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7 - First-Order Methods for Constrained Optimization

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 describe methods for minimizing a smooth function over a closed convex set, using gradient information. We first state results that characterize optimality of points in a way that can be checked, and describe the vital operation of projection onto the feasible set. We next describe the projected gradient algorithm, which is in a sense the extension of the steepest-descent method to the constrained case, analyze its convergence, and describe several extensions. We next analyze the conditional-gradient method (also known as “Frank-Wolfe”) for the case in which the feasible set is compact and demonstrate sublinear convergence of this approach when the objective function is convex.

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Publisher: Cambridge University Press
Print publication year: 2022

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