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Primal Methods

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Linear and Nonlinear Programming

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 228))

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Abstract

In this chapter we initiate the presentation, analysis, and comparison of algorithms designed to solve constrained minimization problems. The four chapters that consider such problems roughly correspond to the following classification scheme. Consider a constrained minimization problem having n variables and m constraints. Methods can be devised for solving this problem that work in spaces of dimension n − m, n, m, or n + m. Each of the following chapters corresponds to methods in one of these spaces. Thus, the methods in the different chapters represent quite different approaches and are founded on different aspects of the theory. However, there are also strong interconnections between the methods of the various chapters, both in the final form of implementation and in their performance. Indeed, there soon emerges the theme that the rates of convergence of most practical algorithms are determined by the Lipschitz constants and the structure of the Hessian of the Lagrangian much like the structure of the Hessian of the objective function determines the rates of convergence for a wide assortment of methods for unconstrained problems. Thus, although the various algorithms of these chapters differ substantially in their motivation, they are ultimately found to be governed by a common set of principles.

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Notes

  1. 1.

    Actually a more standard procedure is to define the pseudoinverse \(\overline {\mathbf {L}}_{k}^{\dagger }\), and then \(\mathbf {z}=\overline {\mathbf {L}}_{k}^{\dagger }{\mathbf {y}}_{k}\).

  2. 2.

    The exact solution is obviously symmetric about the center of the chain, and hence the problem could be reduced to having ten links and only one constraint. However, this symmetry disappears if the first constraint value is specified as nonzero. Therefore for generality we solve the full chain problem.

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Luenberger, D.G., Ye, Y. (2021). Primal Methods. In: Linear and Nonlinear Programming. International Series in Operations Research & Management Science, vol 228. Springer, Cham. https://doi.org/10.1007/978-3-030-85450-8_12

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