Nonlinear Programming

K. Schittkowski, C. Zillober: Encyclopedia of Life Support Systems (EOLSS), UNESCO, Topic: Optimization and Operations Research, 157-177 (2003)
Abstract: Nonlinear programming is a direct extension of linear programming, when we replace linear model functions by nonlinear ones. Numerical algorithms and computer programs are widely applicable and commercially available in form of black box software. However, to understand how optimization methods work, how corresponding programs are organized, how the results are to be interpreted, and, last not least, what are the limitations of the powerful mathematical technology, it is necessary to understand at least the basic terminology. Thus, we present a brief introduction into optimization theory, in particular we introduce optimality criteria for smooth problems. These conditions are extremely important to understand how mathematical algorithms work.

The most popular classes of constrained nonlinear programming algorithms are introduced, i.e., penalty-barrier, interior point, augmented Lagrangian, sequential quadratic programming, sequential linear programming, generalized reduced gradient, and sequential convex programming methods. Common features and methodological differences are outlined. In particular we discuss extensions of these methods for solving large scale nonlinear programming problems.

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