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.     

Nonsmooth variational-like inequalities and nonsmooth vector optimization

In this paper, we consider nonsmooth vector variational-like inequalities and nonsmooth vector optimization problems. By using the scalarization method, we define nonsmooth variational-like inequalities by means of Clarke generalized directional derivative and study their relations with the vector optimizations and the scalarized optimization problems. Some existence results for solutions of our nonsmooth variational-like inequalities are presented under densely pseudomonotonicity or pseudomonotonicity assumption.     

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.     

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.     

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 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 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.     

非光滑多目标规划的不动点算法(英)

本文讨论不动点算法在非光滑多目标规划中的应用，得到了一些新的最优性条件以及不动点与非光滑多目标的解之间的关系，并且给出了解非光滑多目标规划的不动点算法的收敛性．     

Fibrations by nonsmooth genus three curves in characteristic three

Bertini’s theorem on variable singular points may fail in positive characteristic, as was discovered by Zariski in 1944. In fact, he found fibrations by nonsmooth curves. In this work we continue to classify this phenomenon in characteristic three by constructing a two-dimensional algebraic fibration by nonsmooth plane projective quartic curves, that is universal in the sense that the data about some fibrations by nonsmooth plane projective quartics are condensed in it. Our approach has been motivated by the close relation between it and the theory of regular but nonsmooth curves, or equivalently, nonconservative function fields in one variable. Actually, it also provides an understanding of the interesting effect of the relative Frobenius morphism in fibrations by nonsmooth curves. In analogy to the Kodaira-Néron classification of special fibers of minimal fibrations by elliptic curves, we also construct the minimal proper regular model of some fibrations by nonsmooth projective plane quartic curves, determine the structure of the bad fibers, and study the global geometry of the total spaces.     

Perron's method for second order hyperbolic PDE. Part III

We continue Part I of this paper. Here, in Part III, comparison principles are proved for nonsmooth sub and super solutions (with nonsmooth Cauchy data) of semilinear hyperbolic PDE in compact regular domains of R⁺×R⁺ when n?3, and existence of a nonsmooth solution to the nonsmooth Cauchy problem is proved by a Perron-like method.     

Abstract convex approximations of nonsmooth functions

In the article we use abstract convexity theory in order to unify and generalize many different concepts of nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract convex (concave) approximations of a nonsmooth function mapping a topological vector space to an order complete topological vector lattice. We study basic properties of these notions, construct elaborate calculus of abstract codifferentiable functions and discuss continuity of abstract codifferential. We demonstrate that many classical concepts of nonsmooth analysis, such as subdifferentiability and quasidifferentiability, are particular cases of the concepts of abstract codifferentiability and abstract quasidifferentiability. We also show that abstract convex and abstract concave approximations are a very convenient tool for the study of nonsmooth extremum problems. We use these approximations in order to obtain various necessary optimality conditions for nonsmooth nonconvex optimization problems with the abstract codifferentiable or abstract quasidifferentiable objective function and constraints. Then, we demonstrate how these conditions can be transformed into simpler and more constructive conditions in some particular cases.     

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.     

非光滑Lipschitz规划的Mond－Weir对偶

本文建立了非光滑Ｌｉｐｓｃｈｉｔｚ规划的两种Ｍｏｎｄ－Ｗｅｉｒ对偶形式，在引入一定的非光滑广义凸性下证明了相应的对偶定理．     

Eigenvalue Problems for Hemivariational Inequalities

We consider a semilinear eigenvalue problem with a nonsmooth potential (hemivariational inequality). Using a nonsmooth analog of the local Ambrosetti–Rabinowitz condition (AR-condition), we show that the problem has a nontrivial smooth solution. In the scalar case, we show that we can relax the local AR-condition. Finally, for the resonant λ?=?λ ₁ problem, using the nonsmooth version of the local linking theorem, we show that the problem has at least two nontrivial solutions. Our approach is variational, using minimax methods from the nonsmooth critical point theory.     

一类椭圆型半变分不等式解的存在性(英文)

本文研究了一类Dirichlet边界的椭圆型半变分不等式问题.利用非光滑形式的环绕定理和非光滑形式的对称山路定理,得到了在相应假设条件下此不等式问题至少有一个非平凡解和无穷多解.本文中非光滑势能在原点处关于算子+V(x)的第一正特征值λ是不完全共振的.     

Quasidifferentiability in nonsmooth,nonconvex mechanics

Nonconvex and nonsmooth optimization problems arise in advanced engineering analysis and structural analysis applications. In fact the set of inequality and complementarity relations that describe the structural analysis problem are generated as optimality conditions by the quasidifferential potential energy optimization problem. Thus new kind of variational expressions arise for these problems, which generalize the classical variational equations of smooth mechanics, the variational inequalities of convex, nonsmooth mechanics and give a solid, computationally efficient explication of hemivariational inequalities of nonconvex, nonsmooth mechanics. Moreover quasidifferential calculus and optimization software make this approach applicable for a large number of problems. The connection of quasidifferential optimization and nonsmooth, nonconvex mechanics is discussed in this paper. A number of representative examples from elastostatic analysis applications are treated in details. Numerical examples illustrate the theory.     

Subgradient ellipsoid method for nonsmooth convex problems

Mathematical Programming - In this paper, we present a new ellipsoid-type algorithm for solving nonsmooth problems with convex structure. Examples of such problems include nonsmooth convex...     

Smoothing Newton method for nonsmooth second-order cone complementarity problems with application to electric power markets

This paper is dedicated to solving a nonsmooth second-order cone complementarity problem, in which the mapping is assumed to be locally Lipschitz continuous, but not necessarily to be continuously differentiable everywhere. With the help of the vector-valued Fischer-Burmeister function associated with second-order cones, the nonsmooth second-order cone complementarity problem can be equivalently transformed into a system of nonsmooth equations. To deal with this reformulated nonsmooth system, we present an approximation function by smoothing the inner mapping and the outer Fischer-Burmeister function simultaneously. Different from traditional smoothing methods, the smoothing parameter introduced is treated as an independent variable. We give some conditions under which the Jacobian of the smoothing approximation function is guaranteed to be nonsingular. Based on these results, we propose a smoothing Newton method for solving the nonsmooth second-order cone complementarity problem and show that the proposed method achieves globally superlinear or quadratic convergence under suitable assumptions. Finally, we apply the smoothing Newton method to a network Nash-Cournot game in oligopolistic electric power markets and report some numerical results to demonstrate its effectiveness.