Constrained optimum

Real-time synthesis of operating procedures. Most of the ideas and methodologies, presented in this chapter, are applicable to the a priori, off-line, synthesis of operating procedures. There is a need though to address similar problems during the operation of a chemical plant. Typical examples are the synthesis of operational response (i.e., operating procedure) to process upsets, real-time recovery from a fallback position, and supervisory control for constrained optimum operation. 

The Kuhn-Tucker conditions are predicated on this fact At any local constrained optimum, no (small) allowable change in the problem variables can improve the value bf the objective function. To illustrate this statement, consider the nonlinear programming problem ... 

In accordance with the Method of Lagrange, to guarantee that L is independent of the set of state variables, x (i.e. L is unconstrained or in other words, that the optimum of L is equal to the constrained optimum of 0), we must have... 

Table 6.6 Performance under constrained optimum temperature profile.

When the economic trade-off between column height and column diameter is constrained, optimum tray spacing is dictated by cost considerations, while access considerations assume a secondary role. Some situations where the trade-off is constrained are... 

Synthetic rebalancing cannot always be done, however. This is likely to be the case for rttiquid assets and those with legal covenants limiting transfer. For example, an investor may own a partnership or hold a concentrated stock position in a trust whose position cannot be swapped away. In these situations, MV optimization must be amended to include these assets with their weights restricted to the prescribed levels. The returns, risk, and correlation forecasts for the restricted assets must then be incorporated exphcitly in the analysis to take account of their interaction with other assets. The resulting constrained optimum portfolios will comprise a second-best efficient frontier but may not be too far off the unconstrained version. 

For example, adding -log(x,) to the objective function will cause the objective to increase without bound Xj as approaches 0. Of course, if the constrained optimum is on the boundary (i.e., some xf = 0 that is always true for linear programming), then the barrier will prevent us from reaching it. The solution is to use a barrier parameter that balances the contribution of the true objective function against that of the barrier function. 

Arising from 6J = 0, both Equations (3.11) and (3.12) are the necessary conditions for the optimum of J. On the basis of the Lagrange Multiplier Rule, these two equations are also the necessary conditions for the constrained optimum of I. Both optima are still subject to the given initial condition, i. e.. Equation (3.6). The equivalence between the two optima will be shown later in Section 4.3.3. 

In Section 3.2.1 (p. 59), we had asserted the Lagrange Multiplier Rule that the optimum of the augmented J is equivalent to the constrained optimum of I. This rule is based on the Lagrange Multiplier Theorem, which provides the necessary conditions for the constrained optimum. We will first prove this theorem and then apply it to optimal control problems subject to different types of constraints. 

In scientific terms, this is a constrained optimum. It is an approach that has been just as harmful to business practices, in which all the wrong costs are minimized, as it has been to Joints in aircraft structures for the past 70 years. 

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