Necessary Condition for Optimality

Let J be a functionai of a function y. If / is minimum at y, then the values of / for aii other admissibie functions in the vicinity of y cannot be lower than the value of / at y. Precisely, 

Now if I has a variation at y, then from the definition of variation [Liquation (2.11), p. 36] and its homogeneity property 

greater than or equal to zero when a is greater than zero, or 

SI = 0 is the only non-contradicting condition that is applicable for the minimum of I. 

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Analytic solution by solving the first-order necessary conditions for optimality (Section 8.2)... 

First, the first element in the reduced gradient with respect to the superbasic variable y is zero. Second, because the reduced gradient (the derivative with respect to s) is 1, increasing s (the only feasible change to s) causes an increase in the objective value. These are the two necessary conditions for optimality for this reduced problem and the algorithm terminates at (1.5, 1.5) with an objective value of 2.0. 

Using embedding with a structural parameter formulation, Nishida, Liu and Ichikawa (1976) state the necessary conditions for optimality for both the structure and the control of a dynamic process system. They also permit some of the system parameters to take on uncertain values from within allowed ranges. 

Ichikawa, A. and Fan, L.T., "Optimal Synthesis of Process Systems—Necessary Condition for Optimal System and its Use in Synthesis of Systems," Chemical Engineering Science Vol. 28, pp 357-373, 1973. 

V L is equal to the constrained derivatives for the problem, which should be zero at the solution to the problem. Also, these stationarity conditions very neatly provide the necessary conditions for optimality of an equality-constrained problem. 

Assume certain inequality constraints will be active at the final solution. The necessary conditions for optimality are... 

Then one can apply Newton s method to the necessary conditions for optimality, which are a set of simultaneous (non)linear equations. The Newton equations one would write are... 

Derivative methods use the necessary condition for optimality where the derivative of the function with respect to the decision variables is zero at the optimum. Methods include Newton and quasi-Newton methods. 

The necessary conditions for optimization of a system governed by non-linear partial differential equations, such as the equation governing the unsteady state behaviour of tubular reactors, have been derived by Chang and Bankoff (1969). Also, they derived the optimal unsteady state control of a iacketed tubular reactor with and without heat... 

The optimal control of the two-phase tubular reactors had been formulated by Kassem (1977). A distributed minimum principle was presented and the necessary conditions for optimality were derived. Based on these conditions for optimality a functional gradient aigorichm for synthesizing boundary and distributed controls were deduced. 

The values y and y are the current set point values for the two control loops (Figure 25.11a). But when the disturbances change, say dx = d" and d2 = d, the values of yt and y2 that minimize the cost function also change. Let the new optimum values be y" and y. Again they satisfy the following necessary conditions for optimality ... 

In most problems, a Lagrange multiplier can be shown to be related to the rate of change of the optimal objective functional with respect to the constraint value. This is an important result, which will be utilized in developing the necessary conditions for optimal control problems having inequality constraints. 

A necessary condition for optimal solution is that each of the equations (8i) and (8ii) be reduced to zero. As these equations approach zero, the calculated performance data become closer to the history performance data, resulting in optimal parameter point estimates. 

Chapter 4 applies variational calculus to problems that include control variables as well as state variables. Optimal control strategies are developed that extremize precise performance criteria. Necessary conditions for optimization are shown to be conveniently expressed in terms of a mathematical function called the Hamiltonian. Pontryagin s maximum principle is developed for systems that have control constraints. Process applications of optimal control are presented. 

Chapter 9 develops necessary conditions for optimality of discrete time problems. In implementing optimal control problems using digital computers, the control is usually kept constant over a period of time. Problems that were originally described by differential equations defined over a continuous time domain are transformed to problems that are described by a set of discrete algebraic equations. Necessary conditions for optimality are derived for this class of problems and are applied to several process control situations. 

Table 13.1 Necessary Conditions for Optimal Controls (Ogunye and Ray 1971)...

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