weak efficiency for multiobjective variational problems abstract sometimes to locate efficient solutions for multiobjective variational problems mvps is quite costly so in this paper we tackle the study of weakly efficient solutions for mvps a new concept of weak vector critical point which generalizes other ones already existent and a new class of pseudoinvex functions are introduced we will apply a new approach to prove that the new class of pseudoinvex functions is equivalent to the class of functions whose weak vector critical points are weakly efficient solution for mvps

generalized convexities and generalized gradients based on algebraic operations abstract in this paper we investigate properties of generalized convexities based on algebraic operations introduced by ben tal a ben tal on generalized means and generalized convex functions j optim theory appl 21 1977 1 13 and relations between these generalized convexities and generalized monotonicities we also discuss the h generalized directional derivative and gradient and explore the relation between this gradient and the clarke generalized gradient definitions of some generalized averages of the values of a generalized convex function at n equally spaced points based on the algebraic operations are also presented and corresponding results are obtained finally the convexity is defined and some properties of convex functions are derived

optimality conditions for multiobjective programming with generalized i ρ θ convex set functions abstract necessary conditions for pareto optimality in multiobjective programming with subdifferentiable set functions are established in theorem 12 of h c lai and l j lin j math anal appl 132 1988 558 571 in this paper we establish some sufficient conditions under which a feasible solution of such a problem will be pareto optimal provided that a weaker convexity requirement is satisfied for instance i convexity is assumed for both objective and constraint set functions some duality models are also discussed wolfe type and mond weir type duality theorems are proved

optimality and duality for nondifferentiable multiobjective variational problems abstract the concept of efficiency is used to formulate duality for nondifferentiable multiobjective variational problems wolfe and mond weir type vector dual problems are formulated by using the generalized schwarz inequality and a characterization of efficient solution we established the weak strong and converse duality theorems under generalized f convexity assumptions

characterization of vector strict global minimizers of order 2 in differentiable vector optimization problems under a new approximation method abstract in this paper a new approximation method is introduced to characterize a so called vector strict global minimizer of order 2 for a class of nonlinear differentiable multiobjective programming problems with f convex functions of order 2 in this method an equivalent vector optimization problem is constructed by a modification of both the objectives and the constraint functions in the original multiobjective programming problem at the given feasible point in order to prove the equivalence between the original multiobjective programming problem and its associated f approximated vector optimization problem the suitable f convexity of order 2 assumption is imposed on the functions constituting the considered vector optimization problem

on efficiency and duality for multiobjective variational control problems with f ρ convexity abstract the concept of efficiency pareto optimum is used to formulate duality for multiobjective variational control problems wolfe and mond weir type duals are formulated under the generalized f convexity on the functions involved weak and strong duality theorems are proved

mixed type duality for multiobjective variational problems with generalized f ρ convexity a mixed type dual for multiobjective variational problems is formulated several duality theorems are established relating properly efficient solutions of the primal and dual variational problems under generalized f convexity static mixed type dual multiobjective problems are particular cases of these problems

on efficiency and duality for multiobjective variational problems abstract the concept of efficiency pareto optimum is used to formulate duality for multiobjective variational problems wolfe and mond weir type duals are formulated under generalized f convexity assumptions on the functions involved weak and strong duality theorems are proved

duality for nondifferentiable static multiobjective variational problems involving generalized f ϱ convex functions using the generalized schwarz inequality we establish some weak duality theorems for nondifferentiable static multiobjective variational problems involving generalized f convex functions later in the sequel we introduce three dual models for the nondifferentiable static multiobjective fractional variational problems and derive weak duality results for such programs

duality for multiobjective fractional generalized control problems with f ρ f ρ convexity abstract the concept of efficiency is used to formulate duality for multiobjective fractional control problems wolfe and mond weier type duals are formulated using the generalized f convexity on the functions involved we establish the weak and strong duality theorems for multiobjective fractional generalized control problems

mixed l 1 h controllers for siso discrete time systems abstract in this paper we present a methodology for designing mixed l 1 h controllers these controllers allow for minimizing the worst case peak output due to a persistent disturbance while at the same time satisfying an h norm constraint upon some closed loop transfer function of interest thus they are of interest for applications dealing with mixed time frequency domain performance specifications the main result of the paper shows that suboptimal mixed l 1 h controllers can be obtained by solving a sequence of problems each one consisting of a finite dimensional convex optimization and a standard unconstrained h problem

sufficiency and duality in multiobjective programming involving generalized f ρ convexity abstract new classes of generalized f convexity are introduced for vector valued functions examples are given to show their relations with f pseudoconvex f quasiconvex and strictly f pseudoconvex vector valued functions the sufficient optimality conditions and duality results are obtained for multiobjective programming involving generalized f convex vector valued functions

higher order duality in multiobjective fractional programming with support functions abstract in this paper a new class of higher order f type i functions for a multiobjective programming problem is introduced which subsumes several known studied classes higher order mond weir and schaible type dual programs are formulated for a nondifferentiable multiobjective fractional programming problem where the objective functions and the constraints contain support functions of compact convex sets in r n weak and strong duality results are studied in both the cases assuming the involved functions to be higher order f type i a number of previously studied problems appear as special cases

multiobjective control problem with generalized invexity abstract a multiobjective control problem is introduced the definition of invexity for continuous functions is extended to invexity pseudoinvexity and quasi invexity duality results are established for wolfe as well as mond wier type duals

duality for multiobjective variational problems abstract wolfe and mond weir type duals for multiobjective variational problems are formulated under convexity assumptions on the functions involved weak and converse duality theorems are proved to relate properly efficient solutions of the primal and dual problems a close relationship between these variational problems and nonlinear multiobjective programming problems is also indicated

symmetric duality for multiobjective variational problems abstract a mond weir type symmetric dual for a multiobjective variational problem is formulated weak and strong duality theorems under generalized convexity assumptions are proved for properly efficient solutions under an additional condition on the kernel function that occurs in the formulation of the problems a self duality theorem is proved a close relationship between these variational problems and symmetric dual nonlinear multiobjective programming problems is also incorporated

on efficiency and duality for multiobjective programs abstract for a multiobjective nonlinear program which involved inequality and equality constraints wolfe mond weir and general mond weir type duals are formulated and the concept of efficiency pareto optimum is used to state some duality results under generalized f convexity assumptions

efficiency and generalized convex duality for multiobjective programs abstract the concept of efficiency pareto optimum is used to formulate duality for multiobjective non linear programs the results are obtained for convex functions convex i e weakly convex and strongly convex functions and pseudoconvex functions for the convex and convex functions a wolfe type of dual is formulated while for the pseudoconvex and convex functions a mond weir type dual is proposed

second order mixed type duality in multiobjective programming problems abstract second order mixed type dual is introduced for multiobjective programming problems results about weak duality strong duality and strict converse duality are established under generalized second order f convexity assumptions these results generalize the duality results recently given by aghezzaf and hachimi involving generalized first order f convexity conditions

on variational problems characterization of solutions and duality in this paper we introduce a new class of pseudoinvex functions for variational problems using this new concept we obtain a necessary and sufficient condition for a critical point of the variational problem to be an optimal solution illustrated with an example also weak strong and converse duality are established

second order f α ρ d convexity and duality in multiobjective programming a class of second order f a r d convex functions and their generalizations is introduced using the assumptions on the functions involved weak strong and strict converse duality theorems are established for a second order mond weir type multiobjective dual

gradient estimates for solutions of parabolic equations and systems 1 statement of the problem let 52 c r n 2 be a bounded domain with smooth boundary r 2 in this note we study scalar quasilinear parabolic differential equations u div y g ivul vu f u 0 1 u a u f u o 2 and systems u da u f u 0 3 on q x 0 co with initial conditions u 0 u0 resp u 0 u0 and homogeneous dirichlet boundary conditions u i x o co o for 1 and 2 resp 4 u1r x 0 00 0 for 3 5

on minimax fractional optimality conditions with f ρ convexity abstract we establish the sufficient conditions for generalized fractional programming in the framework of f convex functions when the sufficient conditions are utilized one parametric dual problem and two parametric free dual problems may be formulated and duality results are derived

duality for a class of nondifferentiable multiobjective programming problems abstract in this paper both the wolfe type and mond weir type dual problems for a class of nondifferentiable multiobjective programs in which every component of the objective function contains a term involving the support function of a compact convex set are formulated weak duality theorems are established under suitable conditions a special case which appears repeatedly in the literature is that the support function is the square root of a positive semidefinite quadratic form this and other special cases can be readily generated from our results

a necessary and sufficient condition for duality in multiobjective variational problems abstract in this paper we move forward in the study of duality and efficiency in multiobjective variational problems we introduce new classes of pseudoinvex functions and prove that not only it is a sufficient condition to establish duality results but it is also necessary moreover these functions are characterized in order that all kuhn tucker or fritz john points are efficient solutions recent papers are improved we provide an example to show this improvement and illustrate these classes of functions and results

mixed type duality in multiobjective programming problems two mixed type duals are introduced for multiobjective programming and for multiobjective fractional programming respectively results about weak duality strong duality and strictly reverse duality are presented under more suitable conditions

a game theory approach to constrained minimax state estimation this paper presents a game theory approach to the constrained state estimation of linear discrete time dynamic systems in the application of state estimators there is often known model or signal information that is either ignored or dealt with heuristically for example constraints on the state values which may be based on physical considerations are often neglected because they do not easily fit into the structure of the state estimator this paper develops a method for incorporating state equality constraints into a minimax state estimator the algorithm is demonstrated on a simple vehicle tracking simulation

sufficiency and duality in nondifferentiable multiobjective programming involving generalized type i functions abstract recently hachimi and aghezzaf defined generalized f d type i functions a new class of functions that unifies several concepts of generalized type i functions in this paper the generalized f d type i functions are extended to nondifferentiable functions by utilizing the new concepts we obtain several sufficient optimality conditions and prove mixed type and mond weir type duality results for the nondifferentiable multiobjective programming problem

lmi approach to an h sub spl infin control problem with time domain constraints over a finite horizon the discrete time h sub spl infin control problem is considered with time domain constraints over a finite horizon it is shown that a solution of this problem can be obtained from a system of linear matrix inequalities

symmetric duality for multiobjective fractional variational problems involving cones in this paper a pair of multiobjective fractional variational symmetric dual problems over cones is formulated weak strong and converse duality theorems are established under generalized f convexity assumptions moreover self duality theorem is also discussed

a mixed l_ infty cal h _ infty optimization approach to robust controller design in spite of its practical importance the problem of designing controllers capable of satisfying mixed time frequency domain performance requirements under model uncertainty remains to a large extent open in this paper we propose a design procedure for minimizing the maximum amplitude of a regulated error to a specified input while at the same time addressing model uncertainty through bounds on the cal h infty norm of a relevant transfer function this problem is of interest in optimal tracking applications where the objective is to achieve minimum tracking error while at the same time maintaining an adequate robustness level we show that for the siso case the problem can be solved by solving sequence of problems each one consisting of a finite dimensional convex optimization and an unconstrained nehari approximation problem

h sub spl infin optimization with time domain constraints standard h sub spl infin optimization cannot handle specifications or constraints on the time response of a closed loop system exactly in this paper the problem of h sub spl infin optimization subject to time domain constraints over a finite horizon is considered more specifically given a set of fixed inputs w sup i it is required to find a controller such that a closed loop transfer matrix has an h sub spl infin norm less than one and the time responses y sup i to the signals w sup i belong to some prespecified sets spl omega sup i first the one block constrained h sub spl infin optimal control problem is reduced to a finite dimensional convex minimization problem and a standard h sub spl infin optimization problem then the general four block h sub spl infin optimal control problem is solved by reduction to the one block case the objective function is constructed via state space methods and some properties of h sub spl infin optimal constrained controllers are given it is shown how satisfaction of the constraints over a finite horizon can imply good behavior overall an efficient computational procedure based on the ellipsoid algorithm is also discussed

optimality and duality for generalized fractional programming involving nonsmooth f ρ convex functions using a parametric approach we establish necessary and sufficient conditions and derive duality theorems for a class of nonsmooth generalized minimax fractional programming problems containing f r convex functions

second order symmetric duality in mathematical programming with f convexity abstract under second order f convexity f concavity and second order f pseudoconvexity f pseudoconcavity appropriate second order duality results for pair of wolfe and mond weir type second order symmetric dual nonlinear programming problems are established these second order duality results are then used to formulate wolfe type and mond weir type second order minimax mixed integer dual programs and symmetric duality theorem is established under separability and second order f convexity f concavity of the kernel function f x y second order symmetric dual fractional mixed integer programs are studied using the above programs moreover second order self duality theorems for the above pairs are obtained assuming f x y to be skew symmetric

duality for a class of multiobjective control problems with generalized invexity abstract b mond and i smart j math anal appl 136 1988 325 333 defined a kind of invexity and discussed the duality and sufficiency in scalar control problems with such invexity d bhatia and p kumar j math anal appl 189 1995 676 692 defined another kind of invexity corresponding generalized invexity and discussed the duality for multiobjective control problems with such generalized invexity in this paper the duality results for multiobjective control problems with mond and smart s generalized invexity are discussed

weakly efficient solutions and pseudoinvexity in multiobjective control problems abstract in this paper we introduce new pseudoinvexity conditions on functionals involved in a multiobjective control problem called w kt pseudoinvexity and w fj pseudoinvexity we prove that all kuhn tucker or fritz john points are weakly efficient solutions if and only if these conditions are fulfilled we relate weakly efficient solutions to optimal solutions of weighting control problems we generalize recently obtained optimality results of known mathematical programming problems and control problems we illustrate these results with an example

efficient solutions in v kt pseudoinvex multiobjective control problems a characterization in this paper we introduce a new condition on functionals involved in a multiobjective control problem for which we define the v kt pseudoinvex control problem we prove that a v kt pseudoinvex control problem is characterized so that a kuhn tucker point is an efficient solution we generalize recently obtained optimality results of known mathematical programming problems and control problems we illustrate these results with an example

on sufficiency and duality for nonsmooth multiobjective programming problems involving generalized v r invex functions abstract in this paper a class of nonsmooth multiobjective programming problems with inequality constraints is considered we introduce the concepts of v r pseudo invex strictly v r pseudo invex and v r quasi invex functions in which the involved functions are locally lipschitz based upon these generalized v r invex functions sufficient optimality conditions for a feasible point to be an efficient or a weakly efficient solution are derived appropriate duality theorems are proved for a mond weir type dual program of a nonsmooth multiobjective programming under the aforesaid functions

optimality conditions and duality for a minimax fractional programming with generalized convexity we establish the sufficient conditions for generalized fractional programming from a viewpoint of the generalized convexity when the sufficient conditions are utilized the corresponding duality theorems are derived for two types of duals of the generalized fractional programming we extend the corresponding results of several authors

sufficient optimality criteria and duality for multiobjective variational control problems with v invexity abstract the sufficient optimality criteria for a class of multiobjective variational control problems under the v invexity assumption are presented duality results are proved under a variety of v invexity assumptions multiobjective fractional variational control problems are also studied

an exact solution to general siso mixed spl hscr sub 2 spl hscr sub spl infin problems via convex optimization the mixed spl hscr sub 2 spl hscr sub spl infin control problem can be motivated as a nominal lqg optimal control problem subject to robust stability constraints expressed in the form of an spl hscr sub spl infin norm bound a related modified problem consisting on minimizing an upper bound of the spl hscr sub 2 cost subject to spl hscr sub spl infin constraints was introduced by bernstein haddad 1989 although there presently exist efficient methods to solve this modified problem the original problem remains to a large extent still open in this paper we propose a method for solving general discrete time siso spl hscr sub 2 spl hscr sub spl infin problems this method involves solving a sequence of problems each one consisting of a finite dimensional convex optimization and an unconstrained nehari approximation problem

symmetric duality in continuous time programming via minmax duality abstract in this paper we present continuous time generalizations of the symmetric duality formulation of dantzig eisenberg and cottle and self duality formulation of mond and cottle and establish appropriate duality relations we also show that continuous time analogues of several pairs of symmetric dual problems which are well known in the area of finite dimensional nonlinear programming can be obtained as special cases of our general symmetric programs

optimality criteria in nonlinear programming involving nonconvex functions abstract in this paper a new class of functions called convex quasiconvex and pseudoconvex are defined and a number of sufficient optimality criteria are obtained for nonlinear programming problems involving these functions

generalized type i invexity and duality in multiobjective variational problems abstract in this paper we introduce new classes of generalized v type i invex functions for variational problems and consider a multiobjective variational problem vp a number of sufficiency results are established using lagrange multiplier conditions under various types of generalized v type i invexity requirements duality results are obtained for mond weir type duals under the above generalized v type i invexity assumptions and their generalizations

mixed type duality for multiobjective variational problems the concept of mixed type duality has been extended to the class of multiobjective variational problems a number of duality relations are proved to relate the efficient solutions of the primal and its mixed type dual problems the results are obtained for convex generalized convex functions these studies have been generalized to the case of invex generalized invex functions our results apparently generalize a fairly large number of duality results previously obtained for finite dimensional nonlinear programming problems under various convexity assumptions

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

on the design of linear multivariable feedback systems via constrained nondifferentiable optimization in h sup infinity spaces the design discussed is of linear lumped time invariant multivariable feedback systems subject to various frequency and time domain performance specifications the approach is based on the use of stabilizing controller parametrizations which result in the formulation of feedback system design problems as convex nondifferentiable optimization problems these problems are solvable by recently developed nondifferentiable optimization algorithms for the constrained minimization of regular uniformly locally lipschitz continuous functions in r sup n

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

duality in nondifferentiable vector programming in this paper we study the saddle point optimality conditions and lagrange duality in multiobjective optimization for generalized subconvex like functions we obtain results which will allow us to characterize the solutions for multiobjective programming problems from the saddle point conditions and allow us to relate them to the dual problem solutions which will be adequately defined we also define a new dual problem for the multiobjective programming problem with the special property of being a scalar programming problem

estimates for the eigenvalues of the jordan product of hermitian matrices n alikakos and p w bates department of mathematics heriot watt university edinburgh scotland submitted by richard a brualdi abstract given hermitian matrices a and b professor taussky todd posed the problem of estimating the eigenvalues of their jordan product ab ba here we establish bounds for all the eigenvalues of the jordan product when both a and b are positive definite at the same time we give a more straightforward proof and an improvement of estimates given by d w nicholson for the smallest eigenvalue 0 introduction given hermitian matrices a and b their

duality with bf type i functions for a class of nondifferentiable multiobjective variational problems a class of bf type i functions and its extensions are introduced in the continuous case an example is presented in support utilizing these new concepts sufficient optimality conditions and duality results are presented for multiobjective variational problems involving arbitrary norms

three types dual model for minimax fractional programming abstract we establish necessary and sufficient optimality conditions for minimax fractional programming involving nonsmooth generalized t convex functions three types dual model of 1 are considered and certain duality results have been derived in the framework of generalized t convex functions