In the linear **transportation** problem (ordinary **transportation** problem) the cost per unit commodity shipped from a given source to a given destination is constant, regardless of the amount shipped. It is always supposed that the mileage (distance) from every source to every destination is fixed. To solve such **transportation** problem we have the streamlined simplex **algorithm** which is very efficient. However, in reality, we can see at least two cases that the **transportation** problem fails to be linear. First, the cost per unit commodity transported may not be fixed for volume discounts sometimes are available for large shipments. This would make the cost function either piecewise linear or just separable concave function. In this case the problem may be formulated as piecewise linear or concave programming problem with linear constraints.

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Since equilibrium problem (.) provides a uniﬁed model of several **problems** such as variational inequalities, ﬁxed point **problems** and inclusion **problems**. In [], Takahashi and Takahashi further studied ﬁxed points of a nonexpansive mapping and equilibrium problem (.) based on the viscosity approximation method, which was introduced by Moudaﬁ []. To be more clear, they proved the following result.

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inequality **problems** and construct an iterative **algorithm** for approximating the solutions of the system of generalized relaxed cocoercive variational inequalities in Hilbert spaces. We prove the existence of the solutions for the system of generalized relaxed cocoercive variational inequality **problems** and the convergence of iterative sequences generated by the **algorithm**. We also study the convergence and stability of a new perturbed iterative **algorithm** for approximating the solution. The results presented in this paper improve and extend the previously known results in this area.

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known **transportation** problem (TP) in which three item demand and conveyance) are taken into account in the constraint set instead of two (supply and demand), and s of great use in public distribution systems. The STP was first stated by Shell (1955). Although STP was forgotten for long time, because of the new existing solution methodologies, recently it is receiving the attention and the interest of the researchers in this . 1993). Haley (1962) introduced the solution procedure of STP which is an extension of the modified distribution method. Patel and Tripathy (1989)developed a computationally superior method for a STP with mixed index **transportation** problem which is an extension of the modified . (1994) provided an **algorithm** for finding the optimum solution of solid fixed charge linear . (1997) designed a neural network approach for multicriteria STP. Jimenez and Verdegay (1996) developed a parametric approach for solving fuzzy STPs by an evolutionary **algorithm** (EA). Pandian and Natarajan (2010) are d the zeropoint method for finding an optimal solution to a classical TP. Pandian and Auradha, (2010) proposed a new method using the principle of zero point method introduced by Pandian and Natarajan (2010) for finding an optimal solution of . (2010) formulated a STP with discounted costs, fixed charges and vechicle costs as a linear programming problem. Qualitative analysis of some basic notions such as the set of feasible parameters, the solvability set and the stability set ind and the stability set of the second kind are introduced by Osman (1977). Luhandjula (1987) deals with multi- objective programming **problems** with possibilistic coefficients. Hussein (1998) introduced the complete solutions of multi- (1989) provided the concept of -Pareto objective **nonlinear** programming **problems** with differentiability for the considered objective functions. Ammar and Youness (2005) introduced n this paper, we deal with a multi- n problem (Poss MOSTP) with possibilistic coefficients, possibilistic supply values, possibilistic

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According to the table I, the WGA **algorithm** is a completely general method. This paper considered WGA Method in order to solve power system CP **problems**. The CP is a **nonlinear**, non-convex optimization problem. Every well- known heuristic method has some individual features. Meanwhile, some of these predominant features can be categorized as: mutation, swarm intelligence, memory based, and searching in groups. Mutation is the individual feature of GA which also considered in WGA. PSO **algorithm** is firmly based on Swarm intelligence and also it’s a perfectly memory based **algorithm**, both are considered in WGA. Searching in groups is another challenging and perfectly effective feature which is considered in ICA, this also considered in WGA. Also complete and well-sorted detail about WGA would found in [17]. As it obvious, WGA is a really powerful **algorithm** and this is the acceptable reason to consider this **algorithm** for complex non-linear and non-convex **problems** like CP.

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Trust region methods for unconstrained minimization are blessed with both strong theoretical convergence properties and a good accurate results in practice. The trial computational step in these methods is to find an approximate minimizer of some model of the true objective function within a trust region for which a suitable norm of the correction lies inside a given bound. This restriction is known as the trust region constraint, and the bound on the norm is its radius. The radius is adjusted so that successive model **problems** minimized the true objective function within the trust region 7.

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A standard way for solving inverse scattering problem is via optimization, where a sequence of estimates is generated by minimizing a cost function. For ill-posed **problems** with an additive measurement noise model, a cost function usually consists of a quadratic data-fidelity term and a regularization term, which incorporates prior information such as transform-domain sparsity. The challenge of such a formulation for **nonlinear** diffractive imaging is that the data-fidelity term is nonconvex due to the nonlinearity and that sparsity-promoting regularizers are usually nondifferentiable. For such nonsmooth and nonconvex **problems**, the proximal gradient method, also known as iterative shrinkage/thresholding **algorithm** (ISTA) [10, 32, 43], is a natural choice and enjoys convergence guarantees. However, it usually converges slowly. Fast iterative shrink- age/thresholding **algorithm** (FISTA) [8] is an accelerated variant of ISTA, which is proved to converge fast for convex **problems**. Unfortunately, its convergence analysis for nonconvex prob- lems has not been established. This work proposes a relaxed variant of FISTA for the nonsmooth and nonconvex **problems** and provides its convergence guarantee.

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In Section , we deﬁned a Lipschitz continuous mapping and an inverse strongly mono- tone mapping. Inverse strongly monotone mappings arise in various areas of optimization and **nonlinear** analysis (see, for example, [–]). It follows from the Cauchy-Schwarz inequality that if a mapping A : D(A) ⊆ H → R(A) ⊆ H is

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In most real world application and **problems**, a homogeneous product is carried from an origin to a destination by using different **transportation** modes (e.g., road, air, rail and water). This paper investigates a fixed charge **transportation** problem (FCTP), in which there are different routes with different capacities between suppliers and customers. To solve such a NP-hard problem, four meta- heuristic algorithms include Red Deer **Algorithm** (RDA), Stochastic Fractal Search (SFS), Genetic **Algorithm** (GA), and Simulated Annealing (SA) and two new hybrid meta-heuristics include hybrid RDA & GA (HRDGA) **algorithm** and Hybrid SFS & SA (HSFSA) **algorithm** are utilized. Regarding the literature, this is the first attempt to employ such optimizers to solve a FCTP. To tune up their parameters of algorithms, various problem sizes are generated at random and then a robust calibration is applied by using the Taguchi method. The final output shows that Simulated Annealing (SA) **algorithm** is the better than other algorithms for small-scale, medium-scale, and large-scale **problems**. As such, based on the Gap value of algorithms, the results of LINGO software shows that it reveals better outputs in comparison with meta-heuristic algorithms in small- scale and simulated annealing **algorithm** is better than other algorithms in large- scale and medium-scale **problems**. Finally, a set of computational results and conclusions are presented and analyzed.

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This simplex search method, first proposed by Spendley, Hext, and Himsworth [14] and later refined by Nelder and Mead [9]. Their methods is one of the most eﬃcient pattern search method currently available. This method is a derivative-free line search method that was particularly designed for tradi- tional unconstrained minimization scenarios, such as the **problems** of non- linear least squares, **nonlinear** simultaneous equations, and other types of function minimization [10]. In this method for N vertices of an initial sim- plex, evaluate cost function for each vertex at the first. Then the worth vertex replace by newly reflected and better point, which can be approximately lo- cated in the negative gradient direction. In the minimization problem with three initial simplex vertices, the method can be mention as follows [6, 17]: x h : Vertex with highest cost function value.

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In this paper, we consider a special class of **nonlinear** BLPP with nonconvex objective functions, in which the follower’s objective is a function of the linear expression of all variables, and the follower’s constraints are convex with respect to the follower’s variables. There are no restrictions of convexity and differentiability on both the leader’s and the follower’s objective functions, which makes the model different from those proposed in the open literature. In view of the nonconvexity of the leader’s objective, we develop an evolutionary **algorithm** to solve the problem. First, based on the structural fea- tures of the follower’s objective, we give a new decom- position scheme by which the (approximate) optimal solution y to the follower’s problem can be obtained in a finite number of iterations. At the same time, the popula- tions are composed of such points (x, y) satisfying the follower’s optimization problem, that is, y is an optimal solution of the follower’s problem when x is fixed, which improves the feasibility of individuals. Then, to improve the efficiency of proposed **algorithm**, the better individu- als than crossover parents are employed to design new crossover operator, which is helpful to generate better offspring of crossover. Moreover, we design a single- side mutation operator, which can guarantee the diversity of individual in the process of evolution.

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This paper presents a novel conditionally suboptimal filtering **algorithm** on estimation **problems** that arise in discrete **nonlinear** time-varying stochastic difference systems. The suboptimal state estimate is formed by summing of conditionally **nonlinear** filtering estimates that their weights depend only on time instants, in contrast to conditionally optimal filtering, the proposed conditionally suboptimal filtering allows parallel processing of information and reduce online computational requirements in some **nonlinear** stochastic dif- ference system. High accuracy and efficiency of the conditionally suboptimal **nonlinear** filtering are demon- strated on a numerical example.

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Fuzzy **transportation** problem is much more natural to originate them in terms of nodes and arcs, taking advantage of the special structure of the problem. The core objective in a maximum network flow technique is that seek to maximize the flow through a flow network from a single source to a single sink, while minimizing the cost of that flow. As in the MODI method, the maximum flow is evenly distributed to entire cells in our proposed method. This proposed **algorithm** often reaches an optimal solution much faster than the linear programming solvers. In future, many optimized algorithms can be developed to solve the network flow **problems** in more efficient manner.

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In an earlier work, Osman [2] introduced the notions of the stability set of the first kind and the second kind, and analyzed these concepts for parametric convex **nonlinear** programming **problems**. Osman and El-Banna [3] presented the qualitative analysis of the stability set of the first kind for fuzzy parametric multi-objective nonli- near programming **problems**. Kassem [4] dealt with the interactive stability of multi-objective **nonlinear** pro- gramming **problems** with fuzzy parameters in the constraints. Sakawa and Yano [5] introduced the concept of α - multi-objective **nonlinear** programming and α -Pareto optimality. Katagiri and Sakawa [6] dealt with fuzzy ran- dom programming, Loganathan and Sherali [7] presented an interactive cutting plane **algorithm** for determining a best-compromise solution to a multi-objective optimization problem in situations with an implicitly defined utility function. Jameel and Sadeghi [8] solved **nonlinear** programming problem in fuzzy enlivenment. Recently, Elshafei [9] and Parag [10] gave an interactive stability compromise programming method for solving fuzzy multi-objective integer **nonlinear** programming **problems**.

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The study of the TP laid the foundation for further theoretical and algorithmic development of the minimal cost network flow **problems**. **Transportation** problem is famous in operation research for its wide application in real life. This is a special kind of the network optimization **problems** in which goods are transported from a set of sources to a set of destinations subject to the supply and demand of the source and destination, respectively, such that the total cost of **transportation** is minimized.

Abstract The Dynamic Integrated Systems Optimization and Parameter Esti- mation (DISOPE) **algorithm** is an **algorithm** for solving **nonlinear** optimal con- trol **problems** and is of the gradient descent type. The updating step of DISOPE plays an important role in terminating the iterations of the **algorithm** and hence in determining its rate of convergence. In this paper, the mechanism was shown to have Newton-like properties and the order convergence established.

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In this paper, by reformulating the complementarity problem () as an implicit ﬁxed point equation based on splittings of the system matrix A, we establish an accelerated modulus- based matrix splitting iteration **algorithm** and show the convergence analysis when the involved matrix of the WNCP is a P-matrix.

Recently, Yousria Abo-elnaga et al (2012) [12] introduced a trust region globalization strategy to solve multi-objective **transportation**, assignment, and transshipment **problems**. Khurana et al (2011) [13] studied a transshipment problem with mixed constraints. Also. In (2012) Khurana et al [14] they introduced an **algorithm** for solving time minimizing capacitated transshipment problem.

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It is well known that the model of small- and medium-scale smooth functions is simple since it has many optimization algorithms, such as Newton, quasi-Newton, and bundle algorithms. Note that three algorithms fail to eﬀectively address large-scale optimization **problems** because they need to store and calculate relevant matrices, whereas the conju- gate gradient **algorithm** is successful because of its simplicity and eﬃciency.

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