Equality-constrained object

Constrained Derivatives—Equality Constrained Problems Consider minimizing the objective function F written in terms of n variables z and subject to m equahty constraints h z) = 0, or... 

Equality Constrained Problems—Lagrange Multipliers Form a scalar function, called the Lagrange func tion, by adding each of the equality constraints multiplied by an arbitrary iTuiltipher to the objective func tion. 

Inequality Constrained Problems To solve inequality constrained problems, a strategy is needed that can decide which of the inequality constraints should be treated as equalities. Once that question is decided, a GRG type of approach can be used to solve the resulting equality constrained problem. Solving can be split into two phases phase 1, where the go is to find a point that is feasible with respec t to the inequality constraints and phase 2, where one seeks the optimum while maintaining feasibility. Phase 1 is often accomphshed by ignoring the objective function and using instead... 

The next step of SMO is to compute the location of the constrained maximum of the objective function in the following equation while allowing only two Lagrange multipliers to change. Under normal circumstances (it 9 0), there will be a maximum along the direction of the linear equality constrain, and k will be less than zero. In this case, SMO computes the maximum along the direction of the constraint. 

There are many ways to solve this nonlinear equality-constrained optimization problem. One method is the quadratic model approximation [44]. The maximum entropy method can provide the least biased estimates on the basis of available information about the problems. The proven objectivity of the method relies on the fact that the solution is dictated purely by the experimental data and the nature of the problem, not by an arbitrarily introduced differential operator such as the ones may be found in the regularization method. 

Note The L.P. is instructed to maximize the objective vaiue and to appiy ail equations (each row in the matrix represents an equation) in the form LESS THAN,"in contrast to GREATER THAN" or EQUAL TO," except for the objective vaiue row equation, which is set at FREE" (non-constraining, released from inhibition against negative slack value). 

An object is connected with whiffletrees to another object. Their purpose is to distribute forces according to the geometry (often equal) without constraining the relative positions of the horses... 

In parameter estimation we are occasionally faced with an additional complication. Besides the minimization of the objective function (a weighted sum of errors) the mathematical model of the physical process includes a set of constrains that must also be satisfied. In general these are either equality or inequality constraints. In order to avoid unnecessary complications in the presentation of the material, constrained parameter estimation is presented exclusively in Chapter 9. 

An important class of the constrained optimization problems is one in which the objective function, equality constraints and inequality constraints are all linear. A linear function is one in which the dependent variables appear only to the first power. For example, a linear function of two variables x and x2 would be of the general form ... 

To show that this constrained minimization is indeed equivalent to the steady-state formulation, let us adjoin the equality constraints to the objective function to form the Lagrangian function,... 

One method of handling just one or two linear or nonlinear equality constraints is to solve explicitly for one variable and eliminate that variable from the problem formulation. This is done by direct substitution in the objective function and constraint equations in the problem. In many problems elimination of a single equality constraint is often superior to an approach in which the constraint is retained and some constrained optimization procedure is executed. For example, suppose you want to minimize the following objective function that is subject to a single equality constraint... 

Finally in this chapter, an alternative approach for nonlinear dynamic data reconciliation, using nonlinear programming techniques, is discussed. This formulation involves the optimization of an objective function through the adjustment of estimate functions constrained by differential and algebraic equalities and inequalities and thus requires efficient and novel solution techniques. 

This approach operates in two phases. First, a sufficient number of elements is found in order to satisfy the linearization of all of the constraints at the initial point. In this way we guarantee that a feasible QP subproblem exists for (27). Second, to avoid convergence to a suboptimal solution with too few elements, we retain additional dummy elements in the formulation that are constrained to be less than or equal to a negligible element length. These elements can be placed at all nonzero element locations, but in practice they need only be associated with elements that have active error bounds at the QP solution. Now once the QP subproblem is solved, multipliers on the upper bounds of the dummy elements are checked for positive values. These indicate that the objective function can be further improved by relaxing the dummy element. After relaxation (which effectively adds another nonzero element to the problem), another dummy element is added in order to allow for any additional nonzero elements that may be needed. 

After the inequality constraints have been converted to equalities, the complete set of restrictions becomes a set of linear equations with n unknowns. The linear-programming problem then will involve, in general, maximizing or minimizing a linear objective function for which the variables must satisfy the set of simultaneous restrictive equations with the variables constrained to be nonnegative. Because there will be more unknowns in the set of simultaneous equations than there are equations, there will be a large number of possible solutions, and the final solution must be chosen from the set of possible solutions. 

In applying this technique, the Lagrange expression is defined as the real function to be optimized (i.e., the objective function) plus the product of the Lagrangian multiplier (A) and the constraint. The number of Lagrangian multipliers must equal the number of constraints, and the constraint is in the form of an equation set equal to zero. To illustrate the application, consider the situation in which the aim is to find the positive value of variables X and y which make the product xy a maximum under the constraint that x2 + y2 = 10. For this simple case, the objective function is xy and the constraining equation, set equal to zero, is x1 + y2 - 10 = 0. Thus, the Lagrange expression is... 

For this simple ease, the objective function is xy and the constraining equation, set equal to zero, is + y 10 = 0. Thus, the Lagrange expression is... 

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