A descent subgradient method using Mifflin line search for nonsmooth nonconvex optimization

We propose a descent subgradient algorithm for minimizing a real function, assumed to be locally Lipschitz, but not necessarily smooth or convex. To find an effective descent direction, the Goldstein subdifferential is approximated through an iterative process. The method enjoys a new two-point variant of Mifflin line search in which the subgradients are arbitrary. Thus, the line search procedure is easy to implement. Moreover, in comparison to bundle methods, the quadratic subproblems have a simple structure, and to handle nonconvexity the proposed method requires no algorithmic modification. We study the global convergence of the method and prove that any accumulation point of the generated sequence is Clarke stationary, assuming that the objective $f$ is weakly upper semismooth. We illustrate the efficiency and effectiveness of the proposed algorithm on a collection of academic and semi-academic test problems

Optimizing condition numbers

In this paper we study the problem of minimizing condition numbers over a compact convex subset of the cone of symmetric positive semidefinite $n\times n$ matrices. We show that the condition number is a Clarke regular strongly pseudoconvex function. We prove that a global solution of the problem can be approximated by an exact or an inexact solution of a nonsmooth convex program. This asymptotic analysis provides a valuable tool for designing an implementable algorithm for solving the problem of minimizing condition numbers

First-Order Methods for Nonsmooth Nonconvex Functional Constrained Optimization with or without Slater Points

Constrained optimization problems where both the objective and constraints may be nonsmooth and nonconvex arise across many learning and data science settings. In this paper, we show a simple first-order method finds a feasible, $\epsilon$-stationary point at a convergence rate of $O(\epsilon^{-4})$ without relying on compactness or Constraint Qualification (CQ). When CQ holds, this convergence is measured by approximately satisfying the Karush-Kuhn-Tucker conditions. When CQ fails, we guarantee the attainment of weaker Fritz-John conditions. As an illustrative example, our method stably converges on piecewise quadratic SCAD regularized problems despite frequent violations of constraint qualification. The considered algorithm is similar to those of "Quadratically regularized subgradient methods for weakly convex optimization with weakly convex constraints" by Ma et al. and "Stochastic first-order methods for convex and nonconvex functional constrained optimization" by Boob et al. (whose guarantees further assume compactness and CQ), iteratively taking inexact proximal steps, computed via an inner loop applying a switching subgradient method to a strongly convex constrained subproblem. Our non-Lipschitz analysis of the switching subgradient method appears to be new and may be of independent interest

Algorithms for Difference-of-Convex (DC) Programs Based on Difference-of-Moreau-Envelopes Smoothing

In this paper we consider minimization of a difference-of-convex (DC) function with and without linear constraints. We first study a smooth approximation of a generic DC function, termed difference-of-Moreau-envelopes (DME) smoothing, where both components of the DC function are replaced by their respective Moreau envelopes. The resulting smooth approximation is shown to be Lipschitz differentiable, capture stationary points, local, and global minima of the original DC function, and enjoy some growth conditions, such as level-boundedness and coercivity, for broad classes of DC functions. We then develop four algorithms for solving DC programs with and without linear constraints based on the DME smoothing. In particular, for a smoothed DC program without linear constraints, we show that the classic gradient descent method as well as an inexact variant can obtain a stationary solution in the limit with a convergence rate of $\mathcal{O}(K^{-1/2})$, where $K$ is the number of proximal evaluations of both components. Furthermore, when the DC program is explicitly constrained in an affine subspace, we combine the smoothing technique with the augmented Lagrangian function and derive two variants of the augmented Lagrangian method (ALM), named LCDC-ALM and composite LCDC-ALM, focusing on different structures of the DC objective function. We show that both algorithms find an $\epsilon$-approximate stationary solution of the original DC program in $\mathcal{O}(\epsilon^{-2})$ iterations. Comparing to existing methods designed for linearly constrained weakly convex minimization, the proposed ALM-based algorithms can be applied to a broader class of problems, where the objective contains a nonsmooth concave component. Finally, numerical experiments are presented to demonstrate the performance of the proposed algorithms

An adaptive sampling sequential quadratic programming method for nonsmooth stochastic optimization with upper-C2\mathcal{C}^2 objective

We propose an optimization algorithm that incorporates adaptive sampling for stochastic nonsmooth nonconvex optimization problems with upper-$\mathcal{C}^2$ objective functions. Upper-$\mathcal{C}^2$ is a weakly concave property that exists naturally in many applications, particularly certain classes of solutions to parametric optimization problems, e.g., recourse of stochastic programming and projection into closed sets. Our algorithm is a stochastic sequential quadratic programming (SQP) method extended to nonsmooth problems with upper$\mathcal{C}^2$ objectives and is globally convergent in expectation with bounded algorithmic parameters. The capabilities of our algorithm are demonstrated by solving a joint production, pricing and shipment problem, as well as a realistic optimal power flow problem as used in current power grid industry practice.Comment: arXiv admin note: text overlap with arXiv:2204.0963