Quadratic Programming Approach in the Non-Negatif Matrix Factorization

. Non-negative Matrix Factorization is an iteration optimization algorithm. ie to decipher one matrix into several non-negative component matrices. Non-negative Matrix Factorization (FMN) serves to obtain a picture of non-negative data. There is a problem in the Non-negative Matrix Factorization that is optimization at the constraint boundary, where in the optimization solution on the constraint boundary it is necessary to do long iteration and of course very difficult and conquers a long time. Quadratic Programing is an approach to solving linear optimization problems where the constraint is linear function and its purpose function is the square of the decision variable or multiplication of the two decision variables. This method is considered to be an effective method to overcome the optimization in the Non-negative Matrix Factorization.


Introduction
Non-negative Matrix Factorization is an iteration optimization algorithm.ie to decipher one matrix into several non-negative component matrices.Given the  data of the integer V matrix data with   > 0and the positive integer  < min (, ) the non-negative matrix factorization obtains 2 (two) non-negative matrices  ∈   and  ∈   like:

𝑉𝑉 ≈ 𝑊𝑊 𝐻𝐻
If each column V is refresentative on an object; non-negative matrix factorization approximates it with linear combination of r base column W. Conventional approach to obtain W and H by minimizing the difference between   : ≥ 0,   ≥ 0, ∀ , , ,  2 Preliminaries

Non-negative matrix
The non-negative matrix is a real or integer matrix  = �  � where for each element on A is a non-negative number (equal to zero or greater than zero).

Matrix Factorization
Matrix factorization is the process of breaking or decomposition of a matrix into several matrices.
In the Matrix Non-negative matrix V mxn with   ≥ 0, it will be decomposed into two nonnegative matrices  ∈   ∈   with  <  (, ) such that: As an explanation of Matrix Factorization by using Non Negative Matrix Factorization, Given a   matrix, NMF will decipher the matrix V to be: where W and H are matrices with non-negative entries.

Condition of Karush-Kuhn-Tucker
In 1951 Kuhn Tucker proposed an optimization technique that could be used to search the optimum point of a constrained function.Karush Kuhn Tucker method can be used to find the optimum solution of a function regardless of the nature of the function whether linear or non linear.

Quadratic Programing
Quadratic Programing is an approach to solving linear optimization problems where constraints are linear functions and their objective function is the square of decision variables or multiplication of two decision variables (1) min () =    + 1 2    +  with constraints :  ≤ ,  ≥ 0 When the objective function f (x) is the perfect convex for all regions it is reasonable to obtain a point which is a local and also a global minimum.So in such conditions it ensures that Q is a positive definite.

Successive Quadratic Programming
Quadratic sequential programming method (SQP) is a very powerful and popular method class for solving nonlinear programming problems, especially those with strong nonlinear boundaries (3) 3 Results and Discussion

Non-Linear Program
In the application of linear programming, the important assumption to be fulfilled is that all functions are linear.This is what then gave birth to a new concept of nonlinear programming problems.According to (1) the general form of nonlinear programming is finding the value of with constraint   () for every i = 1,2, ..., m and x ≥ 0 (4) The constraint function   () can be a nonlinear function or a linear function.In addition, () and the function   () are functions with  variables.

Quadratic Programming Solution
Nature.1.Complementary slackness in quadratic programming (2) 1)   and   under Kuhn-Tucker and   can not both be positive.
2) The surplus (excess) or slack variables for the i-th constraints and   can not both be positive Evidence of Nature.1.
This applies also to Theorem 2.4, so it is evident that   and   under Kuhn-Tucker and   can not be both positive.
In the same way the     ′ = 0, so it is proved that the surplus (excess) or slack variables for the i-th constraints and   can not both be positive.
The equations derived from the step are a step in the linearity of a nonlinear programming problem by using Kuhn Tucker's condition.

Successive Quadratic Programming
Successive Quadratic Programming (SQP) is a very powerful and popular class of methods for solving nonlinear programming problems, especially those with strong nonlinear boundaries (3).Like sequential linear programming, quadratic programming problems are formed from nonlinear programming problems and solved iteratively until they are optimized.However, iterative procedures are different from successive linear programs.
In quadratic programming, the economic model of quadratic functions, and constraints are all linear equations.To overcome this problem the Lagrangian function is formed, and Kuhn-Tucker's condition is applied to the Lagrangian function to obtain a set of linear equations at this point, it is important to understand the solution of the quadratic programming problem.This is part of the motivation for using quadratic programming which can be demonstrated by the following equation: Kuhn-Tucker's inequality form, equation 6, is used to calculate   > 0. Also, the condition of complementary clearance must be satisfied, ie the slack variable variable  +1 and the Lagrange i multiplier is zero.
If  +1 = 0, then the constraint is active, the equation;   ≠ 0. However, if  +1 ≠ 0, then the constraint is inactive, an inequality and   = 0 The equations of ( 6) and ( 7) can be converted to linear programming problems in the following way.The surplus variable is added to equation (6) as sj, and the slack variable has been added to equation (7) as  + .The slack variable  + can serve as a variable for a base that was originally feasible for equation (7).However, artificial variables are required to have a reasonable basis for the equation (6).Adding artificial variables   with the coefficients   to equation ( 6) is an easy way to start with   .=1Also, the objective function is to minimize the number of artificial variables to ensure that they are not in the final optimal solution As a result of this modification, equation ( 6) and (7) to be:

=
−  substituted to condition 3) = 1,2, … ,    ≥ 0   = 1,2 … ,  where   =   is the second partial derivative with respect to   and   model nonlinear economy.The quadratic programming procedure starts by adding the slack variable  + to the linear constraint equation.No need to use  + 2 because the problem will be solved with linear programming, and all variables must be positive or zero.The Lagrangian function is formed as follows:(, ) = �   +  +1 −  1 �A positive Lagrange multiplier is required, so a negative sign is used in the equation with the constraint equation.Setting the first partial derivative of a Lagrangian function with respect to   and i equal to zero gives the following linear linear algebraic (n + m) set: +  +1 −  1 = 0   = 1,2, … 2 , , . .  ) must satisfy(2)and there are multipliers  1 ,  2 , … ,   as well as slack variable  1 ,  2 , … ,   so that it satisfies Let f() and   () be a problem of patterned drinking.If  = ( 1 ,  2 , , . .  ) is an optimal solution for f() and   (x), then  = ( 1 ,  2 , , . .  )) must satisfy (2) and there are multipliers  1 ,  2 , … ,   as well as the surplus variable  1 ,  2 , … ,   so that it satisfies.