Non-linear programming problem

If the resulting simultaneous equations are linear, their solution is a straightforward matter, but it is easy to see that in our case the equations will not be linear, and thus the minimization of equation (5) subject to equation (9) is an example of a non-linear programming problem. 

Vasant, P. Hybrid LS-SA-PS methods for solving fuzzy non-linear programming problems. Math. Comput. Model. 57(1-2), 180-188 (2013)... 

Costa L. and Oliveira P. (2001). Evolutionary algorithms apporach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering 25, 257-266. 3.3.1... 

Sabri and Beamon (2000) consider a four-stage (suppliers, plants, DCs, and customer zones) problem with both strategic (plant and DC locations) and tactical decisions. Demands for products are deterministic and have to be satisfied. There are fixed costs associated with DCs and transportation links between DCs and customer zones. Production cost is assumed to be linear. Two objectives are considered (a) Total cost, (b) Volume flexibility (difference between plant capacity and its utilization, and difference between DC capacity and its utilization). The strategic sub-model of the problem is formulated as a multi-objective MIR Two operational sub-models (suppliers, production) are formulated and solved as a non-linear programming problem. An overall iterative procedure is proposed which combines the strategic sub-model with the operational sub-models. 

The TMP design optimization is a MINLP (mixed-integer non-linear programming) problem since it has both a discrete, A Ref, and a continuous, Fu,nk, design parameter. The operational optimization subproblem has integer decision variables (number of active refiners in time) affecting the continuous state of intermediate tank volume through process dynamics. The tank volume is constrained to stay between a minimum and a maximum volume. 

Linear Programming.28—A linear programming problem as defined in matrix notation requires that a vector x 0 (non-negativity constraints) be found that satisfies the constraints Ax <, b, and maximizes the linear function cx. Here x = (xx, , xn), A = [aiy] (i = 1,- -,m j = 1,- , ), b - (61 - -,bm), and c = (cu- -,c ) is the cost vector. With the original (the primal) problem is associated the dual problem yA > c, y > 0, bij = minimum, where y yx,- , ym)-A duality theorem 29 asserts that if either the primal or the dual has a solution then the values of the objective functions of both problems at the optimum are the same. It is a relatively easy matter to obtain the solution vector of one problem from that of the other. 

Hence, through the LCAO expansion we have translated the non-linear optimization problem, which required a set of difficult to tackle coupled integro-differential equations, into a linear one, which can be expressed in the language of standard linear algebra and can easily be coded into efficient computer programs. 

Jang, S. S., Josepth, B and Mukai, H. (1986). Comparison of two approaches to on-line parameter and state estimation problem of non-linear systems. Ind. Eng. Chem. Process Des. Dev. 25, 809-814. Jazwinski, A. H. (1970). Stochastic Processes and Filtering Theory. Academic Press, New York. Liebman, M. J., Edgar, T. F., and Lasdon, L. S. (1992). Efficient data reconciliation and estimation for dynamic process using non-linear programming techniques. Comput. Chem. Eng. 16, 963-986. McBrayer, K. F., and Edgar, T. F. (1995). Bias detection and estimation on dynamic data reconciliation. J Proc. Control 15, 285-289. 

As is the case for standard orthogonal-orbital MCSCF calculations, the optimization of VB wavefimctions can be a complicated task, and a program such as CASVB should therefore not be treated as a black box . This is true, to a greater or lesser extent, for most procedures that involve orbital optimization (and, hence, non-linear optimization problems), but these difficulties are compounded in valence bond theory by the... 

The problem described above is a linear programming problem - that is, an optimization problem with a linear objective function and linear constraints. Here the linear object is quite simple (maximize J42). The linear constraints include both linear equalities (SJ = 0) and inequalities (7, >0) yet both sets of constraints are linear in the sense that they involve no non-linear operations on the unknowns (J). 

Abstract. In the present paper the problem of reuse water networks (RWN) have been modeled and optimized by the application of a modified Particle Swarm Optimization (PSO) algorithm. A proposed modified PSO method lead with both discrete and continuous variables in Mixed Integer Non-Linear Programming (MINLP) formulation that represent the water allocation problems. Pinch Analysis concepts are used jointly with the improved PSO method. Two literature problems considering mono and multicomponent problems were solved with the developed systematic and results has shown excellent performance in the optimality of reuse water network synthesis based on the criterion of minimization of annual total cost. 

The formulation of the batch plant retrofit problem generally involves either deterministic methods based on mathematical programming such as LP (Linear Programming), NLP (Non-Linear Programming), MILP or MINLP or stochastic ones such as evolutionary algorithms. The non-linearity aspect may result from the nature of the constraints and/or the objective function. The mixed feature means that continuous and integer variables are involved in the formulation. 

Kocis, G.R. and Grossmann, I.E. (1988) Global optimisation of nonconvex mixed-integer non linear programming (MINLP) problems in process synthesis. Industrial Engineering Chemistry Research, 27 (8), 1407-1421. 

It turns out that these constraints can be formulated as a parametrized linear programming problem [10], in which the feasible set is formed by the context of the result variable and in which the cost function is defined by the lexicographical ordering of the iterations. This latter fact is the key to the solution. Not only are we allowed to define such a non-linear cost function, but it is also smoothly integrated in the way an LP algorithm works. The LP algorithm we are using is called PIP [9]. 

Since the IFT is covered in detail in many non-linear programming books and its application to the GT problems is essentially the same, we do not delve further into this matter. In many practical problems, if JT 0 then it is instrumental to multiply both sides of the expression (2.6) by H. That is justified because the Hessian is assumed to have a non-zero determinant to avoid the cumbersome task of inverting the matrix. The resulting expression is a system of n linear equations which have a closed form solution. See Netes-sine and Rudi (2001b) for such an application of the IFT in a two-player game and Bernstein and Federgruen (2000) in n—player games. 

The general non-linear programming (NLP) problem can be formulated in a similar way as follows ... 

As explained by Cook and Zhu (2005), the non-linear model yields infinite number of solutions. This is due to the fact that if (u, v ) is optimal, then (au, av ) is also optimal for a > 0. The non-linear formula can be converted to the equivalent linear programming problem via the transformation developed by Charnes and Cooper (1962) for linear... 

The gasoline blend problem is formulated as a Mixed Integer Non-Linear Programming (MINLP) problem, where the fuel composition is to be optimised, subject to product attributes (target properties) and process specifications. Considering the multiple types of eonstraint equations, the general gasoline blend problem is formulated as ... 

Non-linear programming technique (NLP) is used to solve the problems resulting from syntheses optimisation. This NLP approach involves transforming the general optimal control problem, which is of infinite dimension (the control variables are time-dependant), into a finite dimensional NLP problem by the means of control vector parameterisation. According to this parameterisation technique, the control variables are restricted to a predefined form of temporal variation which is often referred to as a basis function Lagrange polynoms (piecewise constant, piecewise linear) or exponential based function. A successive quadratic programming method is then applied to solve the resultant NLP. 

Binary variables are used to represent the occurrence of molecular structural groups (e.g. -CH3, -CHO, -OH. ..) found in the group contribution correlations. This allows molecules to be generated according to a set of structural and chemical feasibility constraints. In addition, a variety of pure component physical and environmental property prediction equations, non-ideal multi-component vapour-liquid equilibrium equations (UNIFAC), process operational constraints and an aggregated process model form part of the overall procedure. Finally, the solvent identification task is solved as a mixed integer non-linear programming (MINLP) problem (Buxton et ai, 1999). 

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