Topo-Geometric Model MZ: Feeded Objects

Andrianarizaka Marc Tiana, Doctoral School in Engineering and Innovation Science and Technology (ED-STII) Doctoral Research Team for Cognitive Science and Application (EAD-SCA) University of Antananarivo, MADAGASCAR Robinson Matio, Doctoral School in Engineering and Innovation Science and Technology (ED-STII) Doctoral Research Team for Cognitive Science and Application (EAD-SCA) University of Antananarivo, MADAGASCAR


INTRODUCTION
This paper constitutes the basis of the Topo-geometric model MZ in search of redundant constraints in a problem of linear programming. This article follows the work on the Topo-Geometric MZ model published in MADA-ETI,   ISSN 2220-0673, Vol.2, 2016, www.madarevues.gov.mg which develops the basic concept of Topo-geometric models, and presents the powered objects in search of the redundant constraints of a linear programming problem. All the mathematical object classes will first be presented from the simplest to the most complicated, to model the real concept allowing the decision making. Together with the presentation of each class, the existing operations and relationships will be analyzed; the representation of technical constraints by affine half-spaces is the most important part of this work.

The scalars
A scalar is any real number. Symbolically, a scalar will be represented with a Latin or Greek or lowercase letter.
x y z 

Indexes
It is a positive or null integer that will be attached to an object of a model, as a unique identifier. Symbolically the indexes will be represented by i, j, k...
To represent the possible values of an index, the following notions will be used: i = 1, 2...n meaning that the index i varies from 1 to n.
When an objective x of a model is an index, one writes: which is a condensed representation of :  Conditions (3.03) are called non-negativity constraints.

Definition 3.01: decision-action-point-vectors
We call decision-action any n-tuple : x Whose 12 , ,..., n x x x are elements of It will be noted later : Mathematically, we see that it is an element of n relative to a given base. It is also a point in the affine space n relative to a given landmark. An action decision will be represented by an affine point. Which brings us to the decision-action definition? Notation : an action decision is noted using a capital Greek letter: A, B, X, Y Despite this striking resemblance to Euclidean affine geometry, a limiting aspect will be presented from the start.

Postulate
In the topo-geometric theory MZ, the orthonormal affine frame of the decision space action of the problem LPP to be solved is unique. This assumption means that : -The concept of basic change does not exist.
-The concept of change of origin does not exist. However, following a presolved analysis, it may be possible that n changes, which completely modifies the LPP problem.
4 OPERATIONS ON ACTION-DECISIONS Given an LPP problem, according to definition 3.01, a decision is represented by a point in the decision-action space and / or by a vector of the vector space ℝ n This dual nature (both affine and vectorial) of decisionactions is very important. It allows to express : The creation of other stock-decisions based on existing stock-decisions The search for other decision-actions according to a given direction.

Amplifier of an action decision
Let the decision-action space relate to the reference. Let X0 be an action decision and α a given positive real number. We call amplifier of X0 using α, the generation of an action decision X such that X = αX0 Mathematically, it is therefore the multiplier of vector X0 by the scalar α

Addition of two decision-actions
Let X0 and X1 be two decision-actions in a LPP problem. The addition of X0 and X1 is called the creation of a new action decision X such that X = X0 + X1 Note 4.01 The combination of 2.3.1 and 2.3.2 allows us to model the conic combination concept of two or more action-decisions. Let the decision-actions X 1, X 2... X k.
The conic combination of these decisions is the new decision X defined by: X = α 1 X 1 + α 2 X 2 + ... + α k X k α 1, α 2 ...., α k are positive real numbers or zero. Mathematically, it is the affine ray derived from X0 and the vector V. Conceptually, the research idea is justified by the fact that in practice, any solver is based on a search algorithm starting from an initial point. Thus, from a given decision-action X0, and a given search direction V, one can search the points in the universe of decisions that will satisfy the constraints (3.02) and (3.03). This research idea is not confined to a single research direction. It is also possible to adopt simultaneously two search directions V1 and V2. Hence the concept of "research plan ", Starting from X0 and direction V1 and V2.

THE TECHNICAL CONSTRAINTS
In practice, any action-decision always has an impact. In the field of linear programming, two categories of impacts can be distinguished : -Impact-performance. -Resource impacts.
In this study, we only deal with the impact-resources.
Definition 5.01 resource impact Let X be an action decision point of a given LPP problem. Let V be a null vector of n .We call resource impact according to V, the scalar product of X and V in n noted: ℑ (X, V) = <X, V> Where <X, V> denotes the scalar product U with V. The vector V is called unit consumption vector V of the given resource.

Definition 5.02 constraint resources
In our theory, we assume that a constraint that is logical or material may be associated with a resource that is always limited in the associated LPP problem. Let A = (aj) j = 1... n be a vector V of consumption of a given resource. Let b be the limit value of the resource, the zone of respect of the consumption of the resource limited by b is the set noted ZR (A, b) defined by : In affine geometry, this set is none other than the negative half-space delimited by the noted affine hyperplane. HP (A, b) defined by:

NB:
The vector A is not an action-decision it is linked to a given resource and allows to calculate the impact of an action X decision on this resource 6 The constraints of non-negativity The non-negativity constraints translate the fact that the elementary decisions xi, where i = 1... n must be nonnegatives, is : Let us note immediately that the relations : However the combination of all non-negativity constraints has a particular geometric meaning that we call " nonnegativity cone ".

non-negativity cone
We call non-negativity cone noted 0 C  the cone of search from point-decision O and direction of all decision vectors : , 1, 2,...., i U i n  Geometrically we have:

Non-negativity axes
For all i = 1, 2... n, we call non-negativity axis, ANNi , the set of decision point defined as follows : ANNi =R (O, Ui) i = 1.2... n (6.02) Note that ANNi is none other than the search axis starting from O and with the direction Ui. Recall the concept of redundant constraints.

REDUNDANT CONSTRAINTS
A redundant constraint is a constraint which can be deleted from a system of linear constraints without changing the feasible region or acceptable solution area. If we look at the next system of m and n constraints of linear inequality, no negative (m ≥ n), we can adopt the matrix writing: Let AT i ≤ bi be the i th constraint of system (7.01) and let The acceptable solution area associated with the system (7 .01). Let The acceptable solution area associated to the constraint : is redundant for the system (7.01) if and only if S = Sk.

Definition 7.01
Redondant constraints can be classified as weak and strong redundant constraints. 1.3.2 Low redundancy constraints The constraint ATi X ≤ bi is weakly redundant if she is redundant and ATi X = bi for all X ∈ S.
8 RELATIONSHIP BETWEEN MZ OBJECTS The relationships described in this section are the basic ones. More complex relationships will be discussed in the next chapter. Moreover, the combinations between point-decisions and point-vectors have already been seen. Therefore, only the following points will be dealt with: -The relation between a decision-point and a constrainttechnical zone.
-The relation between a research axis and a zone of technical constraints.
-The relation between a research plan and a zone of technical constraints.

Relationship between a decision-point and a technical constraint zone
Since a point-decision is represented with a point, and a zone of technical constraints is represented by a half-space, a set of decision-point, the essential same basic relation between one of these two object classes, is the ensemblist relation of belonging. Let X0 be a decision point and ZCT (A, b) a constraint zone. The relationship Relationship that is a basis for redundancy and infeasibility analysis. Moreover, when X0 does not belong to a ZCTi, we also say that X0 is exterior to ZCTi as the outer term, and its inner opposite has a topological connotation, the topological definitions of these terms will be recalled.

Projection of a point on a hyperplane
Let ZCTi be the area of technical constraints delimited by the HP hyperplane (Ai, bi) and let X0 be any decision-point.
We call projection of X0 on the hyperplane HP (Ai, bi), the point noted X'0 such that : X0 belongs to HP (Ai, bi) and is collinear to the vector Ai. X'0 is the intersection of HP (Ai, bi) with the straight line passing through X0 and whose direction vector is Ai.
Proof : ♣ Mathematically this right is defined by : The intersection is written, This distance is also called the distance from the point Xo to the hyperplane HP (Ai, bi).
Finally we get the following relation:

.3 Direction of a decision vector with respect to ZCTi
Let V be a decision vector, and let ZCTi be a technical constraint zone V parallel to HP (Ai; bi), that is to say at the ZCTi boundary.

External orientation
We say that V has an external orientation with respect to ZCTi, if <V, Ai> is strictly positive.

Parallel orientation
We say that V has a parallel orientation with respect to ZCTi, if <V, Ai> is equal to 0.
Note 8.01: The qualification "parallel" refers to the fact that V is parallel to HP (Ai ; bi), that is to say at the ZCTi border.

Internal orientation
We say that V has an internal orientation with respect to ZCTi, if <V, Ai> is strictly negative.
Let us show that this is an inside point of ZCTi. Let X0' be the projection of X0 on the hyperplane. We saw that : The distance between X0 and X0' is equal to: We have r> 0 ; and it is obvious that the ball whose center is X0 and radius r is entirely included in ZCTi, which means that X0 is an internal point of ZCTi . ♦ 8. 1.4

.1 S border point
is said to be a border point of S if : X0 is an element of Set ; Whatever the positive real number r is, the open ball of center X0 and radius r contains both elements of S other than X0 and elements of S (the complement of S in Euclidean affine space n ).

S border
Xo be an element of Fr (ZCTi), that means that X0 is also an element of ZCTi so, We can write : The zone of technical constraints ZCTi is topologically closed.
Proof : ♣ Let X0 be a point not belonging to ZCTi (where ZCTi denotes the complement of ZCTi in the affine space). As We showed that : The distance between Xo' and X0 is equal to : The distance between X0 and the border

Relationship between a search axis R (X0, V) and a technical constraint zone ZCTi
Consider an area of ZCTi technical constraints. Let also be the radius R(X0, V) coming from X0 and direction vector V, representing a search axis. This relationship is based on the existence and uniqueness of the solution of the equation : A X ,  is the unknown.
Proof: ♣ Since the two objects are sets of decision points, the main combination that can be imagined between them is the intersection. In addition, since the area of technical constraint is closed and the hyperplane HP (Ai, bi) is closed, we will be much more interested in the intersection of the radius R (X0, V) with this boundary HP (Ai, bi which is an equation where α is the unknown. This equation gives : The existence and uniqueness of α depend on the values of b i -(Ai, X o) and (Ai,V), hence X0 and V.

Where X0 is outside ZCTi and where V is not facing inwards from ZCTi
The equation :

Where X0 is outside ZCTi and where V is inward ZCTi
The equation : which is in addition a unique value. ♦ In this case we say that we have a single point of entry starting from X0, and moving along the axis of R (X0, V).

Where X0 is on ZCTi and where V is outside ZCTi
The equation : The concept of research implies that there is a situation or state of deposition, and that from this situation, one move to another situation or state. And we have already seen that this research, when it is linear, can be modeled by the concept of axis of research starting from a given point and moving according to a given vector   0 , R X V In this section, the starting point of the search is the point X0, which is supposed to be located in the ZNN nonnegativity zone. Since the new decisions found must have remained in ZNN, it is logical to study the conditions under which this search leads us out of ZNN. And as in the case of ZCTi technical constraint zones, we will need to define the boundary concept in ZNN.