“Two Nontrivial Critical Points for Nonsmooth Functionals via Local Linking and Applications”

In this paper, we extend to nonsmooth locally Lipschitz functionals the multiplicity result of Brezis–Nirenberg (Communication Pure Applied Mathematics and 44 (1991)) based on a local linking condition. Our approach is based on the nonsmooth critical point theory for locally Lipschitz functions which uses the Clarke subdifferential. We present two applications. This first concerns periodic systems driven by the ordinary vector p-Laplacian. The second concerns elliptic equations at resonance driven by the partial p-Laplacian with Dirichlet boundary condition. In both cases the potential function is nonsmooth, locally Lipschitz.     

Generalized Newton’s method based on graphical derivatives

This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness and Lipschitzian assumptions, which is illustrated by examples. The algorithm and main results obtained in the paper are compared with well-recognized semismooth and B-differentiable versions of Newton’s method for nonsmooth Lipschitzian equations.     

Existence and concentration of solutions for a class of biharmonic Kirchhoff equations with discontinuous nonlinearity

This paper concerns the existence and concentration of positive solutions for a class of biharmonic Kirchhoff equations with discontinuous nonlinearity. The proof relies on variational method, truncated methods, and nonsmooth critical points theory. Some related results are improved.     

Nonmonotone algorithm for minimax optimization problems

Many real life problems can be stated as a minimax optimization problem, such as the problems in economics, finance, management, engineering and other fields. In this paper, we present an algorithm with nonmonotone strategy and second-order correction technique for minimax optimization problems. Using this scheme, the new algorithm can overcome the difficulties of the Maratos effect occurred in the nonsmooth optimization, and the global and superlinear convergence of the algorithm can be achieved accordingly. Numerical experiments indicate some advantages of this scheme.     

Viscosity Coderivatives and Their Limiting Behavior in Smooth Banach Spaces

This paper concerns with generalized differentiation of set-valued and nonsmooth mappings between Banach spaces. We study the so-called viscosity coderivatives of multifunctions and their limiting behavior under certain geometric assumptions on spaces in question related to the existence of smooth bump functions of any kind. The main results include various calculus rules for viscosity coderivatives and their topological limits. They are important in applications to variational analysis and optimization.     

Epi-convergence Properties of Smoothing by Infimal Convolution

This paper concerns smoothing by infimal convolution for two large classes of functions: convex, proper and lower semicontinous as well as for (the nonconvex class of) convex-composite functions. The smooth approximations are constructed so that they epi-converge (to the underlying nonsmooth function) and fulfill a desirable property with respect to graph convergence of the gradient mappings to the subdifferential of the original function under reasonable assumptions. The close connection between epi-convergence of the smoothing functions and coercivity properties of the smoothing kernel is established.     

非光滑半定规划的一阶最优性条件

首次考虑了非光滑半定规化问题．运用与非线性规划类似的技巧,把现存的理论扩展到约束是结构稀疏矩阵的情况,给出了其一阶最优性条件。考虑了严格互补条件不成立的情形．在约束矩阵为对角阵条件下,所用的正则条件与传统非线性优化意义下的是一致的．     

Necessary Conditions in Nonsmooth Minimization via Lower and Upper Subgradients

The paper concerns first-order necessary optimality conditions for problems of minimizing nonsmooth functions under various constraints in infinite-dimensional spaces. Based on advanced tools of variational analysis and generalized differential calculus, we derive general results of two independent types called lower subdifferential and upper subdifferential optimality conditions. The former ones involve basic/limiting subgradients of cost functions, while the latter conditions are expressed via Fréchet/regular upper subgradients in fairly general settings. All the upper subdifferential and major lower subdifferential optimality conditions obtained in the paper are new even in finite dimensions. We give applications of general optimality conditions to mathematical programs with equilibrium constraints deriving new results for this important class of intrinsically nonsmooth optimization problems.     

The Compatibility with Order of Some Subdifferentials

It is well known that elementary subdifferentials which are the simplest and the most precise among known subdifferentials do not enjoy good calculus rules, whereas more elaborated subdifferentials do have calculus rules but are not as precise and, in particular, do not preserve order. This paper explores an order preservation property for the subdifferentials of the second category. This property concerns the case in which a distance function is involved. It emphasizes the crucial role played by such functions in nonsmooth analysis. The result enables one to get in a simple, unified way the passage from the properties of subdifferentials for Lipschitzian functions to the same properties for the case of lower semicontinuous functions.     

Discrete Gradient Method: Derivative-Free Method for Nonsmooth Optimization

A new derivative-free method is developed for solving unconstrained nonsmooth optimization problems. This method is based on the notion of a discrete gradient. It is demonstrated that the discrete gradients can be used to approximate subgradients of a broad class of nonsmooth functions. It is also shown that the discrete gradients can be applied to find descent directions of nonsmooth functions. The preliminary results of numerical experiments with unconstrained nonsmooth optimization problems as well as the comparison of the proposed method with the nonsmooth optimization solver DNLP from CONOPT-GAMS and the derivative-free optimization solver CONDOR are presented.     

Existence and multiplicity of solutions for second order periodic systems with the p-Laplacian and a nonsmooth potential

In this paper we study nonlinear periodic systems driven by the ordinary p-Laplacian with a nonsmooth potential. We prove an existence theorem using a nonsmooth variant of the reduction method. We also prove two multiplicity results. The first is for scalar problems and uses the nonsmooth second deformation lemma. The second is for systems and it is based on the nonsmooth local linking theorem.     

Existence and multiplicity of solutions for second order periodic systems with the <Emphasis Type="Italic">p</Emphasis>-Laplacian and a nonsmooth potential

In this paper we study nonlinear periodic systems driven by the ordinary p-Laplacian with a nonsmooth potential. We prove an existence theorem using a nonsmooth variant of the reduction method. We also prove two multiplicity results. The first is for scalar problems and uses the nonsmooth second deformation lemma. The second is for systems and it is based on the nonsmooth local linking theorem.     

A verified inexact implicit Runge–Kutta method for nonsmooth ODEs

Structural pounding and oscillations have been extensively investigated by using ordinary differential equations (ODEs). In many applications, force functions are defined by piecewise continuously differentiable functions and the ODEs are nonsmooth. Implicit Runge–Kutta (IRK) methods for solving the nonsmooth ODEs are numerically stable, but involve systems of nonsmooth equations that cannot be solved exactly in practice. In this paper, we propose a verified inexact IRK method for nonsmooth ODEs which gives a global error bound for the inexact solution. We use the slanting Newton method to solve the systems of nonsmooth equations, and interval method to compute the set of matrices of slopes for the enclosure of solution of the systems. Numerical experiments show that the algorithm is efficient for verification of solution of systems of nonsmooth equations in the inexact IRK method. We report numerical results of nonsmooth ODEs arising from simulation of the collapse of the Tacoma Narrows suspension bridge, steel to steel impact experiment, and pounding between two adjacent structures in 27 ground motion records for 12 different earthquakes. This work is partly supported by a Grant-in-Aid from Japan Society for the Promotion of Science and a scholarship from Egyptian Government.     

非光滑非线性互补问题的牛顿法

研究了非光滑的非线性互补问题. 首先将非光滑的非线性互补问题转化为一个非光滑方程组,然后用牛顿法求解这个非光滑方程组. 在该牛顿法中,每次迭代只需一个原始函数B-微分中的一个元素. 最后证明了该牛顿法的超线性收敛性.     

Weak Sharp Solutions for Nonsmooth Variational Inequalities

In this paper, we study the weak sharp solutions for nonsmooth variational inequalities and give a characterization in terms of error bound. Some characterizations of solution set of nonsmooth variational inequalities are presented. Under certain conditions, we prove that the sequence generated by an algorithm for finding a solution of nonsmooth variational inequalities terminates after a finite number of iterates provided that the solutions set of a nonsmooth variational inequality is weakly sharp. We also study the finite termination property of the gradient projection method for solving nonsmooth variational inequalities under weak sharpness of the solution set.     

Coderivative Analysis of Variational Systems

The paper mostly concerns applications of the generalized differentiation theory in variational analysis to Lipschitzian stability and metric regularity of variational systems in infinite-dimensional spaces. The main tools of our analysis involve coderivatives of set-valued mappings that turn out to be proper extensions of the adjoint derivative operator to nonsmooth and set-valued mappings. The involved coderivatives allow us to give complete dual characterizations of certain fundamental properties in variational analysis and optimization related to Lipschitzian stability and metric regularity. Based on these characterizations and extended coderivative calculus, we obtain efficient conditions for Lipschitzian stability of variational systems governed by parametric generalized equations and their specifications.     

Nonconvex energy functions,related eigenvalue hemivariational inequalities on the sphere and applications

The present paper deals with a new type of eigenvalue problems arising in problems involving nonconvex nonsmooth energy functions. They lead to the search of critical points (e.g. local minima) for nonconvex nonsmooth potential functions which in turn give rise to hemivariational inequalities. For this type of variational expressions the eigenvalue problem is studied here concerning the existence and multiplicity of solutions by applying a critical point theory appropriate for nonsmooth nonconvex functionals.     

Multiplicity of nontrivial solutions for elliptic equations with nonsmooth potential and resonance at higher eigenvalues

We consider a semilinear elliptic equation with a nonsmooth, locally Lipschitz potential function (hemivariational inequality). Our hypotheses permit double resonance at infinity and at zero (double-double resonance situation). Our approach is based on the nonsmooth critical point theory for locally Lipschitz functionals and uses an abstract multiplicity result under local linking and an extension of the Castro-Lazer-Thews reduction method to a nonsmooth setting, which we develop here using tools from nonsmooth analysis.     

A weak nonsmooth palais-smale condition and coercivity

In this paper we show that a generally nonsmooth locally Lipschitz function which satisfies the nonsmooth C-condition (nonsmooth Cerami condition) and is bounded from below, is coercive. The Cerami condition is a weak form of the well-known Palais-Smale condition, which suffices to prove minimax principles.     

A smoothing augmented Lagrangian method for solving simple bilevel programs

In this paper, we design a numerical algorithm for solving a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint. We propose to solve a combined problem where the first order condition and the value function are both present in the constraints. Since the value function is in general nonsmooth, the combined problem is in general a nonsmooth and nonconvex optimization problem. We propose a smoothing augmented Lagrangian method for solving a general class of nonsmooth and nonconvex constrained optimization problems. We show that, if the sequence of penalty parameters is bounded, then any accumulation point is a Karush-Kuch-Tucker (KKT) point of the nonsmooth optimization problem. The smoothing augmented Lagrangian method is used to solve the combined problem. Numerical experiments show that the algorithm is efficient for solving the simple bilevel program.