# MINLP Rastrigin#

struct minlp_rastrigin#

A MINLP version of the Rastrigin problem.

This is a scalable, box-constrained, mixed integer nonlinear programmng (MINLP) problem. The objective function is the generalised n-dimensional Rastrigin function:

$F\left(x_1,\ldots,x_n\right) = 10 \cdot n + \sum_{i=1}^n x_i^2 - 10\cdot\cos\left( 2\pi \cdot x_i \right)$

where we constraint the last $$m$$ components of the decision vector to be integers. The variables are box bounded as follows: $$\quad x_i \in [-5.12,5.12], \forall i = 1 .. n-m$$, $$\quad x_i \in [-10,-5], \forall i = m+1 .. n$$

Gradients (dense) are also provided (also for the integer part) as:

$G_i\left(x_1,\ldots,x_n\right) = 2 x_i + 10 \cdot 2\pi \cdot\sin\left( 2\pi \cdot x_i \right)$
And Hessians (sparse as only the diagonal is non-zero) are:
$H_{ii}\left(x_1,\ldots,x_n\right) = 2 + 10 \cdot 4\pi^2 \cdot\cos\left( 2\pi \cdot x_i \right)$

Public Functions

minlp_rastrigin(unsigned dim_c = 1u, unsigned dim_i = 1u)#

Constructor from continuous and integer dimension.

Constructs a MINLP Rastrigin problem.

Parameters
• dim_c – the problem continuous dimension.

• dim_i – the problem continuous dimension.

Throws

std::invalid_argument – if dim_c+ dim_i is < 1

vector_double fitness(const vector_double&) const#

Fitness computation.

Computes the fitness for this UDP.

Parameters

x – the decision vector.

Returns

the fitness of x.

std::pair<vector_double, vector_double> get_bounds() const#

Box-bounds.

It returns the box-bounds for this UDP.

Returns

the lower and upper bounds for each of the decision vector components

inline vector_double::size_type get_nix() const#

Integer dimension.

It returns the integer dimension of the problem.

Returns

the integer dimension of the problem.

It returns the fitness gradient for this UDP.

The gradient is represented in a sparse form as required by problem::gradient().

Parameters

x – the decision vector.

Returns

the gradient of the fitness function

std::vector<vector_double> hessians(const vector_double&) const#

Hessians.

It returns the hessians for this UDP.

The hessians are represented in a sparse form as required by problem::hessians().

Parameters

x – the decision vector.

Returns

the hessians of the fitness function

std::vector<sparsity_pattern> hessians_sparsity() const#

Hessians sparsity (only the diagonal elements are non zero)

It returns the hessian sparisty structure for this UDP.

The hessian sparisty is represented in the form required by problem::hessians_sparsity().

Returns

the hessians of the fitness function

inline std::string get_name() const#

Problem name.

Returns

a string containing the problem name

std::string get_extra_info() const#

Extra info.

Returns

a string containing extra info on the problem