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In this paper, we study the sequential convex programming method withmonotone line search (SCP$_{ls}$) in [46] for a class of difference-of-convex(DC) optimization problems with multiple smooth inequality constraints. TheSCP$_{ls}$ is a representative variant of moving-ball-approximation-typealgorithms [6,10,13,54] for constrained optimization problems. We analyze theconvergence rate of the sequence generated by SCP$_{ls}$ in both nonconvex andconvex settings by imposing suitable Kurdyka-Lojasiewicz (KL) assumptions.Specifically, in the nonconvex settings, we assume that a special potentialfunction related to the objective and the constraints is a KL function, whilein the convex settings we impose KL assumptions directly on the extendedobjective function (i.e., sum of the objective and the indicator function ofthe constraint set). A relationship between these two different KL assumptionsis established in the convex settings under additional differentiabilityassumptions. We also discuss how to deduce the KL exponent of the extendedobjective function from its Lagrangian in the convex settings, under additionalassumptions on the constraint functions. Thanks to this result, the extendedobjectives of some constrained optimization models such as minimizing $\ell_1$subject to logistic/Poisson loss are found to be KL functions with exponent$\frac12$ under mild assumptions. To illustrate how our results can be applied,we consider SCP$_{ls}$ for minimizing $\ell_{1-2}$ [60] subject to residualerror measured by $\ell_2$ norm/Lorentzian norm [21]. We first discuss how thevarious conditions required in our analysis can be verified, and then performnumerical experiments to illustrate the convergence behaviors of SCP$_{ls}$.

Convergence rate analysis of a sequential convex programming method with  line search for a class of constrained difference-of-convex optimization  problems