NAME

nlopt_minimize_constrained - Minimize a multivariate nonlinear function subject to nonlinear constraints

SYNOPSIS

#include <nlopt.h>

nlopt_result nlopt_minimize_constrained(nlopt_algorithm algorithm,


                            int n,
                            nlopt_func f,
                            void* f_data,
                            int m,
                            nlopt_func fc,
                            void* fc_data,
                            ptrdiff_t fc_datum_size,
                            const double* lb,
                            const double* ub,
                            double* x,
                            double* minf,
                            double minf_max,
                            double ftol_rel,
                            double ftol_abs,
                            double xtol_rel,
                            const double* xtol_abs,
                            int maxeval,
                            double maxtime);

You should link the resulting program with the linker flags
-lnlopt -lm on Unix.

DESCRIPTION

nlopt_minimize_constrained() attempts to minimize a nonlinear function f of n design variables, subject to m nonlinear constraints described by the function fc (see below), using the specified algorithm. The minimum function value found is returned in minf, with the corresponding design variable values returned in the array x of length n. The input values in x should be a starting guess for the optimum. The inputs lb and ub are arrays of length n containing lower and upper bounds, respectively, on the design variables x. The other parameters specify stopping criteria (tolerances, the maximum number of function evaluations, etcetera) and other information as described in more detail below. The return value is a integer code indicating success (positive) or failure (negative), as described below.

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 minimum within the given bounds, and others find only a local minimum. Most of the algorithms only handle the case where m is zero (no explicit nonlinear constraints); the only algorithms that currently support positive m are NLOPT_LD_MMA and NLOPT_LN_COBYLA.

The nlopt_minimize_constrained function 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 the nlopt_minimize_constrained interface. However, depending upon the specific function being minimized, the different algorithms will vary in effectiveness. The intent of nlopt_minimize_constrained 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

nlopt_minimize_constrained() minimizes an objective function f of the form:

double f(int 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_minimize_constrained().

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_minimize_constrained(), 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, where lb and ub are the two arrays passed to nlopt_minimize_constrained().

However, a few of the algorithms support partially or totally unconstrained optimization, as noted below, where a (totally or partially) unconstrained design variable is indicated by a lower bound equal to -Inf and/or an upper bound equal to +Inf. Here, Inf is the IEEE-754 floating-point infinity, which (in ANSI C99) is represented by the macro INFINITY in math.h. Alternatively, for older C versions you may also use the macro HUGE_VAL (also in math.h).

With some of the algorithms, especially those that do not require derivative information, a simple (but not especially efficient) way to implement arbitrary nonlinear constraints is to return Inf (see above) whenever the constraints are violated by a given input x. More generally, there are various ways to implement constraints by adding "penalty terms" to your objective function, which are described in the optimization literature. A much more efficient way to specify nonlinear constraints is described below, but is only supported by a small subset of the algorithms.

NONLINEAR CONSTRAINTS

The nlopt_minimize_constrained function also allows you to specify m nonlinear constraints via the function fc, where m is any nonnegative integer. However, nonzero m is currently only supported by the NLOPT_LD_MMA and NLOPT_LN_COBYLA algorithms below.

In particular, the nonlinear constraints are of the form fc(x) <= 0, where the function fc is of the same form as the objective function described above:

double fc(int n,
const double* x,
double* grad,
void* fc_datum);

The return value should be the value of the constraint at the point x, where the dimension n is identical to the one passed to nlopt_minimize_constrained(). As for the objective function, 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 x. (For any algorithm listed as "derivative-free" below, the grad argument will always be NULL and need never be computed.)

The fc_datum argument is based on the fc_data argument passed to nlopt_minimize_constrained(), and may be used to pass any additional data through to the function, and is used to distinguish between different constraints.

In particular, the constraint function fc will be called (at most) m times for each x, and the i-th constraint (0 <= i < m) will be passed an fc_datum argument equal to fc_data offset by i * fc_datum_size. For example, suppose that you have a data structure of type "foo" that describes the data needed by each constraint, and you store the information for the constraints in an array "foo data[m]". In this case, you would pass "data" as the fc_data parameter to nlopt_minimize_constrained, and "sizeof(foo)" as the fc_datum_size parameter. Then, your fc function would be called m times for each point, and be passed &data[0] through &data[m-1] in sequence.

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 minimum 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.)

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_GN_DIRECT, which is the original DIRECT algorithm; NLOPT_GN_DIRECT_L_RAND, a slightly randomized version of DIRECT-L that may be better in high-dimensional search spaces; NLOPT_GN_DIRECT_NOSCAL, NLOPT_GN_DIRECT_L_NOSCAL, and NLOPT_GN_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 supports arbitrary nonlinear constraints as described above.
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++ enabled, and should be linked via -lnlopt_cxx (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 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 discontinuous objectives, but there is no rigorous guarantee that it will converge.) Nonlinear constraints can be crudely supported by returning +Inf when the constraints are violated, as explained above.
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_GD_MLSL_LDS, NLOPT_GN_MLSL_LDS: Global (G) derivative-based (D) or derivative-free (N) optimization using the multi-level single-linkage (MLSL) algorithm with a low-discrepancy sequence (LDS). This algorithm executes a quasi-random (LDS) sequence of local searches, with a clustering heuristic to avoid multiple local searches for the same local minimum. The local search uses the derivative/nonderivative algorithm set by nlopt_set_local_search_algorithm (currently defaulting to NLOPT_LD_MMA and NLOPT_LN_COBYLA for derivative/nonderivative searches, respectively). There are also two other variants, NLOPT_GD_MLSL and NLOPT_GN_MLSL, which use pseudo-random numbers (instead of an LDS) as in the original MLSL algorithm.
NLOPT_LD_MMA: 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.) The NLOPT_LD_MMA algorithm supports both bound-constrained and unconstrained optimization, and also supports an arbitrary number (m) of nonlinear constraints as described above.
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 constraints as described above.
NLOPT_LN_NEWUOA: Local (L) derivative-free (N) optimization using a variant of the 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.

STOPPING CRITERIA

Multiple stopping criteria for the optimization are supported, as specified by the following arguments to nlopt_minimize_constrained(). 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 set the remainder to values where they do nothing (as described below).

minf_max: Stop when a function value less than or equal to minf_max is found. Set to -Inf or NaN (see constraints section above) to disable.
ftol_rel: Relative tolerance on function value: stop when an optimization step (or an estimate of the minimum) changes the function value by less than ftol_rel multiplied by the absolute value of the function value. (If there is any chance that your minimum function value is close to zero, you might want to set an absolute tolerance with ftol_abs as well.) Disabled if non-positive.
ftol_abs: Absolute tolerance on function value: stop when an optimization step (or an estimate of the minimum) changes the function value by less than ftol_abs. Disabled if non-positive.
xtol_rel: Relative tolerance on design variables: stop when an optimization step (or an estimate of the minimum) changes every design variable by less than xtol_rel 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 xtol_abs as well.) Disabled if non-positive.
xtol_abs: Pointer to an array of length n giving absolute tolerances on design variables: stop when an optimization step (or an estimate of the minimum) changes every design variable x[i] by less than xtol_abs[i]. Disabled if non-positive, or if xtol_abs is NULL.
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.) Disabled if non-positive.
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.) Disabled if non-positive.

RETURN VALUE

The value returned is one of the following enumerated constants.

Successful termination (positive return values):

NLOPT_SUCCESS: Generic success return value.
NLOPT_MINF_MAX_REACHED: Optimization stopped because minf_max (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.

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.