Kaul **and** Kaur [7] obtained necessary **optimality** conditions for a non-linear **programming** **problem** 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 [12] 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.

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Abstract: In this paper, we shall establish necessary **and** sufficient conditions for a feasible **solution** to be **efficient** for a nonsmooth **multiobjective** **fractional** **programming** **problem** 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 **duality** **results** for corresponding Mond-Weir subgradient type dual **problem**. These **results** extend **some** earlier **results** on efficiency **and** **duality** to **multiobjective** **fractional** **programming** problems involving pseudolinear **and** η − pseudolinear functions.

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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 **problem** **and** finally these **results** will be simulated using inflows series. This paper is organized as follows: Section 1 presents the formulation **of**

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It is well known that convexity play **an** important role in establishing the sufficient **optimality** conditions **and** **duality** theorems for a **nonlinear** **programming** **problem**. 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 [2]. In [3], Bector **and** Suneja define the class **of** B-invex functions for differentiable numerical functions. The sufficient **optimality** conditions **and** **duality** **results** 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 [15], the author study **some** **of** 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 [8] Preda introduced the class **of** locally Lipschitz (B, ρ ,d ) -preinvex functions **and** extend

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The most important **problem** in group theory in terms **of** quantum algorithms is called hidden subgroup **problem** (HSP) [14]. 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 **of** **efficient** 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 [14]. On the other hand, **an** **efficient** **solution** for a generic non-abelian group is not known. Two important groups in this context are the symmetric **and** the dihedral groups. **An** **efficient** algorithm for solving the HSP for the former group would imply in **an** **efficient** **solution** for the graph isomorphism **problem** [1, 2, 12, 8] **and** for the latter one would solve instances **of** the **problem** **of** finding the shortest vector in a lattice, which has applications in cryptography [16]. 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

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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.”

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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 **of** **efficient** 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 **of** **problem** **and** find better **solution**. Keywords: Bulk Transportation **Problem**, Lexi-Search, Pattern Recognition.

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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 **solution** **of** **nonlinear** 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 **solution** **of** **fractional** order partial differential equations with nonlocal boundary conditions.

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Since 1960 geometric **programming** **problem** 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 **programming** **problem** 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.

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Differential Evolution developed by Storn **and** Price is one **of** the excellent evolutionary algorithms [17] . 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.

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computation over encrypted data a very hard **problem**. On the other hand, the operational details inside the cloud are not transparent enough to customers [4]. Recent researches in both the cryptography **and** the theoretical computer science communities have made steady advances in “secure outsourcing expensive computations” (e.g. [5]– [10] ). Based on Yao’s garbled circuits [11] **and** Gentry’s breakthrough work on fully homomorphic encryption (FHE) scheme [12], a general result **of** secure computation outsourcing has been shown viable in theory [9], 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 [7], [8], or multi- round interactive protocol executions [5], or huge communication complexities [10], are involved. In short, practically **efficient** mechanisms with immediate practices for secure computation outsourcing in cloud are still missing.

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The existence of a solution, as well as some properties of the obtained solution for a Hammerstein type nonlinear integral equation have been investigated.. For a certain class of functi[r]

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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 **of** **an** 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 **multiobjective** **problem**.

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Existence results for some elliptic equations involving the fractional Laplacian operator and critical growth / Yane Lísley Ramos Araújo.- João Pessoa, 2015.. Desigualdade de Trudinger[r]

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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 [13] will be used to prove existence the **solution** for given equation(1.1)

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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 **of** **optimality** 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.

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The use **of** **fractional** orders differential **and** integral operators in mathematical models has become increasingly widespread in recent years (see [11], [22] **and** [26]). Several forms **of** **fractional** differential equations have been proposed in standard models, **and** there has been significant interest in developing numerical schemes for their **solution** (see [11], [14], [22] **and** [26]). 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 [1] **and** [3]-[6]).

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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.

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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 [21], 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.

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