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In this specific example, we will consider a case when c will be equal to zero, c=0. This example dwells on the condition that all the optimization problems are viewed in two different ways, which are either the dual problem or the optimization problem, which is referred to as the duality principle.This theory is most applicable while being used as an interior point in quality programming. The lagrange multiplier is also very efficient in this optimization since it leads to identification of the local maxima and the local minima in any function but is subject to every equality constraint (Xia, 2011).A good example of this application on lagrangian is where we want to maximize a function, and subject is to. After graphically rep[resenting this information, we find that there is a feasible set which is the unit circle and consequently, the level set which is determined to be the diagonal lines w2hich are then calculated to have a slope of -1. After graphically representing this, we find that the minimum and maximum occurs at the pointsAp in this case is = 0. Substituting x = x∗ − p into the objective functional, we getf(x) = 12(x∗ − p)TB(x∗ − p) − (x∗ − p)Tb ==12pTBp − pTBx∗ + pTb+ f(x∗) .it also implies that Bx∗ =( b − ATλ∗). Observing Ap = 0, we havepTBx∗ = pT(b − ATλ∗) = pTb −(Ap)Tλ∗| {z }= 0,whence we find that f(x) = 12pTBp + f(x∗) .In view of p ∈ Ker A, we can write p = Zu , u ∈ lRn−m, and hence,f(x) = 12uTZTBZu + f(x∗) .Since ZTBZ is positive deﬁnitely, we usually deduce f(x) > f(x∗). Eventually, we realize that x∗ is always the quadratic programming unique global minimizer.In this situation, we assume that ˆx ∈ lRn satisﬁes the KKT conditions for the quadratic programming issues and problem in a way that the particular holds true assuming that in all cases the constraint gradients which include ci, 1 ≤ i ≤ m, ai, i ∈ Iac(ˆx) are all linearly independent. Additionally, we also assume that ci, 1 ≤ i ≤ m, ai, i ∈ Iac(ˆx) are the constant gradients and are also linearly independent in all cases. We now suppose that there are s j ∈ Iac(ˆx) such that ˆµj < 0 and then p should represent the quadratic programming sich that we minimize 1/2pTBp − ˆbTp over p ∈ lRn and then subject it to Cp = 0 aTip = 0 , i ∈ Iac(ˆx) \ {j} .; where b is = Bxˆ − b.we always find that p is a feasible direction of constaint j, such that atjp ≤ 0

References

Ahmetoglu, F. (2013). Calculation method for quadratic programming problem in Hilbert spaces, partially ordered by cone with empty interior. Communications series A1 Mathematics &statistics, 62(2), 11-15.

Anstreicher, K. (2012). On convex relations for quadractically constrained quadratic programming. Mathematical programming, 136(2), 233-251. dol:10.1007/s1007-012- 0602-3

Xia, Y. (2011). Global optimization of a class of non convex quardratically constrained programming problems. Acta Mathematica Sinica, 27(9), 1803-1812. dol:10.1007/s10114-011-8351-4

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