In this paper, we discuss about the Augmented Lagrangian Method (ALM) and its algorithm in engineering optimization which has many other interesting properties with supporting numerical and theoretical roles. These methods have been popular for many years because, in part, of their simplicity. The ALM was proposed for equality constraints by Hestenes (1969), and Powell (1969), in the early days, it was known as the “method of multipliers.” A key reference in this area is described by Bertsekas (1982). Chapters 1-3 of that book contain a through motivation of the method that outlines its connection to other approaches. For more discussions, the general constraint optimization problems are given by Fletcher (1975) (6, section 12.2) and Polak (1997) (13, section 2.8). The mathematical technique of ALM has been developed to convert constrained optimization problem into unconstrained optimization problem so that a new problem of higher dimensions made the ALM for equality constraints neatly to the inequality constraint case (In classical problems of optimization, only equality constraints were seriously considered). Here, we discuss the development field of the ALM for general **nonlinear** **programming** (NLP) corresponding to equality or inequality constraint, in particular at the connection with optimality conditions. We also discuss the implementation of Augmented Lagrangian Algorithm (ALA) obtained from ALM in engineering optimization. Now, we define the following NLP optimization problem (P) which is addressed in this paper as follows:

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Fuzzy **nonlinear** **programming** problem (FNLPP) is useful in solving problems which are difficult, impossible to solve due to the imprecise, subjective nature of the problem formulation or have an accurate solution. In this paper, we will discuss the concepts of fuzzy decision making introduced by [1] and the maximum decision [15] that is used in NLPP to find the optimal decision (solution). This decision making used in fuzzy linear **programming** problem [8] and [7]. Furthermore, this problems has fuzzy objective function and fuzzy variables in the constraints [13], [10], and [5], where the fuzzy left and right hand side coefficients on constraints [14]. In addition, the fuzzy NLPP is used in quadratic **programming** [9, 11] which has fuzzy multiobjective function and fuzzy parameters on constraints so in our NLPP that have fuzzy properties on the inequality (≤̃ , ≥̃)and have fuzzy linear membership function. The outline of this paper is as follows: In this section 1, we state in section 2, the earlier method is briefly explained. In section 3, we give our refinement for the same problem by extending it to a general setup of **nonlinear** **programming** problem of this type.

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The main goal of this dissertation is to study the formulation and analysis of primal- dual path-following methods for **nonlinear** **programming** (NLP), which involves the mini- mization or maximization of a **nonlinear** objective function subject to constraints on the variables. Two important types of **nonlinear** program are problems with **nonlinear** equal- ity constraints and problems with **nonlinear** inequality constraints. In this dissertation, two new methods are proposed for **nonlinear** **programming**. The first is a new primal-dual path-following augmented Lagrangian method (PDAL) for solving a **nonlinear** program with equality constraints only. The second is a new primal-dual path-following shifted penalty- barrier method (PDPB) for solving a **nonlinear** program with a mixture of equality and inequality constraints. The method of PDPB may be regarded as an extension of PDAL to handle **nonlinear** inequality constraints.

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We propose an exact penalty approach for solving mixed integer **nonlinear** **programming** (MINLP) problems by converting a general MINLP problem to a finite sequence of **nonlinear** **programming** (NLP) problems with only continuous variables. We express conditions of exactness for MINLP problems and show how the exact penalty approach can be extended to constrained problems.

problem is formalized as a system of linear inequalities [1,2] With the discrete representation of the unknown extremal ( the project line ) with sufficient for practical purposes accuracy we obtain the optimization problem with a few hundred variables and several thousands of constraints – inequalities. This problem is necessary to solve many times with the refinement and detail parameters of the mathematical model because of the interrelation with other design problems [2,4,5 ]. For that reason, the question arises about how to obtain solutions in reasonable time. The standard NLP algorithms was developed for linear constraints of general form. They require too much time in solving problems of large dimension and therefore unsuitable for use in CAD. Therefore still valid implementation of mathematical methods of **nonlinear** **programming** in more efficient algorithms and programs was taken into account the peculiarities of a particular task.

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Separable **programming** deals with such **nonlinear** **programming** problem in which the objective function as well as constraints are separable. For solving Separable **Nonlinear** **Programming** Problem (SNPP) is reduced ﬁrst to Linear **Programming** Problem (LPP) by approximating each separable function by a piecewise linear function and than usual graphical, simplex method applied. A new form of Gauss elimination technique for in- equalities has been proposed for solving a Separable **Nonlinear** **Programming** Problem. The technique is useful than the earlier existing methods because it takes least time and calculations involve in are also simple. The same has been illustrated by a numerical example of SNPP.

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Surprisingly not much attention has been given to maximization of multivariable **nonlinear** **programming** problems by the scholars. So we have chosen to study maximization problems and to our pleasant surprise, the results obtained are compatible with theoretical observations. In gradient search methods, the rate of convergence is slow and the result obtained is approximate to the optimal value. But in Newton’s method and Quasi-Newton methods, the rate of convergence is faster and the results are accurate.

sponse i.e. partial non response was first discussed by Tripathi and Khare (1997). They estimate the population mean in presence Maqbool and Pirzada (2005) discuss it in two variate stratified sample surveys and find out sampling fraction for a fixed budget. In this article, problem of stratified sample surveys in presence of partial response is considered which is formulated as Bi-objective **nonlinear** **programming** problem in section 2. Sectio describes four different optimization techniques to obtain the compromise allocations of the formulated BONLPP. In section 4

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One of the conventional methods for solving zero-one **nonlinear** **programming** problem is to transform it to a linear **programming** problem. The main difficulty of this method is the very large number of variables and con- straints which increases the problem-size.

In general, all the optimization problems that are available in the current scenario, necessitates the basic need for modelling the continuous variables and discrete variables using the Mixed Integer Linear Program (MILP) and Mixed Integer Non- Linear Program (MINLP) methods. Both the objective functions and constraints in MILP problems should be considered as linear, whereas, in the case of MINLP, they should be considered as a non-linear variable. Furthermore, the optimization problems related to MINLP are classified mainly into two groups. The first one is a convex MINLP which deals with the minimization of convex objective function with a feasible convex region. The second group is a non-convex MINLP which indeed deals with the non-convex objective function. At the same time, it also deals with the values that have convex nature, but unidentified in the feasible area. Refs. [1-4] describes MILP techniques elaborately and utilised the same for solving their optimization problems. To solve the MILP problems, different advanced methods had been devised experimentally in [5]. However, many researchers have developed a few commercial and non-commercial solvers to solve both the MILP and MINLP problems. CPLEX [6], Gurobi [7], and XPRESS-MP [8] are the some of the solvers that are developed and utilised in the literature. In [9] John K.Karlof have discussed the integer **programming** theory for solving both MILP and MINLP problems. The methods used to solve MINLP problems has significant advantages when compared to the methods that are used to solve MILP and **Nonlinear** **programming** (NLP) problems. In [10], Belotti et al. have provided a comprehensive review of MINLP problems and also about the methods to solve them.

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A subproblem that has quadratic constraints is more dicult to solve than a subproblem with linear constraints, the latter being the case of Sequential Quadratic **Programming** algorithms [19]. One could of course solve the QCQP with a **nonlinear** **programming** technique. The algorithm in [2] achieves at least linear convergence on the subproblem under the conditions considered here. Since in this work a more accurate model of the constraints is considered, compared to SQP, it would be ex- pected that a smaller number of exterior iterations and thus of function evaluations is needed before completion. However, given the complexity of the subproblem, this will not necessarily results in superior runtime. Nevertheless, algorithms can be derived to deal directly with quadratically constrained problem via semidenite relaxation [16]. Devising methods that specically accommodate quadratic constraints will be the subject of future research.

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Analysis of the results of calculation of coefficients of fractional-linear model throughout the spectrum of observations allows us to conclude the feasibility of imposing restrictions on the area of permissible parameters in order to avoid getting physically incorrect data. Introduction of restrictions casts doubt on the applicability of the method of least squares to determine the coefficients selected mathematical model. The solution is offered to perform by one of the numerical methods of **nonlinear** **programming** − by random search method (RS).

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The purpose of this paper is to introduce, dem- onstrate, test, and use nonlinear programming tech- niques to predict relative chemical composition and energy val[r]

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The paper is organized as follows. In Sect. 2, we construct a general fuzzy **nonlinear** **programming** problem and formulate its Mangasarian type dual. Further, we prove duality theorems using exponential membership functions under convexity assumptions. In the next section, we illustrate a numerical example.

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This API can be used to solve **Nonlinear** **Programming** Problems using the methods referred in section II, as pre- sented in [3] and [4]. Since this API was developed in Java, originally it only allowed the development of applications using this **programming** technology. To cope with this, and to allow remote access to it, it was extended to support Web Services [5] and [6].

Abstract—Optimal control problem, which is a dynamic optimization problem over a time horizon, is a practical problem in determining control and state trajectories to minimize a cost functional. The applications of this optimization problem have been well-defined over past decades. However, the use of **nonlinear** **programming** (NLP) approach for solving optimal control problems is still a potential research topic. In this paper, a formulation of NLP model for optimal control problems is done. In our model, a class of the difference equations, which is nature in discrete time or is discretized by using the approximation scheme, is considered. Based on the control parameterization approach, the optimal control problem is generalized in the canonical form as a mathematical optimization problem. The control variables are defined as control parameters and their values are then calculated. In doing so, the gradient formula of the cost function and the corresponding constraints is derived and is presented as an algorithm. The optimal solution of NLP model approximates closely to the true solution of the original optimal control problem at the end of the computation procedure. For illustration, four examples are studied and the results show the efficiency of the approach proposed.

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We introduce some approximate optimal solutions and a generalized augmented La- grangian in **nonlinear** **programming**, establish dual function and dual problem based on the generalized augmented Lagrangian, obtain approximate KKT necessary optimal- ity condition of the generalized augmented Lagrangian dual problem, prove that the ap- proximate stationary points of generalized augmented Lagrangian problem converge to that of the original problem. Our results improve and generalize some known results. Copyright © 2007 Zhe Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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The paper organized as follow. In the next section we introduce **nonlinear** **programming** problem with bound constraints and its equivalent formulation is described. In section 3, a feedback neural network model with circuit implementation is proposed. Section 4 discusses the stability of the proposed network and analyzes its global convergence. Extension of a proposed neural network for solving **nonlinear** **programming** problems with hybrid constraints is given in Section 5. In Section 6, numerical examples are simulated to show the reasonableness of our theory and demonstrate the high performance of proposed neural networks approach. A comparative analysis is presented in Section 7. Some conclusions are summarized in the last section.

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Linearization methods can be used converting a NLP problem into a LP problem. In this process, extra variables and constraints are in- troduced to construct the original problem. Various methods have been proposed in the literature by linearizing a NLP problem [14], [11]. Sequential Linear **Programming** (SLP) which is one of the direct methods solves NLP problems approximately, and uses a se- ries of LP problems generated by using first order Taylor series expansions of objective functions and constraints. Byrd and No- cedal [3] have presented a new active-set, trust-region algorithm for large-scale optimization using SLP techniques to solve the NLP problems approximately.

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Bozóki et al. [10] developed a Non-Linear **Programming** (NLP) model to minimize max . Since they could not provide an efficient algorithm to solve this NLP model, they proposed the iterative method of cyclic coordinates in which at each step only one variable is allowed to be optimized and the other variables are fixed to the initial values: the optimal solution is then computed by a univariate minimization algorithm. With reference to the local optimality of their proposed algorithm, there was no guaranty for reaching the global optimality of minimizing inconsistency ratio.

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