Kaul and Kaur  obtained necessary optimality conditions for a non-linear programmingproblem by taking the objective and constraint functions to be semilocally convex and their right differentials at a point to be lower semi-continuous. Suneja and Gupta  established the necessary optimality conditions without assuming the semilocal convexity of the objective and constraint functions but their right differentials at the optimal point to be convex.
Abstract: In this paper, we shall establish necessary and sufficient conditions for a feasible solution to be efficient for a nonsmooth multiobjectivefractionalprogrammingproblem involving η − pseudolinear functions. Furthermore, we shall show equivalence between efficiency and proper efficiency under certain boundedness condition. We have also obtained weak and strong dualityresults for corresponding Mond-Weir subgradient type dual problem. These results extend some earlier results on efficiency andduality to multiobjectivefractionalprogramming problems involving pseudolinear and η − pseudolinear functions.
Stochastic dynamic programming (SDP) has been the most suggested technique to solve the MTHS problem since it can adequately cope with the uncertainty of inflows and the nonlinear relations among variables. Although efficient in the treatment of river inflows as random variables described by probability distributions, the SDP technique is limited by the so-called ”curse of dimensionality” since its com- putational burden increases exponentially with the number of hydro plants. In order to overcome this difficulty one common solution adopted is to represent the hydro system by an aggregate model, as it is the case in the Brazilian power system. Alternatives to stochastic models for MTHS can be developed through operational policies based on deterministic models. The advantage of such approaches is their ability to handle multiple reservoir systems without the need of any modeling manipulation. Although some work has been done in the comparison between deterministic and stochastic approaches for MTHS, the discussion about the best approach to the problem is far from ending. The purpose of this paper is to present a discussion about different policies based on Dynamic Programming to solve MTHS. Hydro plants located in different regions of Brazil will be considered as case studies. The uncertainty of inflows will be modelled and the Box-Cox transformation will be used. One determinist model and three other stochastic models will be considered for solving the problemand finally these results will be simulated using inflows series. This paper is organized as follows: Section 1 presents the formulation of
It is well known that convexity play an important role in establishing the sufficient optimality conditions andduality theorems for a nonlinearprogrammingproblem. Several class of functions have been defined for the purpose of weakening the limitations of convexity. Bector and Singh extend the class of convex functions to the class of B-vex functions in . In , Bector and Suneja define the class of B-invex functions for differentiable numerical functions. The sufficient optimality conditions anddualityresults were obtained involving these generalized functions. As so far now, the study about the sufficient optimality conditions and algorithm of semi-infinite programming are under the assumption that the involving functions are differentiable. But non- smooth phenomena in mathematics and optimization occur naturally and frequently, and there is a need to be able to deal with them. In , the author study someof the properties of B- vex functions for locally Lipschitz functions, and extend the class of B-invex, pseudo B-invex and quasi B-invex functions from differentiable numerical functions to locally Lipschitz functions. In [11-13], Preda introduced some classes of V- univex type-I functions , called called ( ρ, ρ')-V- univex type-I, ( ρ, ρ')-quasi V-univex type-I, (ρ, ρ')- pseudo V-univex type-I, ( ρ, ρ')-quasi pseudo V- univex type-I, and ( ρ, ρ')-pseudo quasi V-univex type-I. In  Preda introduced the class of locally Lipschitz (B, ρ ,d ) -preinvex functions and extend
The most important problem in group theory in terms of quantum algorithms is called hidden subgroup problem (HSP) . The HSP can be described as follows: given a group G and a function f : G → X on some set X such that f (x) = f (y) iff x · H = y · H for some subgroup H, the problem consists in determining a generating set for H by querying the function f . We say that the function f hides the subgroup H in G or that f separates the cosets of H in G. A quantum algorithm for the HSP is said to be efficient when the running time is O(poly(log |G|)). There are many examples ofefficient quantum algorithms for the HSP in particular groups [17, 18]. It is known that for finite abelian groups, the HSP can be solved efficiently on a quantum computer . On the other hand, anefficientsolution for a generic non-abelian group is not known. Two important groups in this context are the symmetric and the dihedral groups. Anefficient algorithm for solving the HSP for the former group would imply in anefficientsolution for the graph isomorphism problem [1, 2, 12, 8] and for the latter one would solve instances of the problemof finding the shortest vector in a lattice, which has applications in cryptography . One way to design new quantum algorithms for the HSP is to investigate the structures of all subgroups of a given group, and then to find a quantum algorithm applicable to each subgroup
The most common type of complaint from students was un- questionably related to excessive lecturing on the part of the native TA (about 70% of negative comments related to excessive lecturing, not enough time to talk.) Students often felt shorthanded when not given opportunities to discuss topics in class. (Although this contra- dicts the statements above related to difficult discussion topics, we should assume that students who requested more discussion in class are expecting discussion at an appropriate language level.) One stu- dent said of a native male TA, “I think if there were more opportuni- ties to speak instead of hearing him speak for most of the class, it would have been more beneficial.” Another complained about a na- tive female TA, “She was very enthusiastic about teaching, but activi- ties that actually involved speaking were sparse.” More than anything else, students mentioned discussion in the classroom, and the majori- ty of these complaints and requests were aimed towards native female TAs: “I did not like how little we got to speak as a class. My under- standing increased but I feel my speaking skills went down”; “More class participation would have been good”; “I wish we would have had more opportunities to have class debates and class discussions”; “mostly a lecture [with] little interaction”; “I think the only thing that could be improved is if she would have us speak more in class.” Fi- nally, one student wrote an extensive comment for a female TA that depicted her as an outstanding TA, but then ended with the sugges- tion that “more emphasis on free class discussion might help.”
A large number of real-world planning problems called Combinatorial Optimization Problems share the following properties: They are Optimization Problems, are easy to state, and have a finite but usually very large number of feasible solutions. Lexi-Search is by far the mostly used tool for solving large scale NP-hard Combinatorial Optimization problems. Lexi-Search is, however, an algorithm paradigm, which has to be filled out for each specific problem type, and numerous choices for each of the components exist. Even then, principles for the design ofefficient Lexi-Search algorithms have emerged over the years. Although Lexi-Search methods are among the most widely used techniques for solving hard problems, it is still a challenge to make these methods smarter. The motivation of the calculation of the lower bounds is based on ideas frequently used in solving problems. Computationally, the algorithm extended the size ofproblemand find better solution. Keywords: Bulk Transportation Problem, Lexi-Search, Pattern Recognition.
From analysis and experimental results we observe that the proposed method is simple and provides a very high accurate estimate of the solution. The method can solve nonlocal boundary value problems very easily. In our test problems we consider up to 9-point boundary conditions and observe that the results obtained are satisfactory. The method also works well in solutionofnonlinear FDEs with nonlocal conditions. We observe that by using high scale level the iteration converges more rapidly. As shown above at fifth iteration the norm of error is less than 10 − 8. However by using high scale level, much more accurate results can be achieved. Our future work is related to the extension of method in solutionoffractional order partial differential equations with nonlocal boundary conditions.
Since 1960 geometric programmingproblem has undergone several changes. In most of the engineering problems the parameters are considered as deterministic. In this paper we have discussed the problems by splitting the cost coefficients, constraint coefficients and exponents using binary numbers. Geometric programming has already shown its power in practice in the past. In many real world geometric programmingproblem the parameters may not be known precisely due to insufficient information and hence this paper will help the wider applications in the field of engineering problems.
Differential Evolution developed by Storn and Price is one of the excellent evolutionary algorithms  . DE is a robust statistical method for cost function minimization, which does not make use of a single parameter vector but instead uses a population of equally important vectors. This paper develops an improved DE algorithm to determine the optimum generation schedule of the DED problem that takes into consideration of valve- point effects. In the proposed approach, the search capability of the DE algorithm is enhanced by introducing heuristic crossover operation and gene swap operator, which leads to a higher probability of getting global or near global optimal solutions. The proposed method is tested on five-unit and ten-unit sample test systems and the results are compared with a SA, hybrid EP-SQP, DGPSO and PSO-SQP methods. The effectiveness and potential of the proposed approach to solve DED problem is demonstrated.
computation over encrypted data a very hard problem. On the other hand, the operational details inside the cloud are not transparent enough to customers . Recent researches in both the cryptography and the theoretical computer science communities have made steady advances in “secure outsourcing expensive computations” (e.g. –  ). Based on Yao’s garbled circuits  and Gentry’s breakthrough work on fully homomorphic encryption (FHE) scheme , a general result of secure computation outsourcing has been shown viable in theory , where the computation is represented by an encrypted combinational Boolean circuit that allows to be evaluated with encrypted private inputs. However, applying this general mechanism to our daily computations would be far from practical, due to the extremely high complexity of FHE operation as well as the pessimistic circuit sizes that cannot be handled in practice when constructing original and encrypted circuits. This overhead in general solutions motivates us to seek efficient solutions at higher abstraction levels than the circuit representations for specific computation outsourcing problems. Although some elegant designs on secure outsourcing of scientific computations, sequence comparisons, and matrix multiplication etc. have been proposed in the literature, it is still hardly possible to apply them directly in a practically efficient manner, especially for large problems. In those approaches, either heavy cloud-side cryptographic computations , , or multi- round interactive protocol executions , or huge communication complexities , are involved. In short, practically efficient mechanisms with immediate practices for secure computation outsourcing in cloud are still missing.
This thesis proposes a new necessary condition for the infeasibility of non-linear optimization problems (that becomes necessary under convexity assumption) which is stated as a Pareto-criticality condition ofan auxiliary multiobjective optimization problem. This condition can be evaluated, in a given problem, using multiobjective optimization algorithms, in a search that either leads to a feasible point or to a point in which the infeasibility conditions holds. The resulting infeasibility certificate, which is built with primal variables only, has global validity in convex problems and has at least a local meaning in generic nonlinear optimization problems. In the case of noisy problems, in which gradient information is not available, the proposed condition can still be employed in a heuristic flavor, as a by-product of the expected features of the Pareto-front of the auxiliary multiobjectiveproblem.
In this section we consider the fractional order functional integro-differential equation (1.1).The following hybrid fixed point theorem for three operators in Banach algebras �, due to B.C.Dhage  will be used to prove existence the solution for given equation(1.1)
of broken extremals; that is, they satisfy all the necessary conditions for optimalityof the maximum principle and are diﬀerentiable functions of the parameter between the junction times . The associated ﬂow map of the controlled trajectories is deﬁned as ̥ : (t, p) 7→ ̥(t, p) = (t, x(t, p)), (5.5) i.e., through the graphs of the corresponding trajectories. This is the correct formulation as our problem formulation overall is indeed time-dependent since there exists a ﬁxed ﬁnite terminal time (e.g., see [14, pg. 324]). We emphasize that it is not required for a parameterized family of extremals that this ﬂow deﬁnes an injective mapping. If it does, we call it a field of extremals. Obviously, since the trajectories for diﬀerent parameter values typically obey diﬀerent diﬀerential equations, it is quite possible that these graphs could intersect. In fact, conjugate points and associated loss of local optimalityof extremals precisely correspond to fold singularities in this mapping while the reference controlled extremal will be a strong local minimum if this ﬂow map is a diﬀeomorphism along the reference trajectory t 7→ x ∗ (t) = x(t, p ∗ ) .
Abstract— A fuzzy optimal control model was formulated minimizing the objective function with discounted cost for the length of infinite horizon. We developed an equation ofoptimality in case of fuzzy optimal control problem. We revisited a special fuzzy control model with quadratic objective functional form for linear Liu’s fuzzy control system. As an application, we investigated the infinite horizon production inventory planning problem with nonzero discount rate. We employed fuzzy optimal control to model inventory production planning problem with fuzzy variables and solved.
The use offractional orders differential and integral operators in mathematical models has become increasingly widespread in recent years (see ,  and ). Several forms offractional differential equations have been proposed in standard models, and there has been significant interest in developing numerical schemes for their solution (see , ,  and ). However, much of the work published to date has been concerned with linear single term equations and, of these, equations of order less than unity have been most often investigated (see  and -).
Regarding the problem management process, not all engineers make notes about the problems that they have encountered when testing their own components. Only five out of eight companies record a problem when it gets encountered at this testing level. Two of the three remaining companies only record major and more important problems. The engineers mainly make notes in order not to forget the problem. They use it for planning their next-coming work and for tracking the coding and testing activity.
Convexity and generalized convexity are very important concepts in optimization theory. One reason for this importance is that for these classes of functions it is possible to establish alternative theorems and consequently to obtain necessary and/or sufficient optimality conditions. The generalized convexity notion that we will use here is the generalized subconvex-like functions, which was introduced by Xinmin Yang in , where the author showed that these functions satisfy a Gordan type alternative theorem. He also showed that the generalized subconvex-like class of functions comprise subconvex-like, convex-like and convex class of functions. Thus the generalized subconvex-like is a large class of functions which satisfy a Gordan type alternative theorem.