Search

Given a closed convex set A in a Banach space X, motivated by the continuity and Fréchet differentiability of A introduced, respectively, in [D. Gale and V. Klee, Math. Scand., 7 (1959), pp. 379-391] and [X. Y. Zheng, SIAM J. Optim., 30 (2020), pp. 490-512], this paper considers the Cp-differentiability, subdifferentiability, and G\^ateaux differentiability of A. Using the technique of variational analysis, it is proved that A is Cp-differentiable (resp., subdifferentiable or G\^ateaux differentiable) if and only if for every continuous convex function f : X → R with infx∈A f(x) > infx∈X f(x) the corresponding constrained optimization problem PA(f) is 2/p-order-well-posed solvable (resp., generalized well-posed solvable or weak well-posed solvable). It is also proved that if the conjugate function f* of a continuous convex function f on X is Cp-differentiable on dom(f*), then for every closed convex set A in X with infx∈A f(x) > -∞ the corresponding optimization problem PA(f) is 2/p-order-well-posed solvable. As a byproduct, every constrained convex optimization problem with a strongly convex quadratic objective function is proved to be globally second-order-well-posed solvable. Our main results are new even in the case of finite dimensional spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the sequential convex programming method with monotone line search (SCPls) in [Z. Lu, Sequential Convex Programming Methods for a Class of Structured Nonlinear Programming, https://arxiv.org/abs/1210.3039, 2012] for a class of difference-of-convex optimization problems with multiple smooth inequality constraints. The SCPls is a representative variant of moving-ball-approximation-type algorithms [A. Auslender, R. Shefi, and M. Teboulle, SIAM J. Optim., 20 (2010), pp. 3232-3259; J. Bolte, Z. Chen, and E. Pauwels, Math. Program., 182 (2020) pp. 1-36; J. Bolte and E. Pauwels, Math. Oper. Res., 41 (2016), pp. 442-465; R. Shefi and M. Teboulle, Math. Program., 159 (2016), pp. 137-164] for constrained optimization problems. We analyze the convergence rate of the sequence generated by SCPls in both nonconvex and convex settings by imposing suitable Kurdyka--Löjasiewicz (KL) assumptions. Specifically, in the nonconvex settings, we assume that a special potential function related to the ob jective and the constraints is a KL function, while in the convex settings we impose KL assumptions directly on the extended ob jective function (i.e., sum of the ob jective and the indicator function of the constraint set). A relationship between these two different KL assumptions is established in the convex settings under additional differentiability assumptions. We also discuss how to deduce the KL exponent of the extended ob jective function from its Lagrangian in the convex settings, under additional assumptions on the constraint functions. Thanks to this result, the extended ob jectives of some constrained optimization models such as minimizing\ell1 sub ject to logistic/Poisson loss are found to be KL functions with exponent 1 under mild assumptions. To illustrate how our results can be applied, we consider SCPs for minimizing 1-2 [P-Yin, Y. Lou, Q. He, and J. Xin, SIAM J. Sci. Comput., 37 (2015), pp. A536--A563] subject to residual error measured by norm/Lorentzian norm [R. E. Carrillo, A. B. Ramirez, G. R. Arce, K. E. Barner, and B. M. Sadler, EURASIP J. Adv. Signal Process. (2016), 108]. We first discuss how the various conditions required in our analysis can be verified, and then perform numerical experiments to illustrate the convergence behaviors of SCPls. [ABSTRACT FROM AUTHOR]