The AMPGO Solver¶

AMPGO stands for Adaptive Memory Programming for Global Optimization, and its detailed implementation is described in the paper “Adaptive Memory Programming for Constrained Global Optimization” located here:

The following figure provides a pseudo-code description of the Tabu Tunneling algorithm, which is the basis of the global search in AMPGO:

AMPGO Pseudo code - tabu tunnelling algorithm

AMPGO Function Signature in Python¶

The call signature of AMPGO‘s Python implementation follows very closely the standard signature for the minimization functions in the scipy.optimize package in scipy.

ampgo.AMPGO(objfun, x0, args=(), local='L-BFGS-B', bounds=None, maxfunevals=None, totaliter=20, maxiter=5, glbtol=1e-5, eps1=0.02, eps2=0.1, tabulistsize=5, tabustrategy='farthest', fmin=-numpy.inf, disp=None)

Finds the global minimum of a function using the AMPGO (Adaptive Memory Programming for Global Optimization) algorithm.

Parameters: objfun (callable) – Function to be optimized, in the form f(x, *args). args (tuple) – Additional arguments passed to objfun. local (string) – The local minimization method (e.g. "L-BFGS-B"). It can be one of the available scipy local solvers or OpenOpt solvers. bounds (list) – A list of tuples specifying the lower and upper bound for each independent variable [(xl0, xu0), (xl1, xu1), ...] maxfunevals (integer) – The maximum number of function evaluations allowed. totaliter (integer) – The maximum number of global iterations allowed. maxiter (integer) – The maximum number of Tabu Tunnelling iterations allowed during each global iteration. glbtol (float) – The optimization will stop if the absolute difference between the current minimum objective function value and the provided global optimum (fmin) is less than glbtol. eps1 (float) – A constant used to define an aspiration value for the objective function during the Tunnelling phase. eps2 (float) – Perturbation factor used to move away from the latest local minimum at the start of a Tunnelling phase. tabulistsize (integer) – The size of the tabu search list (a circular list). tabustrategy (string) – The strategy to use when the size of the tabu list exceeds tabulistsize. It can be ‘oldest’ to drop the oldest point from the tabu list or ‘farthest’ to drop the element farthest from the last local minimum found. fmin (float) – If known, the objective function global optimum value. disp (integer) – If zero or defaulted, then no output is printed on screen. If a positive number, then status messages are printed. A tuple of 5 elements, in the following order: best_x (array_like): the estimated position of the global minimum. best_f (float): the value of objfun at the minimum. evaluations (integer): the number of function evaluations. msg (string): a message describes the cause of the termination. tunnel_info (tuple): a tuple containing the total number of Tunnelling phases performed and the successful ones. tuple

Sensitivities on the AMPGO Input Parameters¶

This sections presents a number of sensitivities on the AMPGO input parameters, which may be helpful in guiding the choice of the parameters themselves depending on the difficulty of the optimization problem at hand. The most fundamental argument is evidently the local solver, followed by the maximum number of Tabu Tunnelling iterations allowed and the length of the list containing the “tabu points” (i.e., the previous local minima found by the method).

The Local Solver¶

The AMPGO Python implementation supports a number of local solvers, from scipy and OpenOpt . The available solvers can be categorized as follows:

Scipy-related local solvers supported by AMPGO
Scipy Solver Description
L-BFGS-B L-BFGS-B algorithm
Powell Powell’s conjugate direction method
SLSQP Sequential Least Squares Programming
TNC Truncated Newton algorithm

OpenOpt-related local solvers supported by AMPGO
OpenOpt Solver Description
AUGLAG Augmented Lagrangian method
BOBYQA Powell’s derivative-free method
MMA Method of moving asymptotes
PTN Preconditioned truncated Newton method
RALG R-algorithm with adaptive space dilation
SLMVM2 Shifted limited-memory variable-metric

I applied AMPGO to the entire benchmark suite of N-D optimization problems, considering for every benchmark function 100 random starting points. As there are currently 12 local solvers supported by AMPGO, I re-run the same suite 12 times changing the local solver.

The results obtained are shown in the following table:

Sensitivity on AMPGO local solvers
Optimization Method Overall Success (%) Functions Evaluations
AUGLAG 57.957 1633
BOBYQA 70.418 813
L-BFGS-B 79.027 611
MMA 57.489 1656
Powell 63.636 1248
PTN 60.777 1042
RALG 64.565 952
SLMVM2 67.054 861
SLSQP 65.484 3039
SQLCP 46.902 1231
TNC 63.804 861

So, for example, AMPGO with L-BFGS-B local solver was able to solve, on average, 79.0% of all the test functions for all the 100 random starting points using, on average, 611 functions evaluations. This is also shown graphically in the picture below.

AMPGO - Local solvers sensitivity

Number of Tabu Tunnelling Phases (maxiter)¶

One of the parameters you can supply to the AMPGO optimization routine is the maximum number of Tabu Tunnelling iterations allowed during each global iteration (maxiter). Once a local minimum is found, AMPGO will start a Tunnelling phase to try and move away from this local optimum; if no better point is found, it will start again from another point and try to move away from this local minimum. It will do so for maxiter times.

The table below highlights the various experiments I have run by applying AMPGO to the entire benchmark suite of N-D optimization problems, considering for every benchmark function 100 random starting points. The maxiter parameter has been allowed to range between 1 and 8, but the results are inconclusive: 2 seems to be the absolute minimum to get a higher level of global convergence, but then there isn’t much of a difference between choosing maxiter=3 and maxiter=8.

The results obtained are shown in the following table:

Sensitivity on the number of Tabu Tunnelling phases
Tabu Tunnelling Phases Overall Success (%) Functions Evaluations
1 76.783 635
2 78.837 607
3 78.875 615
4 78.685 619
5 79.027 611
6 78.864 625
7 78.750 623
8 78.647 635

The results appear slightly more interesting if we remove from the analysis all the N-D benchmark functions for which AMPGO achieved global convergence irrespectively of the number of Tabu Tunnelling phases. This information is condensed in the following table:

Sensitivity on the number of Tabu Tunnelling phases
Tabu Tunnelling Phases Overall Success (%) Functions Evaluations
1 49.143 1319
2 53.643 1256
3 53.726 1274
4 53.310 1281
5 54.060 1262
6 53.702 1292
7 53.452 1288
8 53.226 1315

The above results are also shown graphically in the next figure; all in all, the default choice of maxiter=5 in the AMPGO code appears to be a sensible one.

AMPGO - Number of tabu tunnelling phases sensitivity

Size of the Tabu List (tabulistsize)¶

Another parameter you can supply to the AMPGO optimization routine is the maximum size of the Tabu List, which is a list containing the latest tabulistsize local minima found by the optimizer. This is a list of points from which to move away, including both the most recent local solution of the original minimization problem, and recent solutions of tunneling sub-problems which failed to achieve the desired condition.

The table below highlights the various experiments I have run by applying AMPGO to the entire benchmark suite of N-D optimization problems, considering for every benchmark function 100 random starting points.

The tabulistsize parameter has been allowed to range between 2 and 8, but again the results are inconclusive: There appear to be little difference between choosing 2 over 8, even though for real-life problems (with many local minima) I would suggest to keep the tabulistsize number greater than 2.

The results obtained are shown in the following table:

Sensitivity on the size of the tabu list
Tabu List Size Overall Success (%) Functions Evaluations
2 77.272 633
3 78.652 626
4 78.880 618
5 79.027 611
6 79.190 620
7 79.114 617
8 79.527 604

Again, removing from the analysis all the N-D benchmark functions for which AMPGO achieved global convergence irrespectively of the size of the tabu list shows some more interesting trend, as depicted in the following table:

Sensitivity on the size of the tabu list
Tabu List Size Overall Success (%) Functions Evaluations
2 53.011 1243
3 55.865 1232
4 56.337 1217
5 56.640 1202
6 56.978 1223
7 56.820 1215
8 57.674 1190

The above results are also shown graphically in the next figure; all in all, it appears that choosing a high enough size of the tabu list slightly improves the optimization results.

AMPGO - Size of the tabu list sensitivity

Tabu List Strategy (tabustrategy)¶

Another parameter you can supply to the AMPGO optimization routine is the Tabu List strategy, which is the strategy to use when the size of the tabu list exceeds tabulistsize. It can be ‘oldest’ to drop the oldest point from the tabu list or ‘farthest’ to drop the element farthest from the last local minimum found.

The table below highlights the various experiments I have run by applying AMPGO to the entire benchmark suite of N-D optimization problems, considering for every benchmark function 100 random starting points.

The tabustrategy parameter has been allowed to range between ‘oldest’ and ‘farthest’.

The results obtained are shown in the following table:

Sensitivity on the tabustrategy parameter
Tabu Strategy Overall Success (%) Functions Evaluations
farthest 79.027 611
oldest 78.750 630

This time, removing from the analysis all the N-D benchmark functions for which AMPGO achieved global convergence irrespectively of the tabustrategy value shows very little difference between the two strategies, so any user’s choice should be sensible in this respect.

Sensitivity on the tabustrategy parameter
Tabu Strategy Overall Success (%) Functions Evaluations
farthest 50.526 1347
oldest 49.872 1390

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