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Successive quadratic approximations (SQA) are numerically efficient forminimizing the sum of a smooth function and a convex function. The iterationcomplexity of inexact SQA methods has been analyzed recently. In this paper, wepresent an algorithmic framework of inexact SQA methods with four types of linesearches, and analyze its global complexity under milder assumptions. First, weshow its well-definedness and some decreasing properties. Second, under thequadratic growth condition and a uniform positive lower bound condition onstepsizes, we show that the function value sequence and the iterate sequenceare linearly convergent. Moreover, we obtain a o(1/k) complexity without thequadratic growth condition, improving existing O(1/k) complexity results. Atlast, we show that a local gradient-Lipschitz-continuity condition couldguarantee a uniform positive lower bound for the stepsizes.

Global Complexity Analysis of Inexact Successive Quadratic Approximation methods for Regularized Optimization under Mild Assumptions