Nonlinear programming method

The NLP (nonlinear programming) methods to be discussed in this chapter differ mainly in how they generate the search directions. Some nonlinear programming methods require information about derivative values, whereas others do not use derivatives and rely solely on function evaluations. Furthermore, finite difference substitutes can be used in lieu of derivatives as explained in Section 8.10. For differentiable functions, methods that use analytical derivatives almost always use less computation time and are more accurate, even if finite difference approxima-... 

These early approaches suffered from two drawbacks. First, simultaneous approaches lead to much larger nonlinear programs than embedded model approaches. Consequently, nonlinear programming methods available at that time were too slow to compete with smaller feasible path formulations. Second, care must be taken in the formulation in order to yield an accurate algebraic representation of the differential equations. 

L. Pibouleau, P. Floquet, and S. Domenech. Optimal synthesis of reactor separator systems by nonlinear programming method. AlChEJ., 34 163,1988. 

McKinney, D. C., and Lin, M. D. (1995). "Approximate mixed-integer nonlinear programming methods for optimal aquifer remediation design." Water Resour. Res., 31(3), 731-740. 

Biegler, L. T. Advanced Nonlinear Programming Methods for Chemical Process Optimization University of Wisconsin-Madison., O. A. Hougen Lecture, September 22 (2009). 

Several nonlinear programming methods have been applied to... 

Pibouleau. L., Floquet, P., and Domenech, S. "Optimal Synthesis of Reactor Separator Systems by Nonlinear Programming Method, AIChE J. 34, 163 (1988). 

In order to accelerate the process optimization, Kawajiri and Biegler (2006b) have developed an efficient full discretization approach combined with a large-scale nonlinear programming method for the optimization of SMBs. More recently, they have extended this approach to a superstructure SMB formulation and used the e-constraint method to solve the bi-objective problem, where throughput and desorbent consumption were optimized (Kawajiri and Biegler, 2006a). 

Erbil, H.Y. and Meric, R.A. (1988). Determination of surface free energy components of polymers from contact angle data using nonlinear programming methods. Colloids and Surfaces, 33, 85-97. 

Nonlinearities can arise in a number of ways in optimization problems in either the objective function or the constraints. Some of the nonlinearities can be handled by LP methods, whereas the rest have to be solved by specialized nonlinear programming methods. 

Problem Type Nonlinear programs Method Hlipsoid algorithm... 

Problem Type Linear, mixed-integer, nonlinear, dynamic, and mixed-integer nonlinear programs Method Generalized benders decomposition, outer approximation and variants, genertilized cross decomposition... 

Problem Type Large-scale linear and nonlinear programs Method Projected Lagrangian... 

Problem Type Dense linear and nonlinear programs Method Sequential quadratic programming... 

Yuan, X., Zhang, S., and Pibouleau, L. (1988) A Mixed-Integer Nonlinear-Programming Method for Process Design. RAIRO - Recherche Operationnelle-Operations Research, 22, 331-346. 

Contact angles obtained with the Wilhelmy plate method and near surface energies, Yj and y, obtained with nonlinear programming methods were varied with chain length surface concentrations for 72-alkyldimethyl monochlorosilanes (i.e., 72 = 1, 4, 8, and 18) in various solvents. 

The time consumer in calculating chemical equilibria by one of the linear or nonlinear programming methods is usually considerably longer than the time needed for methods based on the theory of Lagrangian multipliers For this reason, linear and non-linear programming are rather seldom employed in calculating chemical equilibria. 

Simplex methods ([72, 71, 73]) move from boundary to boundary within the feasible region. The simplex methods requires initial basic solution to be feasible. There are various variants of simplex methods like dual simplex method, the Big M method, and the two-phase simplex method. Interior point methods on the other hand visit points within the interior of the feasible region more inline with the nonlinear programming methods. In general, good interior point methods perform as well or better than simplex codes on larger problems when no prior information about the solution is available. When such warm start information is available, simplex methods are able to make much better use of it than the interior point methods. 

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