Equality-constrained efficiency

A great number of studies indicated that quadratic approximation methods, which are characterized by solving a sequence of quadratic subproblems recursively belong to the most efficient and reliable nonlinear programming algorithms presently available. This method combines the most efficient characteristics of different optimization techniques (see e.g, [19]). For equality constrained problems, the general nonlinear constrained optimization problem can be formulated by an... 

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. 

Fig. 7. (A) The crosses show a run where library size is constrained, but no constraints are placed on library configuration. The solid squares show the effect of also constraining configuration so that between 15 and 20 reactants are used from each pool. The solid line shows the ideal solution in terms of efficiency, that is, equal numbers of reactants are selected from each reactant pool. (B) No loss of diversity is seen in the configuration-constrained library relative to the less efficient unconstrained solutions.

Mass flow deviations are minimized to enable lower postcombustion O2, CO, and NO. The lowest postcombustion O2 concentration possible is constrained by the burner most starved for air. This starved burner will generate a high CO concentration and consequently the total O2 must be raised to minimize the formation of CO in that burner. By equalizing the air flow to each burner and assuring that the fuel flow is equal, the O2 can be lowered until the CO starts to increase equally for all burners. Lower O2 has additional benefits of lower NO formation and higher thermal efficiency. 

The augmented Lagrangian method is not the only approach to solving constrained optimization problems, yet a complete discussion of this subject is beyond the scope of this text. We briefly consider a popular, and efficient, class of methods, as it is used by fmincon, sequential quadratic programming (SQP). We wUl find it useful to introduce a common notation for the equality and inequality constraints using slack variables. 

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