Nonlinear programs

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). 

The best way to approach the retrofit synthesis of the heat-exchanger network is to model all five tasks simultaneously. A mixed-integer nonlinear programming model is usually formulated to accomplish this goal. 

Kuhn, H. W, and A. W. Tucker. Nonlinear Programming in Neyman, J. (ed.), Froc. Second Berkeley Symp. Mathematical Statistics and Frobabil-ity (1951) 402-411. 

MJNOS5.4. A package available from Stanford Research Institute (affiliated with Stanford University). This package is the state of the ari for mildly nonlinear programming problems. 

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). 

Maximum yield from a four-stage CSTB by nonlinear programming... 

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. 

One important class of nonlinear programming techniques is called quadratic programming (QP), where the objective function is quadratic and the constraints are hnear. While the solution is iterative, it can be obtained qmckly as in linear programming. This is the basis for the newest type of constrained multivariable control algorithms called model predic tive control. The dominant method used in the refining industiy utilizes the solution of a QP and is called dynamic matrix con-... 

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

This is a nonlinear program which can be solved using LINGO to get the following... 

Achenie, L., and Duvedi, A. (1996). A Mixed Integer Nonlinear Programming Modle for the Design of Refrigerant Mixtures, AJChE Annual Meeting, San Francisco. 

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. 

The synthesis of operating procedures for continuous chemical plants can be represented as a mixed-integer nonlinear programming problem, and it has been addressed as such by other researchers. In this chapter we have attempted to present a unifying theoretical framework, which ad-... 

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 ... 

Here the B-spline Bim(zf, Xj) is the ith B-spline basis function on the extended partition Xj (which contains locations of the knots in the Zj direction), and is a coefficient. We use cubic splines and sufficient numbers of uniformly spaced knots so that the estimation problem is not affected by the partition. The estimation problem now involves determining the set of B-spline coefficients that minimizes Eq. (4.1.26), subject to the state equations [Eqs. (4.1.24 and 4.1.25)], for a suitable value of the regularization parameter. At this point, the minimization problem corresponds to a nonlinear programming problem. 

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. 

---