Algebraic Optimization

Algebraic optimization with equality and inequality con.straints... 

The application of simultaneous optimization to reactor-based flowsheets leads us to consider the more general problem of differentiable/algebraic optimization problems. Again, the optimization problem needs to be reconsidered and reformulated to allow the application of efficient nonlinear programming algorithms. As with flowsheet optimization, older conventional approaches require the repeated execution of the differential/algebraic equation (DAE) model. Instead, we briefly describe these conventional methods and then consider the application and advantages of a simultaneous approach. Here, similar benefits are realized with these problems as with flowsheet optimization. 

The rmsteady state nature of batch processes results in differential algebraic optimization (DAOP) problems. Underlying a DAOP is the problem of optimal control where time dependent decisions are decided. A differential algebraic optimization problem in general can be stated as follows. 

Khludneva E. Yu. (1990a) Optimal control of external forces in contact problems for a viscoelastic plate. In Algebra and Math. Analysis. Novosibirsk, 8-14 (in Russian). 

FIG. 8-46 Diagram for selection of optimization techniques with algebraic constraints and objective function. 

Using an algebraic procedure, synthesize an optimal MEN for the benzene recovery example described in Section 3.7 (Example 3.1). 

The foregoing algebraic method can be generalized using optimization techniques. A particularly useful approach is the transshipment formulation (Papoulias and... 

The MEN software is based on the information described in Chapters Five and Six for developing algebraic and optimization-based solutions for the MEN synthesis problem. It can generate composition-interval diagrams, tables of exchangeable loads and optimization formulations for minimizing cost of MSAs. 

Cuthrell, J.E. and Biegler, L.T., 1987. On the optimization of Differential-Algebraic process systems. American Institution of Chemical Engineers Journal, 33(8), 1257. 

To obtain the optimum temperature we differentiate Eq. (15) w.r.t temperature T and equate dtf /dT to zero. After some algebraic manipulations the optimality condition can be written as ... 

The adaptation of the original LJ optimization procedure to parameter estimation problems for algebraic equation models is given next. 

If we have very little information about the parameters, direct search methods, like the LJ optimization technique presented in Chapter 5, present an excellent way to generate very good initial estimates for the Gauss-Newton method. Actually, for algebraic equation models, direct search methods can be used to determine the optimum parameter estimates quite efficiently. However, if estimates of the uncertainty in the parameters are required, use of the Gauss-Newton method is strongly recommended, even if it is only for a couple of iterations. 

The optimality criteria based on which the conditions for the next experiment are determined are the same for dynamic and algebraic systems. However, for a dynamic system we determine the conditions not just of the next measure-... 

In Chapter 4 the Gauss-Newton method for systems described by algebraic equations is developed. The method is illustrated by examples with actual data from the literature. Other methods (indirect, such as Newton, Quasi-Newton, etc., and direct, such as the Luus-Jaakola optimization procedure) are presented in Chapter 5. 

Hence, through the LCAO expansion we have translated the non-linear optimization problem, which required a set of difficult to tackle coupled integro-differential equations, into a linear one, which can be expressed in the language of standard linear algebra and can easily be coded into efficient computer programs. 

Thus, the design equations for a batch reactor for the optimization of a temporal superstructure can be based on differential or algebraic equations. 

All the algebraic and geometric methods for optimization presented so far work when either there is no experimental error or it is smaller than the usual absolute differences obtained when the objective functions for two neighboring points are subtracted. When this is not the case, the direct search and gradient methods can cause one to go in circles, and the geometric method may cause the region containing the maximum to be eliminated from further consideration. 

The ingredients of formulating optimization problems include a mathematical model of the system, an objective function that quantifies a criterion to be extremized, variables that can serve as decisions, and, optionally, inequality constraints on the system. When represented in algebraic form, the general formulation of discrete/continu-ous optimization problems can be written as the following mixed integer optimization problem ... 

Off-line analysis, controller design, and optimization are now performed in the area of dynamics. The largest dynamic simulation has been about 100,000 differential algebraic equations (DAEs) for analysis of control systems. Simulations formulated with process models having over 10,000 DAEs are considered frequently. Also, detailed training simulators have models with over 10,000 DAEs. On-line model predictive control (MPC) and nonlinear MPC using first-principle models are seeing a number of industrial applications, particularly in polymeric reactions and processes. At this point, systems with over 100 DAEs have been implemented for on-line dynamic optimization and control. 

A very popular scheduling framework is based on mixed-integer programming. Herein, the scheduling problem is modeled in terms of variables and algebraic inequalities and solved by mathematical optimization techniques. In opposition to this well-established framework, a different approach is advocated in the paper by Alur and Dill [8] on timed automata (TA). 

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