Non-linear objective functions

In optimizing test intervals based on risk (or unavailability) and cost, like in many engineering optimization problems (i.e. design, reliability etc.), one normally faces multi-modal and non-linear objective functions and a variety of both linear and non-linear constraints. This results in a complex and discrete search space with regions of feasible and unfeasible solutions for a discontinuous objective functions that eventually presents local optima. 

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). 

To solve the above optimization problem, a Multiple-Objective Evolutionary Algorithms (MOEA) is embraced here. MOEA is a term employed in the Evolutionary Multi-criteria Optimization field to refer to a family of evolutionary algorithms formulated to deal with MO. MOEA are able to deal with non-continuos, non-convex and/or non-linear objectives/constraints, and objective functions possibly not explicitly known (e.g. the output of Monte Carlo simulation tuns). 

The performed analysis of problems solved by using MEIS has shown the possibilities for their reduction to convex programming (CP) problems in many important cases. Such reduction is often associated with approximation of dependences among variables. There are cases of multivalued solutions to the formulated CP problems, when the linear objective function is parallel to one of the linear part of set D y). Naturally the problems with non-convex objective functions or non-convex attainability sets became irreducible to CP. Non-convexity of the latter can occur at setting kinetic constraints by a system of linear inequalities, p>art of which is specified not for the whole region D (y), but its individual zones. 

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. 

The activation energy of the propagation reaction (Ep) and that of association equilibriim reaction (Eeq) are reported to be 6.13 Kcal/gmole and 38.6 Kcal/gmole respectively ( ). A non-linear search of the data (Equation 14) will define the constants a b c, and d. Data at 16.6 C and 21 C were incorporated with a least square objective function using Luus and Jaakola s (18) method. The analysis resulted in the following relationships ... 

The objective function, which in this case is the minimization of capital cost, is shown in constraint (3.51). Due to the non-linearity of this constraint the model becomes an MINLP model. 

Secondly, the minimum amount of intermediate storage is determined with and without the PIS operational philosophy. In both cases the production goal was set to that which was achieved when the model was solved with infinite intermediate storage. In the illustrative example a 20% reduction in the amount of intermediate storage is achieved. The design model is an MINLP model due to the non-linear capital cost objective function. This model is applied to an illustrative problem and results in the flowsheet as well as determining the capacities of the required units. 

The best-fitting set of parameters can be found by minimization of the objective function (Section 13.2.8.2). This can be performed only by iterative procedures. For this purpose several minimization algorithms can be applied, for example, Simplex, Gauss-Newton, and the Marquardt methods. It is not the aim of this chapter to deal with non-linear curve-fitting extensively. For further reference, excellent papers and books are available [18]. 

The rate expressions Rj — Rj(T,ck,6m x) typically contain functional dependencies on reaction conditions (temperature, gas-phase and surface concentrations of reactants and products) as well as on adaptive parameters x (i.e., selected pre-exponential factors k0j, activation energies Ej, inhibition constants K, effective storage capacities i//ec and adsorption capacities T03 1 and Q). Such rate parameters are estimated by multiresponse non-linear regression according to the integral method of kinetic analysis based on classical least-squares principles (Froment and Bischoff, 1979). The objective function to be minimized in the weighted least squares method is... 

The objective function (4.1) minimizes the efficiency measure E. For the smallest E obtained the slack variables are maximized. This objective hierarchy is achieved by including the "very small" parameter 8 that subordinates the maximization of the slack variables under the minimization of E. Equations (4.2) and (4.3) specify the output and input factor comparisons. The slack variables contain the surplus of output factors and underconsumption of input factors respectively as compared to the virtual DMU. The weight parameters 7ru are determined by the optimization model and describe the linear combination of real DMUs constituting the virtual DMU. Restriction (4.4) contains non-negativity constraints. 

For the complex non-linear least squares (CNLS) method, the object function, S, is defined as [5, 6]... 

A useful (also extreme) counterpart to the also idealized linear geometry is fractal geometry which plays a key role in many non-linear processes.280 281 If one measures the length of a fractal interface with different scales, it can be seen that it increases with decreasing scale since more and more details are included. The number which counts how often the scale e is to be applied to measure the fractal object, is not inversely proportional to ebut to a power law function of e with the exponent d being characteristic for the self-similarity of the structure d is called the Hausdorff-dimension. Diffusion limited aggregation is a process that typically leads to fractal structures.283 That this is a nonlinear process follows from the complete neglect of the back-reaction. The impedance of the tree-like metal in Fig. 76 synthesized by electrolysis does not only look like a fractal, it also shows the impedance behavior expected for a fractal electrode.284... 

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