On the complexity of solving feasibility problems with regularized models

• Published in: Optimization methods & software Vol. 37; no. 2; pp. 405 - 424
• Main Authors: Birgin, E. G., Bueno, L. F., Martínez, J. M.
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 04.03.2022; Taylor & Francis Ltd
• Subjects: Complexity; Continuity (mathematics); Feasibility; feasibility problem; high order methods; Iterative methods; Lipschitz condition
• ISSN: 1055-6788; 1029-4937
• DOI: 10.1080/10556788.2020.1786564

The complexity of solving feasibility problems is considered in this work. It is assumed that the constraints that define the problem can be divided into expensive and cheap constraints. At each iteration, the introduced method minimizes a regularized pth-order model of the sum of squares of the expensive constraints subject to the cheap constraints. Under a Hölder continuity property with constant $\beta \in (0,1] $ β ∈ ( 0 , 1 ] on the pth derivatives of the expensive constraints, it is shown that finding a feasible point with precision  $\varepsilon > 0 $ ε > 0 or an infeasible point that is stationary with tolerance  $\gamma > 0 $ γ > 0 of minimizing the sum of squares of the expensive constraints subject to the cheap constraints has iteration complexity $O(|\log (\varepsilon )| \, \gamma ^{\zeta (p,\beta )} \, \omega _p^{1 + ({1}/{2}) \zeta (p,\beta )} ) $ O ( | log ⁡ ( ε ) | γ ζ ( p , β ) ω p 1 + ( 1 / 2 ) ζ ( p , β ) ) and evaluation complexity (of the expensive constraints) $O( |\log (\varepsilon )| [ \gamma ^{\zeta (p,\beta )} \, \omega _p^{1 + ({1}/{2}) \zeta (p,\beta )} + {(1-\beta )}/({p+\beta -1} )| \log (\gamma \sqrt {\varepsilon })| ]) $ O ( | log ⁡ ( ε ) | [ γ ζ ( p , β ) ω p 1 + ( 1 / 2 ) ζ ( p , β ) + ( 1 − β ) / ( p + β − 1 ) | log ⁡ ( γ ε ) | ] ) , where $\zeta (p,\beta ) = - (p+\beta )/(p+\beta -1) $ ζ ( p , β ) = − ( p + β ) / ( p + β − 1 ) and $\omega _p = \varepsilon $ ω p = ε if p = 1, while $\omega _p = \Phi (x^0) $ ω p = Φ ( x 0 ) if p>1. Moreover, if the derivatives satisfy a Lipschitz condition and a uniform regularity assumption holds, both complexities reduce to $O(|\log (\varepsilon )|) $ O ( | log ⁡ ( ε ) | ) , independently of p.