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Delivering large horizontal controlled forces for long periods with aerial manipulators is not an easy task to accomplish; several factors including disturbances, reaction forces from the on-board arm or actuator's limitations might diminish their horizontal force delivery capabilities. Aiming to mitigate these drawbacks, this paper presents a comprehensive study of the force delivery capabilities of aerial manipulators leveraging the environment. The methodology to evaluate the aforementioned capabilities is the Generalized-Jacobian-based force ellipsoid, which was applied to different types of both free-flying and attached-to-the-environment aerial manipulators. In addition, large controlled force delivery experiments with the mentioned manipulators were conducted on physics-engine-based robotics simulation software. The obtained results categorically demonstrate that aerial manipulators leveraging the environment are capable of delivering larger forces than their free-flying counterparts, which has not been proved in the literature so far in the field of aerial robotics. [ABSTRACT FROM AUTHOR]

The goal of this note is to point out an erroneous formula for the generalized Hessian of the least squares associated with a system of linear inequalities, that was given in the paper ‘A finite Newton method for classification’ by Mangasarian (Optim. Methods Softw. 17 (2002), pp. 913–929) and reproduced multiple times in other publications. We also provide sufficient contiditions for the validity of Mangasarian's formula and show that Slater's condition guarantees that some particular elements from the set defined by Mangasarian belong to the generalized Hessian of the corresponding function. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Jacobian smoothing inexact Newton-type algorithm for solving the nonlinear complementarity problem by reformulating it as a system of nonlinear equations. We show that the algorithm converges up to q-quadratically and present numerical experiments that show its good local performance. In order to compare in terms of time our algorithm with other known algorithms, we introduce a new normalized measurement that we call time index. [ABSTRACT FROM AUTHOR]

This paper is concerned with the ℓ 1 , ∞ -norm ball constrained multi-task learning problem, which has received extensive attention in many research areas such as machine learning, cognitive neuroscience, and signal processing. To address the challenges of solving large-scale multi-task Lasso problems, this paper develops an inexact semismooth Newton-based augmented Lagrangian (Ssnal) algorithm. When solving the inner problems in the Ssnal algorithm, the semismooth Newton (Ssn) algorithm with superlinear or even quadratic convergence is applied. Theoretically, this paper presents the global and asymptotically superlinear local convergence of the Ssnal algorithm under standard conditions. Computationally, we derive an efficient procedure to construct the generalized Jacobian of the projector onto ℓ 1 , ∞ -norm ball, which is an important component of the Ssnal algorithm, making the computational cost in the Ssn algorithm very cheap. Comprehensive numerical experiments on the multi-task Lasso problems demonstrate that the Ssnal algorithm is more efficient and robust than several existing state-of-the-art first-order algorithms. [ABSTRACT FROM AUTHOR]

Delivering large horizontal controlled forces for long periods with aerial manipulators is not an easy task to accomplish; several factors including disturbances, reaction forces from the on-board arm or actuator’s limitations might diminish their horizontal force delivery capabilities. Aiming to mitigate these drawbacks, this paper presents a comprehensive study of the force delivery capabilities of aerial manipulators leveraging the environment. The methodology to evaluate the aforementioned capabilities is the Generalized-Jacobian-based force ellipsoid, which was applied to different types of both free-flying and attached-to-the-environment aerial manipulators. In addition, large controlled force delivery experiments with the mentioned manipulators were conducted on physics-engine-based robotics simulation software. The obtained results categorically demonstrate that aerial manipulators leveraging the environment are capable of delivering larger forces than their free-flying counterparts, which has not been proved in the literature so far in the field of aerial robotics.

For a smooth projective curve defined over an algebraic number field k, we investigate the finiteness of the set of generalized Jacobians of associated with modules defined over such that a fixed divisor representing a class of finite order in the Jacobian J of provides the torsion class in the generalized Jacobian . Various results on the finiteness and infiniteness of the set of generalized Jacobians with the above property are obtained depending on the geometric conditions on the support of , as well as on the conditions on the field k. These results are applied to the problem of the periodicity of a continued fraction expansion constructed in the field of formal power series for special elements of the field of functions of the hyperelliptic curve . [ABSTRACT FROM AUTHOR]

Due to the importance and effectiveness of a Jacobian smoothing method or smoothing quasi-Newton method for solving the nonlinear complementarity problem and that the Minimum complementarity function has not been used in connection with such methods, in the present work, we propose an algorithm of that type, which uses the minimum function and a smoothing of it. We show that under certain hypotheses the proposed algorithm converges locally and q-quadratically. In addition, we present numerical tests that show a good performance of the algorithm. [ABSTRACT FROM AUTHOR]

The paper is devoted to the question of existence of continuous implicit functions for non-smooth equations. The equation f (x , σ) = 0 is studied with the unknown x and the parameter σ . We assume that the mapping which defines the equation is locally Lipschitzian in the unknown variable, the parameter belongs to a topological space, and both the unknown variable and the value of the mapping belong to finite-dimensional spaces. Sufficient conditions for the existence of a continuous implicit function in a neighbourhood of a given value of the parameter and conditions for the existence of a continuous implicit function on a given subset of the parameter space are obtained in terms of the Clarke generalized Jacobian. The key tool of this development is the smoothing of the initial equation and application of recent results on solvability of smooth nonlinear equations and a priori solution estimates. Applications to control and optimization are discussed. [ABSTRACT FROM AUTHOR]

The objective of this short note is to provide an estimate of the generalized Jacobian of the inverse of a Lipschitzian mapping when Clarke's inverse function theorem applies. Contrary to the classical C 1 case, inverting matrices of the generalized Jacobian is not enough. Simple counterexamples show that our results are sharp. [ABSTRACT FROM AUTHOR]

Featured Application: Inspection of structures, e.g., offshore/nuclear/eolic plants, bridges, and tall buildings. Placement and retrieval of sensors. Assembly of structures in places not accessible/safe for humans. Aerial manipulators are used in industrial and service robotics tasks such as assembly, inspection, and maintenance. One of the main challenges in aerial manipulation is related to the motion of the UAV base caused by manipulator disturbance torques and forces, which jeopardize the precision of the robot manipulator. In this paper, we propose two novel inverse kinematic control methods used to track a trajectory with an aerial manipulator while also considering resultant UAV base motions. The first method is adapted from the generalized Jacobian formulation used in space robotics and includes the change in system momentum resulting from gravity and UAV control forces in the inverse kinematic control equation. This approach is simulated for a 2 and 3 degree-of-freedom aerial manipulator tracking trajectories with the end-effector. Although the end-effector position error is approximately zero throughout the simulated task, we see significant undesired UAV base motions of several centimeters in magnitude. To ameliorate this by exploiting the kinematic redundancy, we modify the generalized Jacobian by adding an additional task constraint which minimizes the reaction torques from the manipulator, to form the extended generalized Jacobian. While the second approach results in improved precision and reduced base motion by an order of magnitude as compared to the generalized Jacobian, a drawback is the reduction in the available workspace as the solution seeks to minimize the manipulator center of gravity translation. We also demonstrate and compare both approaches in a load picking task. All the algorithms are completed computationally faster than real time in the MATLAB simulations, illustrating their potential for application in real-world experiments. [ABSTRACT FROM AUTHOR]

This paper focuses on efficient projection onto the intersection of a half-space and a box-like set and its generalized Jacobian. Based on the Lagrangian duality theory, we deal with the projection problem via a semismooth Newton algorithm with line search safeguard, which admits global and locally quadratic convergence, to solve a univariate semismooth equation. Numerical experiments show that our proposed algorithm outperforms favourably the existing state-of-the-art standard solvers and is able to reliably solve very large-scale projection problems. Besides, we derive an explicit expression of a generalized Jacobian of the studied projection, which is an essential component of second-order nonsmooth methods. [ABSTRACT FROM AUTHOR]

The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a 'complete' answer, obtained independently of model theoretic results on differentially closed fields. Instead, the geometry of curves and generalized Jacobians provides the key ingredient. Classification and formal solutions of autonomous equations are treated. The results are applied to answer a question on D^n-finiteness of solutions of first order differential equations. [ABSTRACT FROM AUTHOR]

Let K be a closed convex cone in R n and let F : R n → R n be a locally Lipschitz map, not necessarily differentiable. We revisit the problem of analyzing whether the flow generated by an autonomous dynamical system φ ˙ (t) = F (φ (t)) is order-preserving with respect to K. This issue has to do with the notion of cross-nonnegativity of a nonlinear map relative to a cone. We study this property in-depth and derive various characterizations of it. We deviate from the classical literature in at least two ways. First of all, K is allowed to be unpointed and nonsolid. These nontrivial relaxations are needed to cover some interesting examples arising in applications. On the other hand, taking into account the possible lack of differentiability of F , we bring the machinery of Clarke's nonsmooth analysis into the picture. [ABSTRACT FROM AUTHOR]

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]

Aerial manipulators are used in industrial and service robotics tasks such as assembly, inspection, and maintenance. One of the main challenges in aerial manipulation is related to the motion of the UAV base caused by manipulator disturbance torques and forces, which jeopardize the precision of the robot manipulator. In this paper, we propose two novel inverse kinematic control methods used to track a trajectory with an aerial manipulator while also considering resultant UAV base motions. The first method is adapted from the generalized Jacobian formulation used in space robotics and includes the change in system momentum resulting from gravity and UAV control forces in the inverse kinematic control equation. This approach is simulated for a 2 and 3 degree-of-freedom aerial manipulator tracking trajectories with the end-effector. Although the end-effector position error is approximately zero throughout the simulated task, we see significant undesired UAV base motions of several centimeters in magnitude. To ameliorate this by exploiting the kinematic redundancy, we modify the generalized Jacobian by adding an additional task constraint which minimizes the reaction torques from the manipulator, to form the extended generalized Jacobian. While the second approach results in improved precision and reduced base motion by an order of magnitude as compared to the generalized Jacobian, a drawback is the reduction in the available workspace as the solution seeks to minimize the manipulator center of gravity translation. We also demonstrate and compare both approaches in a load picking task. All the algorithms are completed computationally faster than real time in the MATLAB simulations, illustrating their potential for application in real-world experiments.

The singly linearly and box constrained least squares regression has diverse applications in various fields. This paper builds upon previous work to develop an efficient and robust semismooth Newton based augmented Lagrangian (Ssnal) algorithm for solving this problem, in which a semismooth Newton (Ssn) algorithm with superlinear or even quadratic convergence is applied to solve the subproblems. Theoretically, the global and asymptotically superlinear local convergence of the Ssnal algorithm hold automatically under standard conditions. Computationally, a generalized Jacobian for the projector onto the feasible set is shown to be either diagonal or diagonal-minus-rank-1, which is a key ingredient for the efficiency of the Ssnal algorithm. Numerical experiments conducted on both synthetic and real data sets demonstrate that the Ssnal algorithm compared to several state-of-the-art first-order algorithms is much more efficient and robust. [ABSTRACT FROM AUTHOR]

This paper describes a control method for an aerial manipulator on an unmanned aerial vehicle (UAV) by using a generalized Jacobian (GJ). Our task is to realize visual check of bridge inspection by employing a UAV with a multi-degree-of-freedom (DoF) manipulator on its top. The manipulator is controlled by using the GJ. Subsequently, by comparing the aerial manipulator control with a conventional Jacobian experimentally, we discovered that the accuracy of the control improved by applying the GJ. The manipulator has three DoFs in the X-Z plane of the UAV coordinate system. The experiment shows that the manipulator controlled with the GJ can compensate for the pose error of the body by 54.5% and 47.7% in the X- and Z-axes, respectively. [ABSTRACT FROM AUTHOR]

Support vector machine (SVM) has proved to be a successful approach for machine learning. Two typical SVM models are the L1-loss model for support vector classification (SVC) and ε-L1-loss model for support vector regression (SVR). Due to the non-smoothness of the L1-loss function in the two models, most of the traditional approaches focus on solving the dual problem. In this paper, we propose an augmented Lagrangian method for the L1-loss model, which is designed to solve the primal problem. By tackling the non-smooth term in the model with Moreau–Yosida regularization and the proximal operator, the subproblem in augmented Lagrangian method reduces to a non-smooth linear system, which can be solved via the quadratically convergent semismooth Newton's method. Moreover, the high computational cost in semismooth Newton's method can be significantly reduced by exploring the sparse structure in the generalized Jacobian. Numerical results on various datasets in LIBLINEAR show that the proposed method is competitive with the most popular solvers in both speed and accuracy. [ABSTRACT FROM AUTHOR]

Spectral operators of matrices proposed recently in [C. Ding, D. F. Sun, J. Sun, and K. C. Toh, Math. Program., 168 (2018), pp. 509–531] are a class of matrix-valued functions, which map matrices to matrices by applying a vector-to-vector function to all eigenvalues/singular values of the underlying matrices. Spectral operators play a crucial role in the study of various applications involving matrices such as matrix optimization problems that include semidefinite programming as one of most important example classes. In this paper, we will study more fundamental first- and second-order properties of spectral operators, including the Lipschitz continuity, ρ-order B(ouligand)-differentiability (0 < ρ ≤ 1), ρ-order G-semismoothness (0 < ρ ≤ 1), and characterization of generalized Jacobians. [ABSTRACT FROM AUTHOR]

We derive an explicit formula, as well as an efficient procedure, for constructing a generalized Jacobian for the projector of a given square matrix onto the Birkhoff polytope, i.e., the set of doubly stochastic matrices. To guarantee the high efficiency of our procedure, a semismooth Newton method for solving the dual of the projection problem is proposed and efficiently implemented. Extensive numerical experiments are presented to demonstrate the merits and effectiveness of our method by comparing its performance against other powerful solvers such as the commercial software Gurobi and the academic code PPROJ (Hager and Zhang in SIAM J Optim 26:1773–1798, 2016). In particular, our algorithm is able to solve the projection problem with over one billion variables and nonnegative constraints to a very high accuracy in less than 15 min on a modest desktop computer. More importantly, based on our efficient computation of the projections and their generalized Jacobians, we can design a highly efficient augmented Lagrangian method (ALM) for solving a class of convex quadratic programming (QP) problems constrained by the Birkhoff polytope. The resulted ALM is demonstrated to be much more efficient than Gurobi in solving a collection of QP problems arising from the relaxation of quadratic assignment problems. [ABSTRACT FROM AUTHOR]

The sparse group Lasso is a widely used statistical model which encourages the sparsity both on a group and within the group level. In this paper, we develop an efficient augmented Lagrangian method for large-scale non-overlapping sparse group Lasso problems with each subproblem being solved by a superlinearly convergent inexact semismooth Newton method. Theoretically, we prove that, if the penalty parameter is chosen sufficiently large, the augmented Lagrangian method converges globally at an arbitrarily fast linear rate for the primal iterative sequence, the dual infeasibility, and the duality gap of the primal and dual objective functions. Computationally, we derive explicitly the generalized Jacobian of the proximal mapping associated with the sparse group Lasso regularizer and exploit fully the underlying second order sparsity through the semismooth Newton method. The efficiency and robustness of our proposed algorithm are demonstrated by numerical experiments on both the synthetic and real data sets. [ABSTRACT FROM AUTHOR]

Support vector machine is an important and fundamental technique in machine learning. In this paper, we apply a semismooth Newton method to solve two typical SVM models: the L2-loss SVC model and the ϵ -L2-loss SVR model. The semismooth Newton method is widely used in optimization community. A common belief on the semismooth Newton method is its fast convergence rate as well as high computational complexity. Our contribution in this paper is that by exploring the sparse structure of the models, we significantly reduce the computational complexity, meanwhile keeping the quadratic convergence rate. Extensive numerical experiments demonstrate the outstanding performance of the semismooth Newton method, especially for problems with huge size of sample data (for news20.binary problem with 19,996 features and 1,355,191 samples, it only takes 3 s). In particular, for the ϵ -L2-loss SVR model, the semismooth Newton method significantly outperforms the leading solvers including DCD and TRON. [ABSTRACT FROM AUTHOR]

We consider the generalized Jacobian J ~ of the modular curve X0(N) of level N with respect to a reduced divisor consisting of all cusps. Supposing N is square free, we explicitly determine the structure of the ℚ-rational torsion points on J ~ up to 6-primary torsion. The result depicts a fuller picture than [18] where the case of prime power level was studied. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first author. We also discuss the Hecke action on J ~ and its Eisenstein property. [ABSTRACT FROM AUTHOR]

A dual-arm space robotic system has large potential in on-orbit servicing missions, such as satellite repairing, large space structure construction and space debris removal. However, there exist multiple dynamic couplings between the base, the manipulators and the target, bringing large challenge to the motion planning and control. In this paper, a virtual-base modeling method (VBM) and coordinated control strategy are proposed for a dual-arm space robot. One of the arms' end-effector is considered as the virtual base, and the other bodies, including each link of the arms and the center body of the spacecraft, form a combined manipulator. Correspondingly, we derive the kinematic and dynamic equations of the combined manipulator with the virtual base, taking it as a hyper-redundant space manipulator. Furthermore, the coordinated trajectory planning and control methods are presented for capturing and manipulating a space target. For the pre-contact stage, the virtual base and end-effector are controlled simultaneously to track the desired trajectory in the inertial space. During contact, the operational force is regulated using admittance control law, by simplifying the operational forces of the two arms as the equivalent operational force of the combined manipulator. After the target is captured and fixed to the two arms, its pose is adjusted by simultaneously controlling the trajectory and the operational force of the virtual base and the combined manipulator. Finally, a co-simulation system is created based on MATLAB/Simulink and MSC.ADAMS. Simulation results verify the effectiveness and robustness of the proposed methods. [ABSTRACT FROM AUTHOR]

The paper provides a step-by-step tutorial on the Generalized Jacobian Matrix (GJM) approach for modeling and simulation of spacecraft-manipulator systems. The General Jacobian Matrix approach describes the motion of the end-effector of an underactuated manipulator system solely by the manipulator joint rotations, with the attitude and position of the base-spacecraft resulting from the manipulator motion. The coupling of the manipulator motion with the base-spacecraft are thus expressed in a generalized inertia matrix and a GJM. The focus of the paper lies on the complete analytic derivation of the generalized equations of motion of a free-floating spacecraft-manipulator system. This includes symbolic analytic expressions for all inertia property matrices of the system, including their time derivatives and joint-angle derivatives, as well as an expression for the generalized Jacobian of a generic point on any link of the spacecraft-manipulator system. The kinematics structure of the spacecraft-manipulator system is described both in terms of direction-cosine matrices and unit quaternions. An additional important contribution of this paper is to propose a new and more detailed definition for the modes of maneuvering of a spacecraft-manipulator. In particular, the two commonly used categories free-flying and free-floating are expanded by the introduction of five categories, namely floating, rotation-floating, rotation-flying, translation-flying, and flying. A fully-symbolic and a partially-symbolic option for the implementation of a numerical simulation model based on the proposed analytic approach are introduced and exemplary simulation results for a planar four-link spacecraft-manipulator system and a spatial six-link spacecraft manipulator system are presented.

In applying the level-set method developed in [E. Van den Berg and M. P. Friedlander, SIAM J. Sci. Comput., 31 (2008), pp. 890{912] and [E. Van den Berg and M. P. Friedlander, SIAM J. Optim., 21 (2011), pp. 1201{1229] to solve the fused lasso problems, one needs to solve a sequence of regularized least squares subproblems. In order to make the level-set method practical, we develop a highly efficient inexact semismooth Newton based augmented Lagrangian method for solving these subproblems. The efficiency of our approach is based on several ingredients that constitute the main contributions of this paper. First, an explicit formula for constructing the generalized Jacobian of the proximal mapping of the fused lasso regularizer is derived. Second, the special structure of the generalized Jacobian is carefully extracted and analyzed for the efficient implementation of the semismooth Newton method. Finally, numerical results, including the comparison between our approach and several state-of-the-art solvers, on real data sets are presented to demonstrate the high efficiency and robustness of our proposed algorithm in solving challenging large-scale fused lasso problems. [ABSTRACT FROM AUTHOR]

In this paper, a simplification of the necessary and sufficient Karush-Kuhn-Tucker (KKT) optimality conditions for the Lagrangian support vector regression in 2-norm resulting a naïve root finding problem of a system of equations in m variables is proposed, where m is the number of training examples. However, since the resulting system of equations contains terms having the nonsmooth “plus” function, two approaches are considered: (i) the problem is reformulated into an equivalent absolute value equation problem and solved by functional iterative and generalized Newton methods and (ii) solve the original root finding problem by generalized Newton method. The proofs of convergence and pseudo-codes of the iterative methods are given. Numerical experiments performed on a number of synthetic and real-world benchmark datasets showing similar generalization performance with much faster learning speed in comparison with support vector regression (SVR) and as good as performance in comparison with the least squares SVR, unconstrained Lagrangian SVR proposed in Balasundaram et al. (Neural Netw 51:67-79, 2014), twin SVR and extreme learning machine clearly indicate the effectiveness and suitability of the proposed problem formulation solved by functional iterative and generalized Newton methods. [ABSTRACT FROM AUTHOR]

In this work, we present a use of nodal curves to detect the compositeness of a given odd integer. The method relies on the extension of Mumford representation and Cantor's algorithm for special singular curves. [ABSTRACT FROM AUTHOR]

New sufficient conditions for injectivity in a finite-dimensional space were developed. They take the form of limits on the derivatives up to and including the second order, and ensure the existence of a homeomorphic continuity of the examined function across the whole plane. We have also analyzed the locally Lipschitz continuity. Conditions for injectivity in this case take the form of limits on the generalized Jacobian. The range of definition of the analyzed functions is typical for the issue of price equation with factors of production. [ABSTRACT FROM AUTHOR]

In this paper, based on a merit function of the split feasibility problem (SFP), we present a Newton projection method for solving it and analyze the convergence properties of the method. The merit function is differentiable and convex. But its gradient is a linear composite function of the projection operator, so it is nonsmooth in general. We prove that the sequence of iterates converges globally to a solution of the SFP as long as the regularization parameter matrix in the algorithm is chosen properly. Especially, under some local assumptions which are necessary for the case where the projection operator is nonsmooth, we prove that the sequence of iterates generated by the algorithm superlinearly converges to a regular solution of the SFP. Finally, some numerical results are presented. [ABSTRACT FROM AUTHOR]

In this work, we establish some relations between several notions of vector critical points and efficient, weak efficient and ideal efficient solutions of a vector optimization problem with a locally Lipschitz objective function. These relations are stated under pseudoinvexity hypotheses and via the generalized Jacobian. We provide a characterization of pseudoinvexity (resp. strong pseudoinvexity) through the property that every vector critical point is a weak efficient (resp. efficient) solution. We also obtain some properties of invex functions in connection with linear scalarizations. Several examples illustrating our results are also provided. [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]