In this specific example, w will consider a case when c will be equal to zero, c0. Tis example dwells on the condition that all the optimization problems are viewed in two different ways, wich are either the dual problem or the optimization problem, wich is referred to as the duality principle. Tis theory is most applicable while being used as an interior point in quality programming. Te lagrange multiplier is also very efficient in this optimization since it leads to identification of the local maxima and the in any function but is subject to every equality constraint (Xia, 2011).

Agood example of this application on lagrangian is where we want to maximize a function, and subject is to. Ater graphically rep[resenting this information, w find that there is a feasible set which is the unit circle and consequently, te level set which is determined to be the diagonal lines w2hich are then calculated to have a slope of -1. Ater graphically representing this, w find that the minimum and maximum occurs at the pointsAp in this = 0.

Sbstituting x = x∗ − p into the objective functional, w getf(x) = 12(x∗ − p)TB(x∗ − p) − (x∗ − p)Tb ==12pTBp − pTBx∗ + pTb+ f(x∗). i also implies that Bx∗ =( b − ATλ∗). Oserving Ap = 0, w havepTBx∗ = pT(b − ATλ∗) = pTb −(Ap)Tλ∗| {z }= 0, wence we find that f(x) = 12pTBp + f(x∗). I view of p ∈ Ker A, w can write p = Zu, u∈ lRn−m, ad hence, fx) = 12uTZTBZu + f(x∗). Snce ZTBZ positive w usually deduce f(x) > f(x∗).

Eentually, w realize that x∗ is always the quadratic programming unique global minimizer. I this situation, w 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, a, i∈ Iac(ˆx) are all linearly independent. Aditionally, w also assume that ci, 1≤ i ≤ m, a, i∈ Iac(ˆx) are the constant and are linearly independent in all cases.

W 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. w always find that p is a feasible direction of constaint j, sch that atjp ≤ 0.. .

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