Optimization nonlinear programming

Unconstrained optimization (nonlinear programming), 2546-2553 classictil methods, 2546-2547 conjugate gradient methods, 2552-2553 golden section method, 2547-2549 line search techniques for, 2547 multidimensional search techniques for, 2549-2552... 

Computer Algorithms Differential Equations Linear Optimization Nonlinear Programming Numerical Analysis... 

Supercomputers, such as the CRAY X-MP, CRAY Y-MP, and CRAY-2, are partially available and used for flow-sheet and optimization studies (7—10). Optimization modules usiag linear and nonlinear programming (LINPRO and UNLPl, based on a revised simplex, and Davidson-Eletcher-PoweU and Broyden methods, respectively) are available ia MicroMENTOR (11). 

With many variables and constraints, linear and nonlinear programming may be applicable, as well as various numerical gradient search methods. Maximum principle and dynamic programming are laborious and have had only limited applications in this area. The various mathematical techniques are explained and illustrated, for instance, by Edgar and Himmelblau Optimization of Chemical Processes, McGraw-Hill, 1988). 

Nonlinear Programming The most general case for optimization occurs when both the objective function and constraints are nonlinear, a case referred to as nonlinear programming. While the idea behind the search methods used for unconstrained multivariable problems are applicable, the presence of constraints complicates the solution procedure. 

All of these methods have been utilized to solve nonlinear programming problems in the field of chemical engineering design and operations (Lasdon and Waren, Oper. Res., 5, 34, 1980). Nonlinear programming is receiving increased usage in the area of real-time optimization. 

Vaidyanathan, R. and El-Halwagi, M. M. (1994). Global optimization of nonconvex nonlinear programs via interval analysis. Comput. Chem. Eng., 18(10), 889-897. 

The term nonlinear in nonlinear programming does not refer to a material or geometric nonlinearity but instead refers to the nonlinearity in the mathematical optimization problem itself. The first step in the optimization process involves answering questions such as what is the buckling response, what is the vibration response, what is the deflection response, and what is the stress response Requirements usually exist for every one of those response variables. Putting those response characteristics and constraints together leads to an equation set that is inherently nonlinear, irrespective of whether the material properties themselves are linear or nonlinear, and that nonlinear equation set is where the term nonlinear programming comes from. 

In an earlier section, we had alluded to the need to stop the reasoning process at some point. The operationality criterion is the formal statement of that need. In most problems we have some understanding of what properties are easy to determine. For example, a property such as the processing time of a batch is normally given to us and hence is determined by a simple database lookup. The optimal solution to a nonlinear program, on the other hand, is not a simple property, and hence we might look for a simpler explanation of why two solutions have equal objective function values. In the case of our branch-and-bound problem, the operationality criterion imposes two requirements ... 

Another method for solving nonlinear programming problems is based on quadratic programming (QP)1. Quadratic programming is an optimization procedure that minimizes a quadratic objective function subject to linear inequality or equality (or both types) of constraints. For example, a quadratic function of two variables x and X2 would be of the general form ... 

Stochastic optimization methods described previously, such as simulated annealing, can also be used to solve the general nonlinear programming problem. These have the advantage that the search is sometimes allowed to move uphill in a minimization problem, rather than always searching for a downhill move. Or, in a maximization problem, the search is sometimes allowed to move downhill, rather than always searching for an uphill move. In this way, the technique is less vulnerable to the problems associated with local optima. 

The methods discussed for linear and nonlinear programming can be adapted to deal with structural optimization by introducing integer (binary) variables that identify whether... 

One of the approaches that can be used in design is to carry out structural and parameter optimization of a superstructure. The structural optimization required can be carried out using mixed integer linear programming in the case of a linear problem or mixed integer nonlinear programming in the case of a nonlinear problem. Stochastic optimization can also be very effective for structural optimization problems. 

As shown in Fig. 3-53, optimization problems that arise in chemical engineering can be classified in terms of continuous and discrete variables. For the former, nonlinear programming (NLP) problems form the most general case, and widely applied specializations include linear programming (LP) and quadratic programming (QP). An important distinction for NLP is whether the optimization problem is convex or nonconvex. The latter NLP problem may have multiple local optima, and an important question is whether a global solution is required for the NLP. Another important distinction is whether the problem is assumed to be differentiable or not. 

Optimization of a distributed parameter system can be posed in various ways. An example is a packed, tubular reactor with radial diffusion. Assume a single reversible reaction takes place. To set up the problem as a nonlinear programming problem, write the appropriate balances (constraints) including initial and boundary conditions using the following notation ... 

In nonlinear programming problems, optimal solutions need not occur at vertices and can occur at points with positive degrees of freedom. It is possible to have no active constraints at a solution, for example in unconstrained problems. We consider nonlinear problems with constraints in Chapter 8. 

Kalitventzeff, B. Mixed Integer Nonlinear Programming and its Application to the Management of Utility Networks. Eng Optim 18 183-207 (1991). 

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