Maratos effect

This undesirable phenomenon is called the Maratos effect, and takes its name from the individual that first observed it (Maratos, 1978). 

The Maratos effect appears when the penalty function is used to control the convergence of an algorithm that is not directly minimizing the same function. However, it does not appear when the penalty function is used for unconstrained minimization. 

The most appealing alternative was proposed by Fletcher (1987). Fletcher demonstrated that if we know the exact values of the Lagrange parameters, Xj, it should be possible to use a merit function that avoids the Maratos effect. This option will be considered in Section 12.2.4. Unfortunately, the exact values of 2 are unknowns in a generic iteration and only their estimations are available therefore, the Maratos effect could also arise with such a function. 

Solve the problem proposed by Powell using a BzzMinimizationMono class object to handle the minimization problem and a BzzFunctionRootRobust class object to handle the nonlinear equation. This ensures the Maratos effect caimot arise since no penalty functions are used and the constraints are satisfied at each iteration of the optimization procedure. 

Fletcher has demonstrated that the Maratos effect is avoided if the exact values of Lagrange parameters, Ij, are inserted into the function (12.23). 

In a generic iteration, Xj used in the function (12.23), can be quite different from the exact ones, especially when many constraints are present. Thus, the Maratos effect could occur. 

Unfortunately, the so[Pg.467]

The Maratos effect was detected by applying the iterations of the SQP method to certain problems. This effect should not be surprising because only one function is used to reconcile two distinct problems minimizing the real function and solving the system of constraints. If the constraints are nonlinear, it may be impossible to reconcile these two problems in one function only. 

In constrained optimization, it is important to distinguish between the Maratos effect and the narrow valley effect (see Section 3.8). Both effects crop in deeming whether x +i is better or not than xj. 

The Maratos effect is related to the fact that only one merit function is used in a constrained optimization problem the function must simultaneously account for... 

Another advantage of SQP methods is that they allow an alternative merit function to be used when the Maratos effect arises because of the nonlinearity of constraints. 

For instance, consider Example 12.4 that leads to the Maratos effect Rather than discard x +i because of the function worsening, if Newton s method is iteratively used on the nonlinear system with the same Jacobian, Maratos effect does not crop up. 

Chapter 13 illustrates the problem of constrained optimization by introducing the active set methods. Successive linear programming (SLP), projection, reduced direction search, SQP methods are described, implemented, and adopted to solve several practical examples of constrained linear/nonlinear optimization, including the solution of the Maratos effect. 

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