Quadratic program conditions

Reformulating the necessaiy conditions as a linear quadratic program has an interesting side effect. We can simply add linearizations of the inactive inequalities to the problem ana let the ac tive set be selected by the algorithm used to solve the linear quadratic program. 

Convex Cases of NLP Problems Linear programs and quadratic programs are special cases of (3-85) that allow for more efficient solution, based on application of KKT conditions (3-88) through (3-91). Because these are convex problems, any locally optimal solution is a global solution. In particular, if the objective and constraint functions in (3-85) are linear, then the following linear program (LP)... 

Quadratic programming (QP) is a special problem including a product of two decision variables in the objective function e.g. maximization of turnover max p x with p and x both variable requiring a concave objective function and that can be solved if the so-called Kuhn-Tucker-Conditions are fulfilled, e g. by use of the Wolf algorithm (Dom-schke/DrexI 2004, p. 192)... 

Successive quadratic programming solves a sequence of quadratic programming problems. A quadratic programming problem has a quadratic economic model and linear constraints. To solve this problem, the Lagrangian function is formed from the quadratic economic model and linear constraints. Then, the Kuhn-Tucker conditions are applied to the Lagrangian function to obtain a set of linear equations. This set of linear equations can then be solved by the simplex method for the optimum of the quadratic programming problem. 

This formulation is a standard quadratic programming problem for which an analytical solution exists from the corresponding Kuhn—Tucker conditions. Different versions of the objective function are sometimes used, but the quadratic version is appealing theoretically because it allows investor preferences to be convex. 

Kuhn-Tucker conditions and reduces the quadratic programming problem to what is referred to as a linear complementarity problem. 

The problem of minimum as formulated above can be solved by sequential methods of nonlinear (in particular quadratic) programming. The idea of the sequential approach consists, most simply, in linearizing the equation g(z) = 0 at point z of the sequence and subjecting the linearized constraint equation to a minimum condition thus the next approximation is found, and so on. Some problems can arise when the whole unmeasured vector y is not observable (not uniquely determined) although the latter case is less frequent in practice, possibly it can happen that the values of some unmeasured variables are not required and admitted as unobservable (undetermined). In what follows we shall outline two methods that do not require the full observability of vector y. 

The above strategy, with possible x < 1 in (10.4.19 and 45), consists of approximating first the sought minimum by a point 2 M (suboptimal reconciliation) and then decreasing the criterion Q (10.4.32) by a series of steps subjected to the condition (10.4.46) where is the k-th value of the criterion. A simpler strategy as follows is, in fact, a version of succesive quadratic programming see Remark (i) below. 

It will be shown that the solution to any mathematical programming problem must satisfy not only the original constraints, but also certain side conditions. These new conditions take the form of inequalities in a new set of non-negative variables, and it will turn out that in quadratic... 

The Eu3+ crystal field parameters, reported in Table 5, have been computed with QCPE program No. 216 (72). This program permits us to find a set of parameters giving the weakest quadratic mean error between experimental and calculated levels, on condition that the error be increasing from 7Fj to 7F4 (the sublevels for small J being more reliable). 

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