Nonsmooth Vector Research Articles

Synchronization and Non-Smooth Dynamical Systems

In this article we establish an interaction between non-smooth systems, geometric singular perturbation theory and synchronization phenomena. We find conditions for a non-smooth vector fields be locally synchronized. Moreover its regularization provide a singular perturbation problem with attracting critical manifold. We also state a result about the synchronization which occurs in the regularization of the fold-fold case. We restrict ourselves to the 3-dimensional systems (l = 3) and consider the case known as a T-singularity.

Proper efficiency conditions and duality results for nonsmooth vector optimization in Banach spaces under [formula omitted]-invexity

In this paper, a new class of nonsmooth nonconvex multiobjective programming problems in Banach spaces is considered. The sufficient optimality conditions for efficiency and proper efficiency are established under nonsmooth (Φ,ρ)-invexity and generalized (Φ,ρ)-invexity notions, after extending these concepts to the functions between Banach spaces. Further, Mond–Weir duality theorems are also established under (Φ,ρ)-invexity for the considered vector optimization problems in Banach spaces.

On a Nonsmooth Vector Optimization Problem with Generalized Cone Invexity

By using Clarke’s generalized gradients we consider a nonsmooth vector optimization problem with cone constraints and introduce some generalized cone‐invex functions called K‐α‐generalized invex, K‐α‐nonsmooth invex, and other related functions. Several sufficient optimality conditions and Mond‐Weir type weak and converse duality results are obtained for this problem under the assumptions of the generalized cone invexity. The results presented in this paper generalize and extend the previously known results in this area.

Higher-order radial derivatives and optimality conditions in nonsmooth vector optimization

We propose notions of higher-order outer and inner radial derivatives of set-valued maps and obtain main calculus rules. Some direct applications of these rules in proving optimality conditions for particular optimization problems are provided. Then we establish higher-order optimality necessary conditions and sufficient ones for a general set-valued vector optimization problem with inequality constraints. A number of examples illustrate both the calculus rules and the optimality conditions. In particular, they explain some advantages of our results over earlier existing ones and why we need higher-order radial derivatives.

Regularization and singular perturbation techniques for non-smooth systems

This paper is concerned with some aspects of the qualitative–geometric theory of non-smooth systems. We present a survey of the state of the art on the connection between the regularization process of non-smooth vector fields and the singular perturbation problems. As a matter of fact, the main results in our setting fill a gap between these areas. Exploiting such results, applications to constrained differential equations are discussed.

Second-order optimality conditions using approximations for nonsmooth vector optimization problems under inclusion constraints

Second-order necessary conditions and sufficient conditions for optimality in nonsmooth vector optimization problems with inclusion constraints are established. We use approximations as generalized derivatives and avoid even continuity assumptions. Convexity conditions are not imposed explicitly. Not all approximations in use are required to be bounded. The results improve or include several recent existing ones. Examples are provided to show that our theorems are easily applied in situations where several known results do not work.

Variational sets: Calculus and applications to nonsmooth vector optimization

We develop elements of calculus of variational sets for set-valued mappings, which were recently introduced in Khanh and Tuan (2008) [1,2] to replace generalized derivatives in establishing optimality conditions in nonsmooth optimization. Most of the usual calculus rules, from chain and sum rules to rules for unions, intersections, products and other operations on mappings, are established. Direct applications in stability and optimality conditions for various vector optimization problems are provided.

On the existence of mean curvature flow with transport term

We prove the global-in-time existence of weak solution for a hypersurface evolution problem where the velocity is the sum of the mean curvature and arbitrarily given non-smooth vector field in a suitable Sobolev space. The approximate solution is obtained by the Allen–Cahn equation with transport term. By establishing the density ratio upper bound on the phase boundary measure it is shown that the limiting surface moves with the desired velocity in the sense of Brakke.

On the lifting and approximation theorem for nonsmooth vector fields

We prove a version of Rothschild-Stein's theorem of lifting and approx- imation and some related results in the context of nonsmooth Hormander's vector fields for which the highest order commutators are only Holder con- tinuous. The theory explicitly covers the case of one vector field having weight two while the others have weight one.

Stochastic flows of SDEs with irregular coefficients and stochastic transport equations

In this article we study (possibly degenerate) stochastic differential equations (SDEs) with irregular (or discontinuous) coefficients, and prove that under certain conditions on the coefficients, there exists a unique almost everywhere stochastic (invertible) flow associated with the SDE in the sense of Lebesgue measure. In the case of constant diffusions and BV drifts, we obtain such a result by studying the related stochastic transport equation. In the case of non-constant diffusions and Sobolev drifts, we use a direct method. In particular, we extend the recent results on ODEs with non-smooth vector fields to SDEs.

A non-smooth version of Minty variational principle

The aim of this article is to study the relationship between generalized Minty vector variational inequalities and non-smooth vector optimization problems. Under pseudoconvexity or pseudomonotonicity, we establish the relationship between an efficient solution of a non-smooth vector optimization problem and a generalized Minty vector variational inequality. This offers a non-smooth version of existing Minty variational principle.

Vector nonsmooth variational-like inequalities and optimization problems

In this paper, we first extend the concepts of Clarke’s generalized derivation and subdifferential for a function whose domain is a subset of a metrizable topological vector space. We also introduce and consider a new class of variational-like inequalities, which are called the vector nonsmooth variational-like inequalities. We also establish the relationship between the nonsmooth variational-like inequalities and vector optimization problems under some suitable conditions. In particular, our results extend and generalize the results of Mishra and Wang [S.K. Mishra, S.Y. Wang, Vector variational-like inequalities and nonsmooth vector optimization problems, Nonlinear Anal. 64 (2006), 1939–1945] and Mishra and Noor [S.K. Mishra, M. Aslam Noor, On vector variational-like inequality problems, J. Math. Anal. Appl. 311 (2005) 69–75].

Necessary optimality conditions for a class of nonsmooth vector optimization

The Kuhn-Tucker type necessary conditions of weak efficiency are given for the problem of minimizing a vector function whose each component is the sum of a differentiable function and a convex function, subject to a set of differentiable nonlinear inequalities on a convex subset C of ℝn, under the conditions similar to the Abadie constraint qualification, or the Kuhn-Tucker constraint qualification, or the Arrow-Hurwicz-Uzawa constraint qualification.

Sliding Vector Fields via Slow--Fast Systems

This paper concerns differential equation systems on $\mathbb R^n$ with discontinuous right--hand sides. We deal with non-smooth vector fields in $\mathbb R^n$ having a codimension-one submanifold $M$ as its discontinuity set. After a regularization of a such system and a global blow-up we are able to bring out some results that bridge the space between discontinuous systems and singularly perturbed smooth systems.

First order optimality conditions in vector optimization involving stable functions

We study a nonsmooth vector optimization problem with an arbitrary feasible set or a feasible set defined by a generalized inequality constraint and an equality constraint. We assume that the involved functions are nondifferentiable. First, we provide some calculus rules for the contingent derivative in which the stability (a local Lipschitz property at a point) of the functions plays a crucial role. Second, another calculus rules are established for steady functions. Third, necessary optimality conditions are stated using tangent cones to the feasible set and the contingent derivative of the objective function. Finally, some necessary and sufficient conditions are presented through Lagrange multiplier rules.

First and Second-Order Approximations as Derivatives of Mappings in Optimality Conditions for Nonsmooth Vector Optimization

First and second-order approximations are used to establish both necessary and sufficient optimality conditions for local weak efficiency and local firm efficiency in nonsmooth set-constrained vector problems. Even continuity and relaxed convexity assumptions are not imposed. Compactness conditions are also relaxed. Examples are provided to show advantages of the presented results over recent existing ones.

Higher-Order Variational Sets and Higher-Order Optimality Conditions for Proper Efficiency in Set-Valued Nonsmooth Vector Optimization

Higher-order variational sets are proposed for set-valued mappings, which are shown to be more convenient than generalized derivatives in approximating mappings at a considered point. Both higher-order necessary and sufficient conditions for local Henig-proper efficiency, local strong Henig-proper efficiency and local λ-proper efficiency in set-valued nonsmooth vector optimization are established using these sets. The technique is simple and the results help to unify first and higher-order conditions. As consequences, recent existing results are derived. Examples are provided to show some advantages of our notions and results.

Generalized nonsmooth invexity over cones in vector optimization

In this paper K-nonsmooth quasi-invex and (strictly or strongly) K-nonsmooth pseudo-invex functions are defined. By utilizing these new concepts, the Fritz–John type and Kuhn–Tucker type necessary optimality conditions and number of sufficient optimality conditions are established for a nonsmooth vector optimization problem wherein Clarke’s generalized gradient is used. Further a Mond Weir type dual is associated and weak and strong duality results are obtained.

Existence of Solutions for Nonconvex and Nonsmooth Vector Optimization Problems

We consider the weakly efficient solution for a class of nonconvex and nonsmooth vector optimization problems in Banach spaces. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings. We prove some existence results concerned with the weakly efficient solution for the nonconvex and nonsmooth vector optimization problems by using the equivalence and Fan-KKM theorem under some suitable conditions.

Existence of weakly efficient solutions in nonsmooth vector optimization

In this paper we study the existence of weakly efficient solutions for some nonsmooth and nonconvex vector optimization problems. We consider problems whose objective functions are defined between infinite and finite-dimensional Banach spaces. Our results are stated under hypotheses of generalized convexity and make use of variational-like inequalities.