An inexact qq-order regularized proximal Newton method for nonconvex composite optimization

This paper concerns the composite problem of minimizing the sum of a twice continuously differentiable function $f$ and a nonsmooth convex function. For this class of nonconvex and nonsmooth problems, by leveraging a practical inexactness criterion and a novel selection strategy for iterates, we propose an inexact $q\in[2,3]$-order regularized proximal Newton method, which becomes an inexact cubic regularization (CR) method for $q=3$. We justify that its iterate sequence converges to a stationary point for the KL objective function, and if the objective function has the KL property of exponent $\theta\in(0,\frac{q-1}{q})$, the convergence has a local $Q$-superlinear rate of order $\frac{q-1}{\theta q}$. In particular, under a locally H\"{o}lderian error bound of order $\gamma\in(\frac{1}{q-1},1]$ on a second-order stationary point set, the iterate sequence converges to a second-order stationary point with a local $Q$-superlinear rate of order $\gamma(q\!-\!1)$, which is specified as $Q$-quadratic rate for $q=3$ and $\gamma=1$. This is the first practical inexact CR method with $Q$-quadratic convergence rate for nonconvex composite optimization. We validate the efficiency of the proposed method with ZeroFPR as the solver of subproblems by applying it to convex and nonconvex composite problems with a highly nonlinear $f$

A Superlinear Convergence Framework for Kurdyka-{\L}ojasiewicz Optimization

This work extends the iterative framework proposed by Attouch et al. (in Math. Program. 137: 91-129, 2013) for minimizing a nonconvex and nonsmooth function $\Phi$ so that the generated sequence possesses a Q-superlinear convergence rate. This framework consists of a monotone decrease condition, a relative error condition and a continuity condition, and the first two conditions both involve a parameter $p\!>0$. We justify that any sequence conforming to this framework is globally convergent when $\Phi$ is a Kurdyka-{\L}ojasiewicz (KL) function, and the convergence has a Q-superlinear rate of order $\frac{p}{\theta(1+p)}$ when $\Phi$ is a KL function of exponent $\theta\in(0,\frac{p}{p+1})$. Then, we illustrate that the iterate sequence generated by an inexact $q\in[2,3]$-order regularization method for composite optimization problems with a nonconvex and nonsmooth term belongs to this framework, and consequently, first achieve the Q-superlinear convergence rate of order $4/3$ for an inexact cubic regularization method to solve this class of composite problems with KL property of exponent $1/2$

A recursively feasible and convergent Sequential Convex Programming procedure to solve non-convex problems with linear equality constraints

A computationally efficient method to solve non-convex programming problems with linear equality constraints is presented. The proposed method is based on a recursively feasible and descending sequential convex programming procedure proven to converge to a locally optimal solution. Assuming that the first convex problem in the sequence is feasible, these properties are obtained by convexifying the non-convex cost and inequality constraints with inner-convex approximations. Additionally, a computationally efficient method is introduced to obtain inner-convex approximations based on Taylor series expansions. These Taylor-based inner-convex approximations provide the overall algorithm with a quadratic rate of convergence. The proposed method is capable of solving problems of practical interest in real-time. This is illustrated with a numerical simulation of an aerial vehicle trajectory optimization problem on commercial-of-the-shelf embedded computers