Duality Gap Estimates for Weak Chebyshev Greedy Algorithms in Banach Spaces

The paper studies weak greedy algorithms for finding sparse solutions of convex optimization problems in Banach spaces. We consider the concept of duality gap, the values of which are implicitly calculated at the step of choosing the direction of the fastest descent at each iteration of the greedy a...

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Bibliographic Details
Published inComputational mathematics and mathematical physics Vol. 59; no. 6; pp. 904 - 914
Main Authors Mironov, S. V., Sidorov, S. P.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.06.2019
Springer Nature B.V
Subjects
Algorithms
Banach spaces
Chebyshev approximation
Computational geometry
Computational Mathematics and Numerical Analysis
Convexity
Greedy algorithms
Mathematics
Mathematics and Statistics
Objective function
Optimization
Upper bounds
greedy algorithms
sparse solution
nonlinear optimization
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Summary:The paper studies weak greedy algorithms for finding sparse solutions of convex optimization problems in Banach spaces. We consider the concept of duality gap, the values of which are implicitly calculated at the step of choosing the direction of the fastest descent at each iteration of the greedy algorithm. We show that these values give upper bounds for the difference between the values of the objective function in the current state and the optimal point. Since the value of the objective function at the optimal point is not known in advance, the current values of the duality gap can be used, for example, in the stopping criteria for the greedy algorithm. In the paper, we find estimates of the duality gap values depending on the number of iterations for the weak greedy algorithms under consideration.
ISSN:0965-5425
1555-6662
DOI:10.1134/S0965542519060113