlopt_opt opt,
double tol);
Set absolute tolerance on function value: stop when an optimization step (or an
estimate of the optimum) changes the function value by less than tol. Criterion is
disabled if tol is non-positive.
nlopt_result nlopt_set_xtol_rel(nlopt_opt opt,
double tol);
Set relative tolerance on design variables: stop when an optimization step (or an
estimate of the optimum) changes every design variable by less than tol multiplied
by the absolute value of the design variable. (If there is any chance that an
optimal design variable is close to zero, you might want to set an absolute
tolerance with nlopt_set_xtol_abs as well.) Criterion is disabled if tol is non-
positive.
nlopt_result nlopt_set_xtol_abs(nlopt_opt opt,
const double* tol);
Set absolute tolerances on design variables. tol is a pointer to an array of
length n giving the tolerances: stop when an optimization step (or an estimate of
the optimum) changes every design variable x[i] by less than tol[i].
For convenience, the following function may be used to set the absolute tolerances
in all n design variables to the same value:
nlopt_result nlopt_set_xtol_abs1(nlopt_opt opt,
double tol);
Criterion is disabled if tol is non-positive.
nlopt_result nlopt_set_maxeval(nlopt_opt opt,
int maxeval);
Stop when the number of function evaluations exceeds maxeval. (This is not a
strict maximum: the number of function evaluations may exceed maxeval slightly,
depending upon the algorithm.) Criterion is disabled if maxeval is non-positive.
nlopt_result nlopt_set_maxtime(nlopt_opt opt,
double maxtime);
Stop when the optimization time (in seconds) exceeds maxtime. (This is not a
strict maximum: the time may exceed maxtime slightly, depending upon the algorithm
and on how slow your function evaluation is.) Criterion is disabled if maxtime is
non-positive.
RETURN VALUE
Most of the NLopt functions return an enumerated constant of type nlopt_result, which
takes on one of the following values:
Successful termination (positive return values):
NLOPT_SUCCESS
Generic success return value.
NLOPT_STOPVAL_REACHED
Optimization stopped because stopval (above) was reached.
NLOPT_FTOL_REACHED
Optimization stopped because ftol_rel or ftol_abs (above) was reached.
NLOPT_XTOL_REACHED
Optimization stopped because xtol_rel or xtol_abs (above) was reached.
NLOPT_MAXEVAL_REACHED
Optimization stopped because maxeval (above) was reached.
NLOPT_MAXTIME_REACHED
Optimization stopped because maxtime (above) was reached.
Error codes (negative return values):
NLOPT_FAILURE
Generic failure code.
NLOPT_INVALID_ARGS
Invalid arguments (e.g. lower bounds are bigger than upper bounds, an unknown
algorithm was specified, etcetera).
NLOPT_OUT_OF_MEMORY
Ran out of memory.
NLOPT_ROUNDOFF_LIMITED
Halted because roundoff errors limited progress.
NLOPT_FORCED_STOP
Halted because the user called nlopt_force_stop(opt) on the optimization's
nlopt_opt object opt from the user's objective function.
LOCAL OPTIMIZER
Some of the algorithms, especially MLSL and AUGLAG, use a different optimization algorithm
as a subroutine, typically for local optimization. You can change the local search
algorithm and its tolerances by calling:
nlopt_result nlopt_set_local_optimizer(nlopt_opt opt,
const nlopt_opt local_opt);
Here, local_opt is another nlopt_opt object whose parameters are used to determine the
local search algorithm and stopping criteria. (The objective function, bounds, and
nonlinear-constraint parameters of local_opt are ignored.) The dimension n of local_opt
must match that of opt.
This function makes a copy of the local_opt object, so you can freely destroy your
original local_opt afterwards.
INITIAL STEP SIZE
For derivative-free local-optimization algorithms, the optimizer must somehow decide on
some initial step size to perturb x by when it begins the optimization. This step size
should be big enough that the value of the objective changes significantly, but not too
big if you want to find the local optimum nearest to x. By default, NLopt chooses this
initial step size heuristically from the bounds, tolerances, and other information, but
this may not always be the best choice.
You can modify the initial step size by calling:
nlopt_result nlopt_set_initial_step(nlopt_opt opt,
const double* dx);
Here, dx is an array of length n containing the (nonzero) initial step size for each
component of the design parameters x. For convenience, if you want to set the step sizes
in every direction to be the same value, you can instead call:
nlopt_result nlopt_set_initial_step1(nlopt_opt opt,
double dx);
STOCHASTIC POPULATION
Several of the stochastic search algorithms (e.g., CRS, MLSL, and ISRES) start by
generating some initial "population" of random points x. By default, this initial
population size is chosen heuristically in some algorithm-specific way, but the initial
population can by changed by calling:
nlopt_result nlopt_set_population(nlopt_opt opt,
unsigned pop);
(A pop of zero implies that the heuristic default will be used.)
PSEUDORANDOM NUMBERS
For stochastic optimization algorithms, we use pseudorandom numbers generated by the
Mersenne Twister algorithm, based on code from Makoto Matsumoto. By default, the seed for
the random numbers is generated from the system time, so that they will be different each
time you run the program. If you want to use deterministic random numbers, you can set
the seed by calling:
void nlopt_srand(unsigned long seed);
Some of the algorithms also support using low-discrepancy sequences (LDS), sometimes known
as quasi-random numbers. NLopt uses the Sobol LDS, which is implemented for up to 1111
dimensions.
AUTHORS
Written by Steven G. Johnson.
Copyright (c) 2007-2012 Massachusetts Institute of Technology.
SEE ALSO
nlopt_minimize(3)