On the complexity of solving feasibility problems with regularized models. (4th March 2022) - Birgin, E. G.; Bueno, L. F.; Martínez, J. M.
Abstract : 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 p th-order model of the sum of squares of the expensive constraints subject to the cheap constraints. Under a Hölder continuity property with constant β ∈ ( 0, 1 ] on the p th derivatives of the expensive constraints, it is shown that finding a feasible point with precision ε > 0 or an infeasible point that is stationary with tolerance γ > 0 of minimizing the sum of squares of the expensive constraints subject to the cheap constraints has iteration complexity O ( | log ⁡ ( ε ) | γ ζ ( p, β ) ω p 1 + ( 1 / 2 ) ζ ( p, β ) ) and evaluation complexity (of the expensive constraints) O ( | log ⁡ ( ε ) | [ γ ζ ( p, β ) ω p 1 + ( 1 / 2 ) ζ ( p, β ) + ( 1 − β ) / ( p + β − 1 ) | log ⁡ ( γ ε ) | ] ), where ζ ( p, β ) = − ( p + β ) / ( p + β − 1 ) and ω p = ε if p = 1, while 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 ⁡ ( ε ) | ), independently of p .