Gradient sampling (GS) methods for the minimization of objective functions that may be nonconvex and/or nonsmooth are proposed, analyzed, and tested. One of the most computationally expensive components of contemporary GS methods is the need to solve a convex quadratic subproblem in each iteration. By contrast, the methods proposed in this paper allow the use of inexact solutions of these subproblems, which, as proved in the paper, can be incorporated without the loss of theoretical convergence guarantees. Numerical experiments show that, by exploiting inexact subproblem solutions, one can consistently reduce the computational effort required by a GS method. Additionally, a strategy is proposed for aggregating gradient information after a subproblem is solved (potentially inexactly) as has been exploited in bundle methods for nonsmooth optimization. It is proved that the aggregation scheme can be introduced without the loss of theoretical convergence guarantees. Numerical experiments show that incorporating this gradient aggregation approach can also reduce the computational effort required by a GS method.
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This content will become publicly available on March 25, 2025
A new inexact gradient descent method with applications to nonsmooth convex optimization
The paper proposes and develops a novel inexact gradient method (IGD) for minimizing smooth
functions with Lipschitzian gradients. We show that the sequence of
gradients generated by IGD converges to zero. The convergence of iterates to stationary points is guaranteed
under the Kurdyka- Lojasiewicz property of the objective function with convergence rates depending on
the KL exponent. The newly developed IGD is applied to designing two novel gradient-based methods of nonsmooth convex optimization such as the inexact proximal point methods (GIPPM) and the inexact augmented
Lagrangian method (GIALM) for convex programs with linear equality constraints. These two methods inherit
global convergence properties from IGD and are confirmed by numerical experiments to have practical advantages over some well-known algorithms of nonsmooth convex optimization
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- Award ID(s):
- 2204519
- PAR ID:
- 10515872
- Publisher / Repository:
- Taylor & Francis
- Date Published:
- Journal Name:
- Optimization Methods and Software
- ISSN:
- 1055-6788
- Page Range / eLocation ID:
- 1 to 29
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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