Separation Functions and Optimality Conditions in Vector Optimization

In this paper, we propose weak separation functions in the image space for general constrained vector optimization problems on strong and weak vector minimum points. Gerstewitz function is applied to construct a special class of nonlinear separation functions as well as the corresponding generalized Lagrangian functions. By virtue of such nonlinear separation functions, we derive Lagrangian-type sufficient optimality conditions in a general context. Especially for nonconvex problems, we establish Lagrangian-type necessary optimality conditions under suitable restriction conditions, and we further deduce Karush–Kuhn–Tucker necessary conditions in terms of Clarke subdifferentials.     

Nonlinear Separation Approach to Constrained Extremum Problems

In this paper, by virtue of a nonlinear scalarization function, two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function are first introduced, respectively. Then, by the image space analysis, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, some necessary and sufficient optimality conditions are obtained for constrained extremum problems.     

Nonlinear separation approach to inverse variational inequalities

In this paper, we employ the image space analysis to investigate an inverse variational inequality (for short, IVI) with a cone constraint. By virtue of the nonlinear scalarization function commonly known as the Gerstewitz function, three nonlinear weak separation functions, two nonlinear regular weak separation functions and a nonlinear strong separation function are first introduced. Then, by these nonlinear separation functions, theorems of the weak and strong alternative and some optimality conditions for IVI with a cone constraint are derived without any convexity. In particular, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, two gap functions and an error bound for IVI with a cone constraint are obtained.     

Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming

This paper aims at showing that the class of augmented Lagrangian functions for nonlinear semidefinite programming problems can be derived, as a particular case, from a nonlinear separation scheme in the image space associated with the given problem. By means of the image space analysis, a global saddle point condition for the augmented Lagrangian function is investigated. It is shown that the existence of a saddle point is equivalent to a regular nonlinear separation of two suitable subsets of the image space. Without requiring the strict complementarity, it is proved that, under second order sufficiency conditions, the augmented Lagrangian function admits a local saddle point. The existence of global saddle points is then obtained under additional assumptions that do not require the compactness of the feasible set. Motivated by the result on global saddle points, we propose two modified primal-dual methods based on the augmented Lagrangian using different strategies and prove their convergence to a global solution and the optimal value of the original problem without requiring the boundedness condition of the multiplier sequence.     

Separations and Optimality of Constrained Multiobjective Optimization via Improvement Sets

In this paper, we investigate the separations and optimality conditions for the optimal solution defined by the improvement set of a constrained multiobjective optimization problem. We introduce a vector-valued regular weak separation function and a scalar weak separation function via a nonlinear scalarization function defined in terms of an improvement set. The nonlinear separation between the image of the multiobjective optimization problem and an improvement set in the image space is established by the scalar weak separation function. Saddle point type optimality conditions for the optimal solution of the multiobjective optimization problem are established, respectively, by the nonlinear and linear separation methods. We also obtain the relationships between the optimal solution and approximate efficient solution of the multiobjective optimization problem. Finally, sufficient and necessary conditions for the (regular) linear separation between the approximate image of the multiobjective optimization problem and a convex cone are also presented.     

Separation Approach for Augmented Lagrangians in Constrained Nonconvex Optimization

This paper aims at showing that the class of augmented Lagrangian functions, introduced by Rockafellar and Wets, can be derived, as a particular case, from a nonlinear separation scheme in the image space associated with the given problem; hence, it is part of a more general theory. By means of the image space analysis, local and global saddle-point conditions for the augmented Lagrangian function are investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Under second-order sufficiency conditions in the image space, it is proved that the augmented Lagrangian admits a local saddle point. The existence of a global saddle point is then obtained under additional assumptions that do not require the compactness of the feasible set.     

Image Space Analysis for Constrained Inverse Vector Variational Inequalities via Multiobjective Optimization

In this paper, we employ the image space analysis to study constrained inverse vector variational inequalities. First, sufficient and necessary optimality conditions for constrained inverse vector variational inequalities are established by using multiobjective optimization. A continuous nonlinear function is also introduced based on the oriented distance function and projection operator. This function is proven to be a weak separation function and a regular weak separation function under different parameter sets. Then, two alternative theorems are established, which lead directly to sufficient and necessary optimality conditions of the inverse vector variational inequalities. This provides a partial answer to an open question posed in Chen et al. (J Optim Theory Appl 166:460–479, 2015).     

Reduction of multidimensional first order equations with multi-homogeneous function of derivatives

We present an analysis of solutions to multidimensional first order equation order with several independent variables under assumption that the nonlinear part of the equation is a multi-homogeneous function of derivatives. The reduction of the original equation is performed for the class of solutions depending on linear combinations of prescribed groups of initial variables. We obtain solutions to the reduced equation. We consider also the cases of additional, multiplicational and combined separation of variables.     

Nonlinear separation in the image space with applications to constrained optimization

In this paper, by means of the image space analysis, we obtain optimality conditions for vector optimization of objective multifunction with multivalued constraints based on disjunction of two suitable subsets of the image space. By the oriented distance function a nonlinear regular separation is introduced and some optimality conditions for the constrained extremum problem are obtained. It is shown that the existence of a nonlinear separation is equivalent to a saddle point condition for the generalized Lagrangian function.     

Nonlinear separation functions,optimality conditions and error bounds for Ky Fan quasi-inequalities

In this paper, we employ the image space analysis to investigate a Ky Fan quasi-inequality with cone constraints. By means of the oriented distance function, a new nonlinear weak (regular) separation function is introduced. Some necessary and sufficient optimality conditions, especially, a saddle-point sufficient optimality condition for the Ky Fan quasi-inequality with cone constraints, are obtained. By virtue of the nonlinear regular weak separation function, a gap function for the Ky Fan quasi-inequality with cone constraints is obtained. Moreover, we get an error bound for the solution set of the Ky Fan quasi-inequality with respect to the gap function under strongly monotone assumptions.     

The nonlinear bilevel programming problem:formulations,regularity and optimality conditions

In this paper, we study regularity and optimality conditions for the BLPP by using a marginal function formulation, where the marginal function is defined by the optimal value function of the lower problem. We address the regularity issue by exploring the structure of the tangent cones of the feasible set of the BLPP. These regularity results indicate that the nonlinear/nonlinear BLPP is most likely degenerate and a class of nonlinear/linear BLPP is regular in the conventional sense. Existence of exact penalty function is proved for a class of nonlinear/linear BLPP. Fritz-John type optimality conditions are derived for nonlinear BLPP, while KKT type conditions are obtained for a class of nonlinear/linear BLPP in the framework of nonsmooth analysis. A typical example is examined for these conditions and some applications of these conditions are pointed out     

Evolution equations associated with non-isothermal phase separation: Subdifferential approach

This paper is concerned with a non-isothermal phase separation model, for instance, describing component separation dynamics for a class of binary systems (solid-solid systems). Our model is a couple of nonlinear parabolicPDE's which are possibly degenerate and singular at the same time. It will be shown that our model can be reformulated as a nonlinear evolution equation governed by subdifferential operator in a Hilbert space, and its mathematically basic questions, such as existence and uniqueness, can be handled in the abstract theory of nonlinear evolution equations in Hilbert spaces. Entrata in Redazione il 15 luglio 1997 e, in versione riveduta, il 25 ottobre 1997.     

A fixed-point algorithm for blind source separation with nonlinear autocorrelation

This paper addresses blind source separation (BSS) problem when source signals have the temporal structure with nonlinear autocorrelation. Using the temporal characteristics of sources, we develop an objective function based on the nonlinear autocorrelation of sources. Maximizing the objective function, we propose a fixed-point source separation algorithm. Furthermore, we give some mathematical properties of the algorithm. Computer simulations for sources with square temporal autocorrelation and the real-world applications in the analysis of the magnetoencephalographic recordings (MEG) illustrate the efficiency of the proposed approach. Thus, the presented BSS algorithm, which is based on the nonlinear measure of temporal autocorrelation, provides a novel statistical property to perform BSS.     

Optimality Conditions for Generalized Ky Fan Quasi-Inequalities with Applications

In this paper, the image space analysis is employed to study a generalized Ky Fan quasi-inequality with cone constraints. By virtue of a nonlinear scalarization function and a positive linear operator, a nonlinear (regular) weak separation function and a linear regular weak separation function are introduced. Nonlinear and, in particular, linear separations for the generalized Ky Fan quasi-inequality with cone constraints are characterized. Some necessary and sufficient optimality conditions, especially a saddle-point sufficient optimality condition for the generalized Ky Fan quasi-inequality with cone constraints, are obtained. As applications, some sufficient conditions for (weak) vector equilibrium flows of vector traffic equilibrium problems with capacity arc constraints, are derived.     

Nonlinear separation concerning <Emphasis Type="Italic">E</Emphasis>-optimal solution of constrained multi-objective optimization problems

This paper aims at investigating optimality conditions in terms of E-optimal solution for constrained multi-objective optimization problems in a general scheme, where E is an improvement set with respect to a nontrivial closed convex point cone with apex at the origin. In the case where E is not convex, nonlinear vector regular weak separation functions and scalar weak separation functions are introduced respectively to realize the separation between the two sets in the image space, and Lagrangian-type optimality conditions are established. These results extend and improve the convex ones in the literature.     

A parameter estimation method for model analysis

To solve a class of nonlinear parameter estimation problems, a method combining the regularized structured nonlinear total least norm (RSNTLN) method and parameter separation scheme is suggested. The method guarantees the convergence of parameters and has an advantages in reducing the residual norm over the use of RSNTLN only. Numerical experiments for two models appeared in signal processing show that the suggested method is more effective in obtaining solution and parameter with minimum residual norm.     

具任意次非线性项的Camassa-Holm方程的双孤子新解

给出辅助方程、函数变换与变量分离解相结合的方法,构造了具任意次非线性项的Camassa-Holm方程的双孤子和双周期新解.首先,通过两个辅助方程、函数变换与变量分离解,将具任意次非线性项的Camassa-Holm方程的求解问题转化为非线性代数方程的求解问题.然后,借助符号计算系统Mathematica求出该方程组的解,并用辅助方程的相关结论,构造了双周期解和双孤子新解.     

非光滑向量均衡问题近似拟全局真有效解的最优性条件

本文研究一类非光滑向量均衡问题(Vector Equilibrium Problem)(VEP)关于近似拟全局真有效解的最优性条件.首先,利用凸集的拟相对内部型分离定理和Clarke次微分的性质,得到了问题(VEP)关于近似拟全局真有效解的最优性必要条件.其次,引入近似伪拟凸函数的概念,并给出具体实例验证其存在性,且在该凸性假设下建立了问题(VEP)关于近似拟全局真有效解的充分条件.最后,利用Tammer函数以及构建满足一定性质的非线性泛函,得到了问题(VEP)近似拟全局真有效解的标量化定理.     

一类非线性系统的局部镇定

研究一类非线性系统的局部状态反馈镇定问题．基于中心流形理论,给出一类非线性系统渐近镇定的充分条件,并设计出镇定系统的反馈控制律．文中利用具有齐次导数的Lyapunov函数方法,特别研究了一类平面非线性系统及具有二重零特征值的一类非线性系统的渐近镇定问题,给出了系统镇定的若干充分条件,并构造出控制律．文中的例表明了所得结果的有效性．