Calculation Method for Quadratic Programming - Case Study Example

Quadratic programming: of Definition A quadratic program is an independent discipline with many applications that have linearly constrained problems with a quadratic function that is optimized (Xia, 2011). According to Jensen and Bard (2012), linear programming in most cases forms the basis of many non-linear programming algorithms. Quadratic programming is a discipline that enhances proper understanding of the values that determine the real values and subject them to the linear inequality constraints. Consequently, this leads to a successful yield of an extreme value of the quadratic function. Other than linear programming being a great step that enables the solving of elaborate problems involved in economic models and other non linear programming problems, it is also applicable in solving a number of problems that are of best interests to itself (Ahmetoglu, 2013). According to Wolfe (1959), the problems that were of great interest to quadratic programming include regression, efficient production, “portelolio” problem, and convex programming. Wolfe (1959) studied that in regression, it is essential to find the least square that perfectly fits into any given statistical data in cases where certain parameters can satisfy constraints that have inequalities such as being non-negative (Xia, 2011). Additionally, he describes the efficient production to be the maxima ion of all profits. However, efficient production assumes that all the marginal costs vary, whereas the linear production is functioning at all cases. The “portelolio” problem is a problem that combines different random variables together but has a minimum variance and a given expectation. Lastly, convex programming involves identification of a general convex function using quadratic approximations that falls under linear constrains Example (Wolfe, 1959) Let the variables of the problem constitute the n-vector x = (xi, .,x.) ( will denote transposition; we take x to be a column vector, that is, an n by 1 matrix). We Lett A be an m by n matrix, whereas b to be an m by 1 martix, we express the constraint of this problem by stating that; (1) x>O, Ax b, that is, x ?0 (> - 1 .,n), ZXnI aijx= bi (i - 1, . . ). (p. 2) Important theoriems There are many theories associated with quadratic programming in mathematical concepts wich include, the global minimize, the reduction of objective functions, and the lagrangian duality method of finding the maxima and the minima. The Lagrangian duality. 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. Consequently, the lagrangian function is defined as As explained above, the dual function of Langrangian is defined to be . Consequently, its infimum while we employ , The dual function henceforth is defined to be . Subjected to, we hence conclude that the lagrangian duality theory is; 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 points , respectively. The image is drawn as The global minimizer In this case, we assume A ∈ lRm×n, such that m ≤ n and the reduced Hessian is positive Z TBZ. The matrix K is then non singular. Through this, let there be (x∗, λ∗) uniquely. In this case, the x* is unique solution of QP(3.2a), (3.2b) Proof: In this proof, we are to let x ∈ F be a feasible point, such that in this case, Ax = c, and p := x∗ − x. Then we find that; Ap 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. The descent and the feasible direction for an active strategy 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 . Properties of the logarithmic barrier function In this theorem, we assume that the set S of certain solutions is a nonempty set and is bounded, and that the interior Fint of the feasible set is nonempty, letting {βk}lN be a decreasing sequence of barrier with the parameters βk → 0 as k → ∞. In this case, there holds: For any β > 0 the logarithmic barrier function B(β)(x) is convex in FInt and it observed to attains a minimizer x(β) on Fintn putting in mind that there is a local minimizer x(β) which is also a global minimizer of B(β)(x). If {x(βk)}lN is a sequence of minimizers, then there also exists a lN0 ⊂ lN such that x(βk) → x∗ ∈ S, k ∈ lN0. The conditions to solve quadratic problems Quadratic programming is very efficient in ensuring that problems that are related it are solved effectively. The quadratic programming employs various conditions through which problems can be solved which include the active set solution method, the conjugate gradient solution method, the interior point solution method, and the augmented lagrangian solution method (Anstreicher, 2012). Interior point solution method The interior solution methods are those barrier methods that solve the linear and non-linear convex optimization of certain algorithms classes. The method employs convex optimization concepts while solving problems by easily transforming the problem into a maxima or minima linear function convex. An example of this type of solution represented in a convex epigraph form is shown below (Anstreicher, 2012) he most used method of solving the non linear optimization is through the primal dual method. In this case, we minimize f(x) to be the subject to Through research, one identifies that In this case , represents a small scalar which is positive which in most cases, it is referred to as the barrier parameter and it is should to converge to zero, the minimum of . The gradient of the barrier function in this case is And are both gradients representing the slope of function f(x) and respectively. Consequently, we introduce the slack variable so that the function appears as . The next step involves applying the fourth to the third equation so that one can get the gradient (Ahmetoglu, 2013). Applying this equation, we find that; And in this case, we find that the matrix is a constraint for . The theory behind (5) is that when we calculate, we find the gradient of   should lie in the subspace spanned by the constraints gradients and the boundary  or that the projection of the gradient  on the constraint component  normal should be almost or nearing to zero. When we apply the Newton’s method to equation 4 and equation 5, we usually get an equation that for there is  update : And in which   is the hessian matrix that is  and  is the diagonal matrix  and  is a diagonal matrix where  is  since the equation 1and equation 4 are in the condition that and should be reignforced at each and every step in the process and can opnly be done by choosing the best : . Active set method A problem can be defined through an objective function in mathematical optimization and also use the objective function to minimize or maximize, incorporating a set of constraints which define the feasible region which is the set x setting all optimal solutions available. In cases where the point in the feasible region is given, there is always a constraint which is active at point  if  and consequently, the point is inactive at the point only if   in quadratic programming, the equality constraints are always found to be active and consequently, the active set at point  is made up of the constraints that are current at the point. The constraint is given as  . The active set is very much important especially in the optimization process because this optimization theory is a great determinant of the constraints that are going to greatly influence the results of the optimization process. This method is used in solving most of the encountered linear programming problems as the active set usually gives the hyper planes that always intersect at the solution point. Many assumptions and estimations of active sets are however made since the solution is not very necessary in bounding polygon edge which in return leads to an inequality in the subsets being searched, leading to an easier search. The conjugate gradient The conjugate gradient method is very efficient in solving numerical data in the linear equations systems; however, this applies to the matrixes, which are both positively defined, and symmetric (Anstreicher, 2012). Most commonly, the conjugate gradient method is used in ways that ensures its implementation in an iterative algorithm and very applicable to systems that are very large in such that a way that they cannot be handled through a direct method or implementation which may be, for example, Cholesky decomposition. While handling most of the optimization problems or the partial differential equation, most large sparse systems arise. Eventually, the method is also applicable in cases where optimization problems are unconstrained such as the energy minimization. In this case, we find that since A is symmetric and positive definite, the inner product is mostly defined by the left hand side. We consider the two cases of vector to be conjugate, if the vectors are orthogonal with respect to this inner product, however, through research, being conjugate is a symmetric relation, for example, we find that, consequently if v is conjugate to u, then if u is conjugate to v. Consequently, we suppose that  is a set of n mutually conjugate directions and then  is the basis of  and consequently, within  we can expand the solution to be   of : In this case, we see that For any , (because  are mutually conjugate) Augmented lagrangian method This method is a problem solving method that solves all the constrained optimization problems in certain classes of algorithms (Anstreicher, 2012). They are however very similar to the penalty methods because both in most cases normally replaces the constrained optimization problem, however, they are different in that an additional term is added to the unconstrained objective by the Lagrangian method. In an example to show this method, we will solve this constrained problem whereby the problem is; Subjected to this case, In this case, a problem can be solved as a series of unconstrained minimization problems whereby, the first step involves listing a penalty method approach since it this method approach solves the problem adequately. As it is observed, the penalty method solves the constrained problem adequately and, in the next iteration, the approach is able to resolve the problems through a value, which is larger, like,   and the use the old solution as the initial guess. In the augmented Lagrangia n method, there is the use of the unconstrained objectives such as; Consequently, it is essential and vital after each and every alliteration, in addition to updating , the variable  is also updated according to the rule which states that: where  is the solution to the unconstrained problem at the kth step, i.e.  lastly, this variable  is an usually an estimate of the langrange multiplier and hence all the calculations should be very accurate and should improve in each and every step. Additionally, this method has a very important advantage that outweighs the penalty method that it is not essential at all to take    in order to solve the original constrained problem. However, instead, stays a much longer time due to the  langrange multiplier’s presence. 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 Read More

In this case, we minimize f(x) to be the subject to Through research, one identifies that In this case , represents a small scalar which is positive which in most cases, it is referred to as the barrier parameter and it is should to converge to zero, the minimum of . The gradient of the barrier function in this case is And are both gradients representing the slope of function f(x) and respectively. Consequently, we introduce the slack variable so that the function appears as . The next step involves applying the fourth to the third equation so that one can get the gradient (Ahmetoglu, 2013).

Applying this equation, we find that; And in this case, we find that the matrix is a constraint for . The theory behind (5) is that when we calculate, we find the gradient of   should lie in the subspace spanned by the constraints gradients and the boundary  or that the projection of the gradient  on the constraint component  normal should be almost or nearing to zero. When we apply the Newton’s method to equation 4 and equation 5, we usually get an equation that for there is  update : And in which   is the hessian matrix that is  and  is the diagonal matrix  and  is a diagonal matrix where  is  since the equation 1and equation 4 are in the condition that and should be reignforced at each and every step in the process and can opnly be done by choosing the best : .

Active set method A problem can be defined through an objective function in mathematical optimization and also use the objective function to minimize or maximize, incorporating a set of constraints which define the feasible region which is the set x setting all optimal solutions available. In cases where the point in the feasible region is given, there is always a constraint which is active at point  if  and consequently, the point is inactive at the point only if   in quadratic programming, the equality constraints are always found to be active and consequently, the active set at point  is made up of the constraints that are current at the point.

The constraint is given as  . The active set is very much important especially in the optimization process because this optimization theory is a great determinant of the constraints that are going to greatly influence the results of the optimization process. This method is used in solving most of the encountered linear programming problems as the active set usually gives the hyper planes that always intersect at the solution point. Many assumptions and estimations of active sets are however made since the solution is not very necessary in bounding polygon edge which in return leads to an inequality in the subsets being searched, leading to an easier search.

The conjugate gradient The conjugate gradient method is very efficient in solving numerical data in the linear equations systems; however, this applies to the matrixes, which are both positively defined, and symmetric (Anstreicher, 2012). Most commonly, the conjugate gradient method is used in ways that ensures its implementation in an iterative algorithm and very applicable to systems that are very large in such that a way that they cannot be handled through a direct method or implementation which may be, for example, Cholesky decomposition.

While handling most of the optimization problems or the partial differential equation, most large sparse systems arise. Eventually, the method is also applicable in cases where optimization problems are unconstrained such as the energy minimization. In this case, we find that since A is symmetric and positive definite, the inner product is mostly defined by the left hand side. We consider the two cases of vector to be conjugate, if the vectors are orthogonal with respect to this inner product, however, through research, being conjugate is a symmetric relation, for example, we find that, consequently if v is conjugate to u, then if u is conjugate to v.

Consequently, we suppose that  is a set of n mutually conjugate directions and then  is the basis of  and consequently, within  we can expand the solution to be   of : In this case, we see that For any , (because  are mutually conjugate) Augmented lagrangian method This method is a problem solving method that solves all the constrained optimization problems in certain classes of algorithms (Anstreicher, 2012).

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