# Monotonic Basin Hopping (MBH) - Generalized

class pagmo::mbh

Monotonic Basin Hopping (generalized).

Monotonic basin hopping, or simply, basin hopping, is an algorithm rooted in the idea of mapping the objective function $$f(\mathbf x_0)$$ into the local minima found starting from $$\mathbf x_0$$. This simple idea allows a substantial increase of efficiency in solving problems, such as the Lennard-Jones cluster or the MGA-1DSM interplanetary trajectory problem that are conjectured to have a so-called funnel structure.

In pagmo we provide an original generalization of this concept resulting in a meta-algorithm that operates on any pagmo::population using any suitable user-defined algorithm (UDA). When a population containing a single individual is used and coupled with a local optimizer, the original method is recovered. The pseudo code of our generalized version is:

> Create a pagmo::population
> Select a UDA
> Store best individual
> while i < stop_criteria
> > Perturb the population in a selected neighbourhood
> > Evolve the population using the algorithm
> > if the best individual is improved
> > > increment i
> > > update best individual
> > else
> > > i = 0


pagmo::mbh is a user-defined algorithm (UDA) that can be used to construct pagmo::algorithm objects.

See: https://arxiv.org/pdf/cond-mat/9803344.pdf for the paper introducing the basin hopping idea for a Lennard-Jones cluster optimization.

Public Types

typedef std::tuple<unsigned long long, double, vector_double::size_type, double, unsigned> log_line_type

Single entry of the log (feval, best fitness, n. constraints violated, violation norm, trial).

typedef std::vector<log_line_type> log_type

The log.

Public Functions

mbh()

Default constructor.

The default constructor will initialize the algorithm with the following parameters:

• inner algorithm: pagmo::compass_search;

• consecutive runs of the inner algorithm that need to result in no improvement for pagmo::mbh to stop: 5;

• scalar perturbation: 1E-2;

• seed: random.

Throws

unspecified – any exception thrown by the constructor of pagmo::algorithm.

template<typename T, ctor_enabler<T> = 0>
inline explicit mbh(T &&a, unsigned stop, double perturb, unsigned seed = pagmo::random_device::next())

Constructor (scalar perturbation).

This constructor will construct a monotonic basin hopping algorithm using a scalar perturbation.

Note

This constructor is enabled only if T can be used to construct a pagmo::algorithm.

Parameters
• a – a user-defined algorithm (UDA) or a pagmo::algorithm that will be used to construct the inner algorithm.

• stop – consecutive runs of the inner algorithm that need to result in no improvement for pagmo::mbh to stop.

• perturb – the perturbation to be applied to each component of the decision vector of the best population found when generating a new starting point. These are defined relative to the corresponding bounds.

• seed – seed used by the internal random number generator (default is random).

Throws
• unspecified – any exception thrown by the constructor of pagmo::algorithm.

• std::invalid_argument – if perturb is not in the (0,1] range.

template<typename T, ctor_enabler<T> = 0>
inline explicit mbh(T &&a, unsigned stop, vector_double perturb, unsigned seed = pagmo::random_device::next())

Constructor (vector perturbation).

This constructor will construct a monotonic basin hopping algorithm using a vector perturbation.

Warning

This constructor is enabled only if T, after the removal of cv/reference qualifiers, is not pagmo::algorithm.

Parameters
• a – a user-defined algorithm (UDA) or a pagmo::algorithm that will be used to construct the inner algorithm.

• stop – consecutive runs of the inner algorithm that need to result in no improvement for pagmo::mbh to stop.

• perturb – a pagmo::vector_double with the perturbations to be applied to each component of the decision vector of the best population found when generating a new starting point. These are defined relative to the corresponding bounds.

• seed – seed used by the internal random number generator (default is random).

Throws
• unspecified – any exception thrown by the constructor of pagmo::algorithm.

• std::invalid_argument – if not all the elements of perturb are in the (0,1] range.

population evolve(population) const

Evolve method.

This method will evolve the input population up to when stop consecutve runs of the internal algorithm do not improve the solution.

Parameters

pop – population to be evolved.

Throws

std::invalid_argument – if the problem is multi-objective or stochastic, or if the perturbation vector size does not equal the problem size.

Returns

evolved population.

void set_seed(unsigned)

Set the seed.

Parameters

seed – the seed controlling the algorithm’s stochastic behaviour.

inline unsigned get_seed() const

Get the seed.

Returns

the seed controlling the algorithm’s stochastic behaviour.

inline void set_verbosity(unsigned level)

Set the algorithm verbosity.

This method will sets the verbosity level of the screen output and of the log returned by get_log(). level can be:

• 0: no verbosity,

• >0: will print and log one line at the end of each call to the inner algorithm.

Example (verbosity 100):

Fevals:          Best:      Violated:    Viol. Norm:         Trial:
105        110.395              1      0.0259512              0 i
211        110.395              1      0.0259512              1 i
319        110.395              1      0.0259512              2 i
422        110.514              1      0.0181383              0 i
525         111.33              1      0.0149418              0 i
628         111.33              1      0.0149418              1 i
731         111.33              1      0.0149418              2 i
834         111.33              1      0.0149418              3 i
937         111.33              1      0.0149418              4 i
1045         111.33              1      0.0149418              5 i

Fevals is the number of fitness evaluations, Best is the objective function of the best fitness currently in the population, Violated is the number of constraints currently violated by the best solution, Viol. Norm is the norm of the violation (discounted already by the constraints tolerance) and Trial is the trial number (which will determine the algorithm stop). The small i appearing at the end of the line stands for “infeasible” and will disappear only once Violated is 0.

Parameters

level – verbosity level.

inline unsigned get_verbosity() const

Get the verbosity level.

Returns

the verbosity level.

inline const vector_double &get_perturb() const

Get the perturbation vector.

Returns

a const reference to the perturbation vector.

void set_perturb(const vector_double&)

Set the perturbation vector.

Parameters

perturb – the perturbation vector.

Throws

std::invalid_argument – if not all the components of the perturbation vector are in the (0,1] range.

The thread safety of this meta-algorithm is the minimum between the thread safety of the internal pagmo::algorithm and the basic thread safety level. I.e., this algorithm never provides more than the basic thread safety level.

Returns

the thread safety level of this algorithm.

inline const algorithm &get_inner_algorithm() const

Getter for the inner algorithm.

Returns a const reference to the inner pagmo::algorithm.

Returns

a const reference to the inner pagmo::algorithm.

inline algorithm &get_inner_algorithm()

Getter for the inner algorithm.

Returns a reference to the inner pagmo::algorithm.

Warning

The ability to extract a non const reference is provided only in order to allow to call non-const methods on the internal pagmo::algorithm instance. Assigning a new pagmo::algorithm via this reference is undefined behaviour.

Returns

a reference to the inner pagmo::algorithm.

inline const log_type &get_log() const

Get log.

A log containing relevant quantities monitoring the last call to mbh::evolve(). Each element of the returned std::vector is a mbh::log_line_type containing: Fevals, Best, Violated, Viol.Norm and Trial as described in mbh::set_verbosity().

Returns

an std::vector of mbh::log_line_type containing the logged values Fevals, Best, Violated, Viol.Norm and Trial.

inline std::string get_name() const

Algorithm name.

Returns

a string containing the algorithm name.

std::string get_extra_info() const

Extra info.

Returns

a string containing extra info on the algorithm.