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Regional complexity analysis of algorithms for nonconvex smooth optimization
- Source :
- Mathematical Programming. 187:579-615
- Publication Year :
- 2020
- Publisher :
- Springer Science and Business Media LLC, 2020.
-
Abstract
- A strategy is proposed for characterizing the worst-case performance of algorithms for solving nonconvex smooth optimization problems. Contemporary analyses characterize worst-case performance by providing, under certain assumptions on an objective function, an upper bound on the number of iterations (or function or derivative evaluations) required until a pth-order stationarity condition is approximately satisfied. This arguably leads to conservative characterizations based on anomalous objectives rather than on ones that are typically encountered in practice. By contrast, the strategy proposed in this paper characterizes worst-case performance separately over regions comprising a search space. These regions are defined generically based on properties of derivative values. In this manner, one can analyze the worst-case performance of an algorithm independently from any particular class of objectives. Then, once given a class of objectives, one can obtain an informative, fine-tuned complexity analysis merely by delineating the types of regions that comprise the search spaces for functions in the class. Regions defined by first- and second-order derivatives are discussed in detail and example complexity analyses are provided for a few fundamental first- and second-order algorithms when employed to minimize convex and nonconvex objectives of interest. It is also explained how the strategy can be generalized to regions defined by higher-order derivatives and for analyzing the behavior of higher-order algorithms.
- Subjects :
- Class (set theory)
Mathematical optimization
021103 operations research
Optimization problem
General Mathematics
Numerical analysis
0211 other engineering and technologies
Regular polygon
010103 numerical & computational mathematics
02 engineering and technology
Function (mathematics)
Space (mathematics)
01 natural sciences
Upper and lower bounds
Optimization and Control (math.OC)
FOS: Mathematics
0101 mathematics
Mathematics - Optimization and Control
Software
Analysis of algorithms
Mathematics
Subjects
Details
- ISSN :
- 14364646 and 00255610
- Volume :
- 187
- Database :
- OpenAIRE
- Journal :
- Mathematical Programming
- Accession number :
- edsair.doi.dedup.....5d0dea941aefa4d98d1fb3a96e74ae05