sufficiency and duality in nonsmooth multiobjective optimization involving generalized f α π d type i functions in this paper new classes of generalized f a p d type i functions are introduced for a nonsmooth multiobjective programming problem based upon these generalized functions sufficient optimality conditions are established weak strong and strict converse duality theorems are also derived for wolfe and mond weir type multiobjective dual programs in order to relate the efficient and weak efficient solutions of primal and dual problems

azumaya algebras with involutions res 2wr zbr s cores br r 1 one could ask whether this sequence is exact indeed if r and s are fields and char r 2 the exactness of the above sequence is proved in cl corollary 4 6 p 4731 let a be an azumaya s algebra whose class a belongs to kernel cores br s br r for a to belong to res br r it is necessary and sufficient that there exists an azumaya s algebra b brauer equivalent to a admitting involutions of first and second kinds which commute results of saltman assert lo that there exist algebras bi i 1 2 brauer equivalent to a which support involutions of the first and second kind respectively we construct section 4 a homomorphism h ker cores br s br r pic rjcores pit s whose kernel consists precisely of classes of azumaya s algebras which support commuting involutions of both kinds thus h is a measure of the obstruction to the sequence 1 being exact in particular if pit r cores pit s 1 is exact

a variational like inequality for multifunctions with applications abstract we study a variational like inequality problem for a multifunction a certain relevant complementarity problem is defined we establish some existence theorems for the above two problems an application to convex mathematical programming is also shown

a modified objective function method for solving nonlinear multiobjective fractional programming problems abstract a new method is used for solving nonlinear multiobjective fractional programming problems having v invex objective and constraint functions with respect to the same function in this approach an equivalent vector programming problem is constructed by a modification of the objective fractional function in the original nonlinear multiobjective fractional problem furthermore a modified lagrange function is introduced for a constructed vector optimization problem by the help of the modified lagrange function saddle point results are presented for the original nonlinear fractional programming problem with several ratios finally a mond weir type dual is associated and weak strong and converse duality results are established by using the introduced method with a modified function to obtain these duality results between the original multiobjective fractional programming problem and its original mond weir duals a modified mond weir vector dual problem with a modified objective function is constructed

minimizing maximum promptness and maximum lateness on a single machine we consider the following single machine bicriteria scheduling problem a set of n independent jobs has to be scheduled on a single machine that can handle only one job at a time and that is available from time zero onwards each job jj requires processing during a given positive uninterrupted time pj and has a given target start time sj and due date dj with a dj sj a pj for some constant a for each job jjj 1 n its promptness pj is defined as the difference between the target start time and the actual start time and its lateness lj as the difference between the completion time and the due date we consider the problem of finding a schedule that minimizes a function f of maximum promptness pmax max1 j npj and maximum lateness lmax max1 j nlj we assume that f is nondecreasing in both arguments we present an on2 algorithm for the variant in which idle time is not allowed and an on2 log n algorithm for the special case in which the objective function is linear we prove that the problem is np hard if neither of these restrictions is imposed as a side result we prove that the special case of minimizing maximum lateness subject to release dates that lie in the interval dj pj a dj a for all j 1 n and for some constant a is solvable in on log n time if no machine idle time is allowed and in on2 log n time if machine idle time is allowed

a hybrid multi objective immune algorithm for a flow shop scheduling problem with bi objectives weighted mean completion time and weighted mean tardiness this paper investigates a novel multi objective model for a no wait flow shop scheduling problem that minimizes both the weighted mean completion time c w and weighted mean tardiness t w obtaining an optimal solution for this type of complex large sized problem in reasonable computational time by using traditional approaches and optimization tools is extremely difficult this paper presents a new hybrid multi objective algorithm based on the features of a biological immune system is and bacterial optimization bo to find pareto optimal solutions for the given problem to validate the performance of the proposed hybrid multi objective immune algorithm hmoia in terms of solution quality and diversity level various test problems are examined further the efficiency of the proposed algorithm based on various metrics is compared against five prominent multi objective evolutionary algorithms ps nc ga nsga ii spea ii moia and misa our computational results suggest that our proposed hmoia outperforms the five foregoing algorithms especially for large sized problems

efficiency through variational like inequalities with lipschitz functions in this work first we introduce several notions of invexity and pseudoinvexity for a locally lipschitz function by means of the generalized jacobian we study relationships between these concepts in particular the implications between preinvexity and invexity next we obtain necessary and sufficient optimality conditions for efficient and weak efficient solutions of finite dimensional non necessarily pareto vector optimization problems with locally lipschitz objective functions through solutions of vector variational like inequality problems these conditions are stated via the generalized jacobian and under pseudoinvexity hypotheses and they show that a vector optimization problem can be reformulated as a vector variational like inequality problem this work extends and improves several previous papers where the objective function of the vector optimization problem is assumed to be differentiable or being locally lipschitz the authors consider the componentwise subdifferential based on the clarke s generalized gradients of the components of the objective function throughout the paper some simple examples are given in order to illustrate the main concepts and results

an algorithm for the multiple objective integer linear programming problem abstract a technique is presented for solving the multiple objective integer linear programming problem the technique can be used to identify some or all efficient solutions while the technique is applicable with any integer programming algorithm it is well suited for implementation using integer postoptimality techniques such an implementation based on balas additive algorithm is described for problems with zero one variables

relationships between pre invex concepts abstract it is well known that some properties of convex optimization problems especially sufficient conditions for optimality and duality conditions are preserved when convexity is weakened to pre invexity however there are several distinct pre invex concepts b vex b pre invex pre univex p r pre invex and these are not equivalent the pre invexity concepts mentioned above have also some properties which are characteristic and invariant for functions belonging to these classes of functions

earliness tardiness scheduling problems ii derivation of completion times about a restrictive common due date a companion paper part i considers the problem of minimizing the weighted earliness and tardiness of jobs scheduled on a single machine around a common due date d which is unrestrictively late this paper part ii considers the problem of minimizing the unweighted earliness and tardiness of jobs allowing the possibility that d is early enough to constrain the scheduling decision we describe several optimality conditions the recognition version of the problem is shown to be np complete in the ordinary sense confirming a well known conjecture moreover this complexity definition is precise since we describe a dynamic programming algorithm which runs in pseudopolynomial time this algorithm is also extremely efficient computationally providing an improvement over earlier procedures of almost two orders of magnitude in the size of instance that can be solved finally we describe a special case of the problem which is polynomially solvable

taylor series approach to fuzzy multiobjective linear fractional programming this paper presents the use of a taylor series for fuzzy multiobjective linear fractional programming problems fmolfp the taylor series is a series expansion that a representation of a function in the proposed approach membership functions associated with each objective of fuzzy multiobjective linear fractional programming problem transformed by using a taylor series are unified thus the problem is reduced to a single objective practical applications and numerical examples are used in order to show the efficiency and superiority of the proposed approach

a note on invex problems with nonnegative variable we provide sufficient optimality conditions involving invexity for mathematical programming problems with nonnegative variable we also show that upon certain assumptions on the scaling function these problems are convex transformable hence certain associated convex problems should be solved some applications are shown

new solution methods for single machine bicriteria scheduling problem minimization of average flowtime and number of tardy jobs we consider the bicriteria scheduling problem of minimizing the number of tardy jobs and average flowtime on a single machine this problem which is known to be np hard is important in practice as the former criterion conveys the customer s position and the latter reflects the manufacturer s perspective in the supply chain we propose four new heuristics to solve this multiobjective scheduling problem two of these heuristics are constructive algorithms based on beam search methodology the other two are metaheuristic approaches using a genetic algorithm and tabu search our computational experiments indicate that the proposed beam search heuristics find efficient schedules optimally in most cases and perform better than the existing heuristics in the literature

two machine flow shop scheduling with two criteria maximum earliness and makespan abstract in this study we address the problem of minimizing makespan and maximum earliness simultaneously in a two machine flow shop environment we develop a branch and bound procedure that generates all efficient solutions with respect to two criteria we propose several lower and upper bounding schemes to enhance the efficiency of the algorithm we also propose a heuristic procedure that generates approximate efficient solutions our computational results reveal that the branch and bound procedure is capable of solving problems with up to 25 jobs and the heuristic procedure produces approximate efficient solutions that are very close to exact efficient solutions in very small computation times

earliness tardiness scheduling problems i weighted deviation of completion times about a common due date this paper and its companion part ii concern the scheduling of jobs with cost penalties for both early and late completion in part i we consider the problem of minimizing the weighted sum of earliness and tardiness of jobs scheduled on a single processor around a common due date d we assume that d is not early enough to constrain the scheduling decision the weight of a job does not depend on whether the job is early or late but weights may vary between jobs we prove that the recognition version of this problem is np complete in the ordinary sense we describe optimality conditions and present a computationally efficient dynamic programming algorithm when the weights are bounded by a polynomial function of the number of jobs a fully polynomial approximation scheme is given we also describe four special cases for which the problem is polynomially solvable part ii provides similar results for the unweighted version of this problem where d is arbitrary

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

the arason invariant and mod 2 algebraic cycles introduction 73 1 review of the arason invariant 75 2 the special clifford group 76 3 k cohomology of split reductive algebraic groups 78 4 k cohomology of bg 86 5 gl n and cliff n n 92 6 two invariants for clifford bundles 95 7 snaking a bloch ogus differential 100 8 proof of theorem 1 101 9 application to quadratic forms 102 appendix a toral descent 104 appendix b the rost invariant 108 appendix c an amusing example 113 acknowledgements 116 references 116

theory and methodology a bicriteria approach to the two machine flow shop scheduling problem in this paper we address the problem of minimizing makespan and sum of completion times simultaneously in a two machine flow shop environment we formulate the problem as a bicriteria scheduling problem and develop a branch and bound procedure that iteratively solves restricted single objective scheduling problems until the set of efficient solutions is completely enumerated we report computational results and explore certain properties of the set of efficient solutions we then discuss their implications for the decision maker

nonsmooth invex functions and sufficient optimality conditions in this paper the various definitions of nonsmooth invex functions are gathered in a general scheme by means of the concept of k directional derivative characterizations of nonsmooth k invexity are derived as well as results concerning constrained optimization without any assumption of convexity of the k directional derivatives

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

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

nondifferentiable multiobjective programming under generalized d univexity abstract in this paper we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints we introduce four new classes of generalized convex functions by combining the concepts of weak strictly pseudoinvex strong pseudoinvex weak quasi invex weak pseudoinvex and strong quasi invex functions in aghezzaf and hachimi numer funct anal optim 22 2001 775 d invex functions in antczak europ j oper res 137 2002 28 and univex functions in bector et al univex functions and univex nonlinear programming proc admin sci assoc canada 1992 p 115 by utilizing the new concepts we derive a karush kuhn tucker sufficient optimality condition and establish mond weir type and general mond weir type duality results for the nondifferentiable multiobjective programming problem

multi objective infinite programming using the theory of abstract optimization problems in infinite dimensional spaces we provide necessary optimality conditions of first and second order for weakly efficient solutions of the multi objective infinite programming problem sufficient conditions are given under invexity assumptions we generalize the notion of kkt invexity for the multi objective infinite problem and show that this notion is a necessary and sufficient condition for every vector kkt solution to be a weakly efficient solution moreover we develop a theorem of the alternative useful for proving some of our results

generalized proper efficiency and duality for a class of nondifferentiable multiobjective variational problems withv invexity abstract a mond weir type dual for a class of nondifferentiable multiobjective variational problems in which every component of the objective function contains a term involving the square root of a certain positive semidefinite quadratic form is considered and various duality results viz weak strong and converse duality theorems are developed for conditionally properly efficient solutions these results are obtained under v invexity assumptions and its generalizations on objective and constraint functions this work extends many results on variational problems established earlier

multiobjective programming under d invexity abstract in this paper a generalization of convexity is considered in the case of nonlinear multiobjective programming problem where the functions involved are nondifferentiable by considering the concept of pareto optimal solution and substituting d invexity for convexity the fritz john type and karush kuhn tucker type necessary optimality conditions and duality in the sense of mond weir and wolfe for nondifferentiable multiobjective programming are given

a bicriteria two machine permutation flowshop problem the problem attacked in this paper is the scheduling of n jobs on two machines which are assumed to be continuously available all jobs are available at the beginning of the scheduling period setup times are included in the processing times no preemption of jobs is allowed the objective is the minimization of a weighted combination of the average job flowtime and the schedule makespan three branch and bound bb and each one of these problems is solved for five different levels of the weighting factor an analysis into the resulting efficient points is also included an improved version of fsh1 called fsh2 is developed and both heuristics are subjected to extensive numerical experimentation

pre invex functions in multiple objective optimization and called such functions convexlike if s is a convex set and if f is a convex function then clearly f is convexlike elster and nehse obtained a saddlepoint optimality condition for convexlike mathematical programs hayashi and komiya 9 also considered convexlike functions and developed a gordan type theorem of the alternative involving convexlike functions and in addition considered lagrangian duality for convexlike programs hanson 7 considered differentiable functions f s r for which there exists an n dimensional vector function q q u such that for all x u e s