NLopt Algorithms
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===Multi-level single-linkage (MLSL)=== | ===Multi-level single-linkage (MLSL)=== | ||
+ | This is my implementation of the "Multi-Level Single-Linkage" (MLSL) algorithm for global optimization by a sequence random local optimizations, proposed by: | ||
+ | |||
+ | * A. H. G. Rinnooy Kan and G. T. Timmer, "Stochastic global optimization methods," ''Mathematical Programming'', vol. 39, p. 27-78 (1987). (Actually 2 papers — part I: clustering methods, p. 27, then part II: multilevel methods, p. 57.) | ||
+ | |||
+ | We also include a modification of MLSL use a Sobol low-discrepancy sequence (LDS) instead of pseudorandom numbers, which was argued to improve the convergence rate by: | ||
+ | |||
+ | * Sergei Kucherenko and Yury Sytsko, "Application of deterministic low-discrepancy sequences in global optimization," ''Computational Optimization and Applications'', vol. 30, p. 297-318 (2005). | ||
+ | |||
+ | In either case, MLSL is a "multistart" algorithm: it works by doing a sequence of local optimizations (using some other local optimization algorithm) from random or low-discrepancy starting points. MLSL is distinguished, however by a "clustering" heuristic that helps it to avoid repeated searches of the same local optima, and has some theoretical guarantees of finding all local optima in a finite number of local minimizations. | ||
+ | |||
+ | The local-search portion of MLSL can use any of the other algorithms in NLopt, and in particular can use either gradient-based (<code>D</code>) or derivative-free algorithms (<code>N</code>) The local search uses the derivative/nonderivative algorithm set by nlopt_set_local_search_algorithm (currently defaulting to <code>NLOPT_LD_MMA</code> and <code>NLOPT_LN_COBYLA</code> for derivative/nonderivative searches, respectively). | ||
+ | |||
+ | LDS-based MLSL with gradient-based local search is specified as <code>NLOPT_GD_MLSL_LDS</code>, while LDS-based MLSL with derivative-free local search is specified by <code>NLOPT_GN_MLSL_LDS</code>. The non-LDS original MLSL (using pseudo-random numbers, currently via the Mersenne twister algorithm) is indicated by <code>NLOPT_GD_MLSL</code> and <code>NLOPT_GN_MLSL</code>, respectively. | ||
===StoGO=== | ===StoGO=== |
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NLopt includes implementations of a number of different optimization algorithms. These algorithms are listed below, including links to the original source code (if any) and citations to the relevant articles in the literature (see Citing NLopt).
Even where I found available free/open-source code for the various algorithms, I modified the code at least slightly (and in some cases noted below, substantially) for inclusion into NLopt. I apologize in advance to the authors for any new bugs I may have inadvertantly introduced into their code.
Contents |
Nomenclature
Each algorithm in NLopt is identified by a named constant, which is passed to the NLopt routines in the various languages in order to select a particular algorithm. These constants are of the form NLOPT_{G,L}{N,D}_xxxx
, where G
/L
denotes global/local optimization and N
/D
denotes derivative-free/gradient-based algorithms, respectively.
For example, the NLOPT_LN_COBYLA
constant refers to the COBYLA algorithm (described below), which is a local (L
) derivative-free (N
) optimization algorithm.
Many of the algorithms have several variants, which are grouped together below.
Global optimization
DIRECT and DIRECT-L
DIRECT is the DIviding RECTangles algorithm for global optimization, described in:
- D. R. Jones, C. D. Perttunen, and B. E. Stuckmann, "Lipschitzian optimization without the lipschitz constant," J. Optimization Theory and Applications, vol. 79, p. 157 (1993).
and DIRECT-L is the "locally biased" variant proposed by:
- J. M. Gablonsky and C. T. Kelley, "A locally-biased form of the DIRECT algorithm," J. Global Optimization, vol. 21 (1), p. 27-37 (2001).
These is are deterministic-search algorithms based on systematic division of the search domain into smaller and smaller hyperrectangles. The Gablonsky version makes the algorithm "more biased towards local search" so that it
is more efficient for functions without too many local minima. NLopt contains several implementations of both of these algorithms. I would tend to try NLOPT_GN_DIRECT_L
first; YMMV.
First, it contains a from-scratch re-implementation of both algorithms, specified by the constants NLOPT_GN_DIRECT
and NLOPT_GN_DIRECT_L
, respectively.
Second, there is a slightly randomized variant of DIRECT-L, specified by NLOPT_GLOBAL_DIRECT_L_RAND
, which uses some randomization to help decide which dimension to halve next in the case of near-ties.
The DIRECT and DIRECT-L algorithms start by rescaling the bound constraints to a hypercube, which gives all dimensions equal weight in the search procedure. If your dimensions do not have equal weight, e.g. if you have a "long and skinny" search space and your function varies at about the same speed in all directions, it may be better to use unscaled variants of these algorthms, which are specified as NLOPT_GLOBAL_DIRECT_NOSCAL
, NLOPT_GLOBAL_DIRECT_L_NOSCAL
, and NLOPT_GLOBAL_DIRECT_L_RAND_NOSCAL
, respectively. These implementations have a number of hard-coded limitations on things like the number of function evaluations; I removed several of these limitations, but some remain.
Finally, NLopt also includes separate implementations based on the original Fortran code by Gablonsky et al., which are specified as NLOPT_GN_ORIG_DIRECT
and NLOPT_GN_ORIG_DIRECT_L
.
Most of the above algorithms only handle bound constraints, and in fact require finite bound constraints (they are not applicable to unconstrained problems). They do not handle arbitrary nonlinear constraints. However, the ORIG
versions by Gablonsky et al. include some support for arbitrary constraints, if you represent constraint violations by having your objective function return NaN or infinity (HUGE_VAL
, in C).
Controlled Random Search (CRS) with local mutation
My implementation of the "controlled random search" (CRS) algorithm (in particular, the CRS2 variant) with the "local mutation" modification, as defined by:
- P. Kaelo and M. M. Ali, "Some variants of the controlled random search algorithm for global optimization," J. Optim. Theory Appl. 130 (2), 253-264 (2006).
The original CRS2 algorithm was described by:
- W. L. Price, "A controlled random search procedure for global optimization," in Towards Global Optimization 2, p. 71-84 edited by L. C. W. Dixon and G. P. Szego (North-Holland Press, Amsterdam, 1978).
The CRS algorithms are sometimes compared to genetic algorithms, in that they start with a random "population" of points, and randomly "evolve" these points by heuristic rules. In this case, the "evolution" somewhat resembles a randomized Nelder-Mead algorithm.
Only bound-constrained problems are supported by this algorithm.
Multi-level single-linkage (MLSL)
This is my implementation of the "Multi-Level Single-Linkage" (MLSL) algorithm for global optimization by a sequence random local optimizations, proposed by:
- A. H. G. Rinnooy Kan and G. T. Timmer, "Stochastic global optimization methods," Mathematical Programming, vol. 39, p. 27-78 (1987). (Actually 2 papers — part I: clustering methods, p. 27, then part II: multilevel methods, p. 57.)
We also include a modification of MLSL use a Sobol low-discrepancy sequence (LDS) instead of pseudorandom numbers, which was argued to improve the convergence rate by:
- Sergei Kucherenko and Yury Sytsko, "Application of deterministic low-discrepancy sequences in global optimization," Computational Optimization and Applications, vol. 30, p. 297-318 (2005).
In either case, MLSL is a "multistart" algorithm: it works by doing a sequence of local optimizations (using some other local optimization algorithm) from random or low-discrepancy starting points. MLSL is distinguished, however by a "clustering" heuristic that helps it to avoid repeated searches of the same local optima, and has some theoretical guarantees of finding all local optima in a finite number of local minimizations.
The local-search portion of MLSL can use any of the other algorithms in NLopt, and in particular can use either gradient-based (D
) or derivative-free algorithms (N
) 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).
LDS-based MLSL with gradient-based local search is specified as NLOPT_GD_MLSL_LDS
, while LDS-based MLSL with derivative-free local search is specified by NLOPT_GN_MLSL_LDS
. The non-LDS original MLSL (using pseudo-random numbers, currently via the Mersenne twister algorithm) is indicated by NLOPT_GD_MLSL
and NLOPT_GN_MLSL
, respectively.