Search

The Fantope-constrained sparse principal subspace estimation problem is initially proposed by Vu et al. (Vu et al., 2013). This paper investigates a semismooth Newton based proximal point (Ppassn) algorithm for solving the equivalent form of this problem, where a semismooth Newton (Ssn) method is utilized to optimize the inner problems involved in the Ppassn algorithm. Under standard conditions, the Ppassn algorithm is proven to achieve global convergence and an asymptotic superlinear convergence rate. Computationally, we derive nontrivial expressions for the Fantope projection and its generalized Jacobian, which are key ingredients for the Ppassn algorithm. Some numerical results on synthetic and real data sets are presented to illustrate the effectiveness of the proposed Ppassn algorithm for large-scale problems and superiority over the alternating direction method of multipliers (ADMM). [ABSTRACT FROM AUTHOR]

As applying the nonsmooth Newton’s method to the equivalent reformulation of the linear complementarity problem, a modulus-based nonsmooth Newton’s method is established and its locally quadratical convergence conditions are presented. In the implementation, local one step convergence is discussed by properly choosing the initial vector and the generalized Jacobian, and a mixed algorithm is given for finding an initial vector. Numerical experiments show that the proposed methods are efficient and accelerate the convergence performance of the modulus-based matrix splitting iteration methods. [ABSTRACT FROM AUTHOR]

In this work, first we introduce several notions of invexity and pseudoinvexity for a locally Lipschitz function by means of the generalized Jacobian. We study relationships between these concepts, in particular the implications between preinvexity and invexity. Next, we obtain necessary and sufficient optimality conditions for efficient and weak efficient solutions of finite-dimensional (non necessarily Pareto) vector optimization problems with locally Lipschitz objective functions through solutions of vector variational-like inequality problems. These conditions are stated via the generalized Jacobian and under pseudoinvexity hypotheses, and they show that a vector optimization problem can be reformulated as a vector variational-like inequality problem. This work extends and improves several previous papers, where the objective function of the vector optimization problem is assumed to be differentiable, or being locally Lipschitz, the authors consider the componentwise subdifferential based on the Clarke’s generalized gradients of the components of the objective function. Throughout the paper some simple examples are given in order to illustrate the main concepts and results. [ABSTRACT FROM AUTHOR]

With the ultimate goal of iteratively solving piecewise smooth (PS) systems, we consider the solution of piecewise linear (PL) equations. As shown in [7] PL models can be derived in the fashion of automatic or algorithmic differentiation as local approximations of PS functions with a second order error in the distance to a given reference point. The resulting PL functions are obtained quite naturally in what we call the abs-normal form, a variant of the state representation proposed by Bokhoven in his dissertation [27] . Apart from the tradition of PL modelling by electrical engineers, which dates back to the Master thesis of Thomas Stern [26] in 1956, we take into account more recent results on linear complementarity problems and semi-smooth equations originating in the optimization community [3,25,5] . We analyze simultaneously the original PL problem (OPL) in abs-normal form and a corresponding complementary system (CPL), which is closely related to the absolute value equation (AVE) studied by Mangasarian and Meyer [14] and a corresponding linear complementarity problem (LCP). We show that the CPL, like KKT conditions and other simply switched systems, cannot be open without being injective. Hence some of the intriguing PL structure described by Scholtes in [25] is lost in the transformation from OPL to CPL. To both problems one may apply Newton variants with appropriate generalized Jacobians directly computable from the abs-normal representation. Alternatively, the CPL can be solved by Bokhoven's modulus method and related fixed point iterations. We compile the properties of the various schemes and highlight the connection to the properties of the Schur complement matrix, in particular its partial contractivity as analyzed by Rohn and Rump [23] . Numerical experiments and suitable combinations of the fixed point solvers and stabilized generalized Newton variants remain to be realized. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we describe H-differential of the implicit Lagrangian function. We show how, under appropriate regularity conditions on an H-differential of f, minimizing the implicit Lagrangian function corresponding to f leads to a solution of the nonlinear complementarity problem. Our results give a unified treatment of such results for C 1-functions and for locally Lipschitzian functions. [Copyright &y& Elsevier]

Abstract: This paper deals with characterizations of strictly semi-monotone (semi-monotone) properties in nonsmooth functions that are H-differentiable. These results give a unified treatment of such results for C 1-functions, semi-smooth functions, and for locally Lipschitzian functions. Illustrations to nonlinear complementarity problems are given to show the usefulness of our results. [Copyright &y& Elsevier]

In these notes, we present a general result concerning the Lipschitz regularity of a certain type of set-valued maps often found in constrained optimization and control problems. The class of multifunctions examined in this paper is characterized by means of a set of Lipschitz continuous constraint functions defined on some Lipschitz manifold. The proof of the regularity result for this class of multifunctions is based on a quantitative version of the Implicit Function Theorem for Lipschitzian maps which provides estimates for the neighborhoods where the implicit map can be defined. [Copyright &y& Elsevier]