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10 - Duality and Algorithms

Published online by Cambridge University Press:  31 March 2022

Stephen J. Wright
Affiliation:
University of Wisconsin, Madison
Benjamin Recht
Affiliation:
University of California, Berkeley
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Summary

Here, we discuss concepts of duality for convex optimization problems, and algorithms that make use of these concepts. We define the Lagrangian function and its augmented Lagrangian counterpart. We use the Lagrangian to derive optimality conditions for constrained optimization problems in which the constraints are expressed as linear algebraic conditions. We introduce the dual problem, and discuss the concepts of weak and strong duality, and show the existence of positive duality gaps in certain settings. Next, we discuss the dual subgradient method, the augmented Lagrangian method, and the alternating direction method of multipliers (ADMM), which are useful for several types of data science problems.

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Publisher: Cambridge University Press
Print publication year: 2022

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