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1 results

  • 7 - First-Order Methods for Constrained Optimization

  • Stephen J. Wright, University of Wisconsin, Madison, Benjamin Recht, University of California, Berkeley
  • Book:
    Optimization for Data Analysis
    Published online:
    31 March 2022
    Print publication:
    21 April 2022, pp 118-131

  • 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.