Infine Functions, Nonsmooth Alternative Theorems and Vector Optimization Problems

In this paper we introduce a new notion of infine nonsmooth functions and give several characterizations of infineness property. We prove alternative theorems with mixed constraints (i.e., inequality and equality constraints) being described by invex-infine nonsmooth functions. We establish a necessary and sufficient condition for a solution of a vector optimization problem involving mixed constraints to be a properly efficient solution.     

An efficient synthesis of highly functionalized chiral lactams

A new method was developed to synthesize highly functionalized lactams via a one pot reductive amination/lactam formation reaction. This methodology is amenable for parallel synthesis and was used to prepare a large number of lactam analogs in a library format with good ee (de) retention.     

Scalable synthesis of 1-bicyclo[1.1.1]pentylamine via a hydrohydrazination reaction

The reaction of [1.1.1]propellane with di-tert-butyl azodicarboxylate and phenylsilane in the presence of Mn(dpm)(3) to give di-tert-butyl 1-(bicyclo[1.1.1]pentan-1-yl)hydrazine-1,2-dicarboxylate is described. Subsequent deprotection gives 1-bicyclo[1.1.1]pentylhydrazine followed by reduction to give 1-bicyclo[1.1.1]pentylamine. The reported route marks a significant improvement over the previous syntheses of 1-bicyclo[1.1.1]pentylamine in terms of scalability, yield, safety, and cost.     

Upper semicontinuity result for the solution mapping of a mixed parametric generalized vector quasiequilibrium problem with moving cones

In this paper, we give sufficient conditions for the upper semicontinuity property of the solution mapping of a parametric generalized vector quasiequilibrium problem with mixed relations and moving cones. The main result is proven under the assumption that moving cones have local openness/local closedness properties and set-valued maps are cone-semicontinuous in a sense weaker than the usual sense of semicontinuity. The nonemptiness and the compactness of the solution set are also investigated.     

New Scalarizing Approach to the Stability Analysis in Parametric Generalized Ky Fan Inequality Problems

This paper gives sufficient conditions for the upper and lower semicontinuities of the solution mapping of a parametric mixed generalized Ky Fan inequality problem. We use a new scalarizing approach quite different from traditional linear scalarization approaches which, in the framework of the stability analysis of solution mappings of equilibrium problems, were useful only for weak vector equilibrium problems and only under some convexity and strict monotonicity assumptions. The main tools of our approach are provided by two generalized versions of the nonlinear scalarization function of Gerstewitz. Our stability results are new and are obtained by a unified technique. An example is given to show that our results can be applied, while some corresponding earlier results cannot.     

New Results on Henig Proper Generalized Vector Quasiequilibrium Problems

New sufficient conditions are given for the existence of solutions of a Henig proper generalized vector quasiequilibrium problem with moving cones. They are established by a new scalarizing approach, which is based on a suitable nonlinear scalarization function, proposed recently for set-valued maps in Sach and Tuan (J. Optim. Theory Appl. 157:347–364 (2013)). Examples are given to illustrate our main results.     

Invex multifunctions and duality

For optimization problems with multifunction objective and constraints, duality theorems are proved for analogs of the Wolfe and Mond–Weir dual problems, assuming that the multifunctions satisfy a generalization of the invex property for functions. Several characterizations of generalized invexity are obtained.     

Convexity Criteria for Set-Valued Maps

The paper gives necessary and sufficient conditions for a set-valued map F between Banach spaces X and Y to be convex with respect to a convex cone KY These conditions are written in terms of the contingent derivative of the map      

Efficiency in vector quasi-equilibrium problems and applications

In this paper, we give sufficient conditions for the existence of efficient solutions of a generalized vector quasi-equilibrium problem in topological vector spaces. The motivations for introducing this problem come from practical problems in traffic networks and the optimal control theory for discrete-time dynamical systems. The main results of the paper are proven with the help of a strongly monotonic function which can be constructed from the data of the problem under consideration. Some notions of cone-semicontinuity of set-valued maps, weaker than the usual concepts of semicontinuity, are also used in our study. As applications, we obtain existence results in vector quasi-optimization problems, Stampacchia set-valued vector quasi-variational inequality problems and Pareto vector quasi-saddle point problems. All these results are different from the corresponding ones in the literature.