MONET Definitions

Target Namespace: http://monet.nag.co.uk/problems/

Problem . . .
constrained_minimisation
Header . . .
Taxonomy: http://gams.nist.gov
Code: GamsG
Body . . .
Input . . .
The objective function.
objective ∈ R n → R
The bounds on the variables.
simple_bounds ∈ list ⁡ ( interval ⁡ ( R ) )
The set of linear constraint functions
linear_constraint_functions ∈ list ⁡ ( R n → R )
The bounds on the linear constraint functions
linear_constraint_bounds ∈ list ⁡ ( interval ⁡ ( R ) )
The set of non-linear constraint functions
non_linear_constraint_functions ∈ list ⁡ ( R n → R )
The bounds on the non-linear constraint functions
non_linear_constraint_bounds ∈ list ⁡ ( interval ⁡ ( R ) )
Output . . .
The point at which the minimum occurs
minimum ∈ R n
Pre-Condition . . .
size ⁡ ( simple_bounds ) = n
size ⁡ ( linear_constraint_functions ) = size ⁡ ( linear_constraint_bounds )
size ⁡ ( non_linear_constraint_functions ) = size ⁡ ( non_linear_constraint_bounds )
∀ f . f ∈ linear_constraint_functions ⇒ linear ⁡ ( f )
∀ f . f ∈ non_linear_constraint_functions ⇒ non_linear ⁡ ( f )
Post-Condition . . .
This states that the simple bounds are satisfied at the minimum.
∀ i . i ∈ [ 1 , n ] ⇒ minimum i ∈ simple_bounds i
This states that the linear constraints are satisfied at the minimum.
∀ i . i ∈ [ 1 , size ⁡ ( linear_constraint_functions ) ] ⇒ ( linear_constraint_functions i ) ⁡ ( minimum ) ∈ linear_constraint_bounds i
This states that the non-linear constraints are satisfied at the minimum.
∀ i . i ∈ [ 1 , size ⁡ ( non_linear_constraint_functions ) ] ⇒ ( non_linear_constraint_functions i ) ⁡ ( minimum ) ∈ non_linear_constraint_bounds i
This states that for all values x which satisfy all three sets of constraints, the value of objective evaluated at x is greater than or equal to the minimum.
∀ x . x ∈ R n ∧ ∀ i . i ∈ [ 1 , n ] ⇒ x i ∈ simple_bounds i ∧ ∀ i . i ∈ [ 1 , size ⁡ ( linear_constraint_functions ) ] ⇒ ( linear_constraint_functions i ) ⁡ ( x ) ∈ linear_constraint_bounds i ∧ ∀ i . i ∈ [ 1 , size ⁡ ( non_linear_constraint_functions ) ] ⇒ ( non_linear_constraint_functions i ) ⁡ ( x ) ∈ non_linear_constraint_bounds i ⇒ objective ⁡ ( x ) ≥ objective ⁡ ( minimum )