Optimality conditions and duality in subdifferentiable multiobjective fractional programming

Fritz John and Kuhn-Tucker necessary and sufficient conditions for a Pareto optimum of a subdifferentiable multiobjective fractional programming problem are derived without recourse to an equivalent convex program or parametric transformation. A dual problem is introduced and, under convexity assumptions, duality theorems are proved. Furthermore, a Lagrange multiplier theorem is established, a vector-valued ratio-type Lagrangian is introduced, and vector-valued saddle-point results are presented.The authors are thankful to the referees and Professor P. L. Yu for their many useful comments and suggestions which have improved the presentation of the paper.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant No. A-5319. The authors are also thankful to the Dean's Office, Faculty of Management, University of Manitoba, for the financial support provided for the third author's visit to the Faculty.     

Sufficient optimality conditions and duality for a quasiconvex programming problem

Under differentiability assumptions, Fritz John Sufficient optimality conditions are proved for a nonlinear programming problem in which the objective function is assumed to be quasiconvex and the constraint functions are assumed to quasiconcave/strictly pseudoconcave. Duality theorems are proved for Mond-Weir type duality under the above generalized convexity assumptions.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant No. A-5319. The authors are thankful to Professor B. Mond for suggestions that improved the original draft of the paper.     

Generalized B-vex functions and generalized B-vex programming

A class of functions called pseudo B-vex and quasi B-vex functions is introduced by relaxing the definitions of B-vex, pseudoconvex, and quasiconvex functions. Similarly, the class of B-invex, pseudo B-invex, and quasi B-invex functions is defined as a generalization of B-vex, pseudo B-vex, and quasi B-vex functions. The sufficient optimality conditions and duality results are obtained for a nonlinear programming problem involving B-vex and B-invex functions.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant A-5319. The second author is grateful to the Faculty of Management, University of Manitoba for the financial support provided for her visit. The authors are thankful to Prof. R. N. Kaul, Department of Mathematics, Delhi University for his constructive criticism of the paper.     

Models with few isomorphic expansions

We shall characterize the countable modelsM, with only countably many expansions by a one place predicate. The second author is thankful for NSF Grant 43901, by which he was partially supported.     

Analysis of cyclic queueing networks with parallelism and vacation

The aim of this paper is to improve the machine interference model with vacation to deal with more recent problems of the communication area. To this scope the model is extended to include parallelism in the vacation station. The underlying Markov process is analyzed and a state arrangement is found that yields an efficient matrix-analytic technique that substantially lowers down the time- and space-complexity of standard methods. A numerical example of the method effectiveness is presented, and an example of resource allocation is introduced that finds applications in the QoS management of wireless networks. The author is thankful to the anonymous referees for the improvements their comments have earned to the quality of the presentation and to the completeness of the paper. The author is thankful to Giuseppe Iazeolla, whose careful reading of the original draft of this paper led to significant improvements in its overall quality. This work was partially supported by funds from the FIRB project “Performance Evaluation of Complex Systems: Techniques Methodologies and Tools” and by the University of Roma TorVergata project on High Performance ICT Platforms.     

Generalized second-order characterizations of convex functions

We introduce a second-order upper Dini-directional derivative and obtain a second-order characterization for a continuously Gâteaux differentiable function to be a convex function.This research was done under the supervision of Dr. V. Jeyakumar whose helpful guidance and valuable suggestions are much appreciated. The author is thankful to the referee for providing a simplication of the original proof of Theorem 2.1.     

On groups of automorphisms of a class of surfaces

In this note we describe the group of automorphisms of a commutative algebra with three generatorsx, y andz satisfying a relationxy= P(z), whereP(z) is a polynomial. The author was visiting Bar-Ilan and Tel-Aviv Universities while working on this project. He is deeply thankful for hospitality rendered.     

On various duality theorems in nonlinear programming

Recently, Gulati and Craven and Mond and Egudo established strict converse duality theorems for some of Mond-Weir duals for nonlinear programming problems. Here, we establish various duality theorems under weaker convexity conditions that are different from those of Gulati and Craven, Mond and Weir, and Mond and Egudo.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant A-5319.     

An existence theorem for a fractional control problem

Computational algorithms in mathematical programming have been much in use in the theory of optimal control (see, for example Refs. 1–2). In the present work, we use the algorithm devised by Dinkelback (Ref. 3) for a nonlinear fractional programming problem to prove an existence theorem for a control problem with the cost functional having a fractional form which subsumes the control problem considered by Lee and Marcus (Ref. 4) as a particular case.The author is thankful to the referee for suggestions.     

Semi-B-Preinvex Functions

In this note, a class of functions, called semi-B-preinvex function, which are a generalization of the semipreinvex functions and the B-vex functions, is introduced. Examples are given to show that there exist functions which are semi-B-preinvex functions, but are neither semipreinvex nor B-vex. A property of the semi-B-preinvex functions is obtained.Communicated by F. GiannessiThis research was partially supported by a Science Committee Project, Research Foundation of Chongqing, Grant 8409. The authors are thankful to the referees and Prof. F. Giannessi for suggestions and comments which helped to give the present form to this paper.     

On the method of complementary functions for nonlinear boundary-value problems

We show that, like the method of adjoints, the method of complementary functions can be effectively used to solve nonlinear boundary-value problems.This work was supported by the Alexander von Humboldt Foundation. The author is thankful to Prof. G. Hämmerlin for providing the facilities and to Miss J. Gumberger for performing numerical tests. The author is also indebted to Dr. S. M. Roberts for his suggestions on the first draft of this paper.     

An estimate of the curvature of level curves in a class of univalent functions

In this note a class of normalized functions which are univalent in the unit disk is considered. An exact solution of the problem of an upper estimate of curvature of level curves for functions of this class is obtained. In particular, an exact solution of this problem for univalent functions with real coefficients at the points of the interval (0, 1) is obtained.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 381–388, March, 1976.The author is thankful to the referee for taking interest in this note.     

Optimality conditions in multiobjective differentiable programming

Necessary conditions not requiring convexity are based on the convergence of a vector at a point and on Motzkin's theorem of the alternative. A constraint qualification is also involved in the establishment of necessary conditions. Three theorems on sufficiency require various levels of convexity on the component of the functions involved, and the equality constraints are not necessarily linear. Scalarization of the objective function is used only in the last sufficiency theorem.The author is thankful to the unknown referce whose comments improved the quality of the paper.     

Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints

We present some necessary and sufficient conditions for null controllability for a class of general linear evolution equations on a Banach space with constraints on the control space. We also present a result on the existence of time-optimal controls and some partial results on the maximum principle. Some interesting insights that can be obtained from these results are discussed, and the paper is concluded with an application to a boundary control problem.This work was supported in part by the National Science and Engineering Council of Canada under Grant No. 7109.The author is thankful to Professor L. Cesari for many helpful suggestions and also for calling his attention to the recent papers of Professor K. Narukawa.     

Necessary conditions for nonlinear suboptimization over the weakly-efficient set

The optimization problem of a nonlinear real function over the weakly-efficient set associated to a nonlinear multi-objective program is examined. Necessary first-order conditions for a suboptimal solution are proposed, assuming the convexity of the multi-objective program. Estimations of the optimal value are established and an algorithm for finding suboptimal solutions is proposed. The optimal value is approximated to any prescribed degree of accuracy using a weakly-efficient suboptimal solution.This work was done while the author was preparing his Ph.D. Thesis at the University of Melbourne, Australia. The author is thankful to Dr. B. D. Craven for his suggestions and helpful discussions and to Professor W. Stadler and the anonymous referees for their helpful comments and corrections.     

Max-min sum minimization transportation problem

A non-convex optimization problem involving minimization of the sum of max and min concave functions over a transportation polytope is studied in this paper. Based upon solving at most (g+1)(< p) cost minimizing transportation problems with m sources and n destinations, a polynomial time algorithm is proposed which minimizes the concave objective function where, p is the number of pairwise disjoint entries in the m× n time matrix {t _ij } sorted decreasingly and T ^g is the minimum value of the max concave function. An exact global minimizer is obtained in a finite number of iterations. A numerical illustration and computational experience on the proposed algorithm is also included. We are thankful to Prof. S. N. Kabadi, University of New Brunswick-Fredericton, Canada, who initiated us to the type of problem discussed in this paper. We are also thankful to Mr. Ankit Khandelwal, Ms. Neha Gupta and Ms. Anuradha Beniwal, who greatly helped us in the implementation of the proposed algorithm.     

Existence of Solutions to Some Equilibrium Problems

The concept of a monotone family of functions, which need not be countable, and the solution of an equilibrium problem associated with the family are introduced. A fixed-point theorem is applied to prove the existence of solutions to the problem.¹The authors are thankful to the referees and Professor S. Schaible for valuable suggestions and comments to make the paper in the present form. The first author thanks the Department of Mathematics, University of Queensland for the kind hospitality shown during his stay ²This research was supported in part by grant No. 83470019 from IPM     

On the imbedding problem for local fields

The imbedding problem of local fields is considered for the case where the whole of the group is a p-group having as many generators as the Galois group of the extension and the extension consists of a primitive root of 1 of degree equal to the period of the kernel. It is proved that it is necessary and sufficient for the solvability of this problem that a concordance condition (and even a weaker condition) be satisfied (see [4]).Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 91–94, July, 1972.The author is thankful to A. V. Yakovlev for his valuable advice.When this article was in press, the author proved that Theorem 1 is valid even without Condition IV. He has also found an example showing that Condition III cannot be discarded.     

Hybrid artificial neural networks for efficient valuation of real options and financial derivatives

A hybrid valuation methodology is proposed and tested for improving the efficiency of contingent claims pricing by combining Artificial Neural Networks (ANN) and conventional parametric option pricing techniques. With one application on financial derivatives and one on real options the methods superiority is demonstrated. The resulting efficiency is instrumental for real time applications.MSC code: 90-08 Acknowledgements: Both authors are thankful for partial financial support to the HERMES European Center of Excellence on Computational Finance and Economics of the University of Cyprus and a University of Cyprus grant for research in ANNs and Derivatives, and to the anonymous referees for their helpful comments and discussions.