Nonlinear variable bilinear

In practice, mode-mode coupling calculations are almost always performed with bilinear variables alone, and with one or at most two different bilinear products at that. We are aware of no attempt to justify such extreme simplification. Our feeling is that in many cases the use of bilinear variables alone is completely correct, but that in other cases the possible importance of other nonlinear variables deserves further study. All the problems that we discuss in detail will involve only bilinear variables. 

Bilinear, and higher, nonlinear variables containing Sn are also negligi-ble near a critical point. In estimating critical point ( ) behavior of various contributions to we have been approximating various correlation... 

The above formulation is a mixed-integer nonlinear programming MINLP model and has the following characteristics. The binary variables appear linearly and separably from the continuous variables in both the objective and constraints, by defining a new set of variables Wj = tij/Tu and including the bilinear constraints WiTu = Uj- The continuous variables nitBi,Tu,Wi appear nonlinearly. In particular, we have bilinear terms of riiBi in the objective and constraints, bilinear terms of niT i and WiTu in the constraints. The rest of the continuous variables Vj, (V fc)ra, Vfjk appear linearly in the objective function and constraints. 

Note that the first set of constraints are linear while the rest are nonlinear. Also note that in the objective function we have sum of products of integers times concave functions, while the second set of constraints has bilinear terms of integer and continuous variables. The third set of constraints has sum of fractional terms. 

Remark 1 The resulting optimization model is an MINLP problem. The objective function is linear for this illustrative example (note that it can be nonlinear in the general case) and does not involve any binary variables. Constraints (i), (v), and (vi) are linear in the continuous variables and the binary variables participate separably and linearly in (vi). Constraints (ii), (iii), and (iv) are nonlinear and take the form of bilinear equalities for (ii) and (iii), while (iv) can take any nonlinear form dictated by the reaction rates. If we have first-order reaction, then (iv) has bilinear terms. Trilinear terms will appear for second-order kinetics. Due to this type of nonlinear equality constraints, the feasible domain is nonconvex, and hence the solution of the above formulation will be regarded as a local optimum. 

Remark 1 The mathematical model is an MINLP problem since it has both continuous and binary variables and nonlinear objective function and constraints. The binary variables participate linearly in the objective and logical constraints. Constraints (i), (iv), (vii), and (viii) are linear while the remaining constraints are nonlinear. The nonlinearities in (ii), (iii), and (vi) are of the bilinear type and so are the nonlinearities in (v) due to having first-order reactions. The objective function also features bilinear and trilinear terms. As a result of these nonlinearities, the model is nonconvex and hence its solution will be regarded as a local optimum unless a global optimization algorithm is utilized. 

This second-level modeling of the feedback mechanisms leads to nonlinear models for processes, which, under some experimental conditions, may exhibit chaotic behavior. The previous equation is termed bilinear because of the presence of the b [y (/,)] r (I,) term and it is the general formalism for models in biology, ecology, industrial applications, and socioeconomic processes [601]. Bilinear mathematical models are useful to real-world dynamic behavior because of their variable structure. It has been shown that processes described by bilinear models are generally more controllable and offer better performance in control than linear systems. We emphasize that the unstable inherent character of chaotic systems fits exactly within the complete controllability principle discussed for bilinear mathematical models [601] additive control may be used to steer the system to new equilibrium points, and multiplicative control, either to stabilize a chaotic behavior or to enlarge the attainable space. Then, bilinear systems are of extreme importance in the design and use of optimal control for chaotic behaviors. We can now understand the butterfly effect, i.e., the extreme sensitivity of chaotic systems to tiny perturbations described in Chapter 3. 

Water allocation problems have nonlinearities and non-convexities due to bilinear terms. To address this issue we propose to discretize one of the variables of the bilinear terms. As a result an MILP model is generated, which provides a lower bound. To reduce the gap between this lower bound and the upper bound (a feasible solution found using the original NLP model), an interval elimination procedure is proposed. As a result, the feasible space shrinks after each iteration and the global optimum is identified. We illustrate the methodology for minimum water allocation problems. 

Equations (9) and (10) constitute the fundament of all QSAR studies. Since 1964, they have remained essentially unchanged, with the exception of two minor modifications. Improvements resulted from the combination of Hansch equations with indicator variables [22], which may be considered as a mixed Hansch/Free-Wilson model (Eq. (11)) [23], and from the formulation of a theoretically derived nonlinear model for transport and distribution of drugs in a biological system, the bilinear model (Sec. 4 Eq. (30)) [24] ... 

In addition, the biological data should cover a range of at least one, better two or even more logarithmic units they should be well distributed over the whole distance i.e., no clustering of activity values should occur, as discussed in chapter 2). Also the physicochemical parameters should be spread over a certain range and should be more or less evenly distributed if a certain parameter has identical values for all but one or two objects, then this parameter must be considered as a hidden indicator variable and should be replaced by such a term. In parabolic and especially in bilinear equations the nonlinear parameter should cover a range of at least two logarithmic units, in order to justify the presence of a nonlinear term. 

For certain cases, it is possible to represent the time derivative of a linear variable exactly, or at least quite reasonably, in terms of a bilinear variable. No expansion or limiting process is involved the bilinear variable just happens to be well suited to express or /k. Under these circumstances, the nonlinear Langevin equation may be deduced from simple physical arguments. The cleanest examples of problems where bilinear variables arise in a fairly obvious way are diffusion problems. In any diffusion problem, self, mutual, or whatever, the linear variable of interest is a concentration, nZ [see Eq. (22)]. The time derivative of a concentration is a momentum density/mass. 

Only bilinear variables with intermediate wave vector kc are to be included in the nonlinear Langevin equation, while the relations just derived contain sums over all wave vectors. Let us split up the sum in Eq. (67) and use the definition of the diffusion flux = ik /t, to obtain... 

Sometimes only a part of the nonlinear Langevin equation can be rigorously expressed in bilinear form. Consider an arbitrary conserved variable. 

The underlying notion in bilinear modeling is that something causes the systematic variabilities in the X data. But we may not correctly know what it is there may be surprises in the data due to unexpected interferents, chemical interactions, nonlinear responses, etc. An approximate model of the subspace spanned by these phenomena in X is created. This X model is used for stabilizing the calibration modeling. The PLS regression primarily models the most dominant and most y-relevant of these X phenomena. Thus neither the manifest measured variables nor our causal assumptions about physical laws are taken for granted. Instead we tentatively look for systematic patterns in the data, and if they seem reasonable, we use them in the final calibration model. 

When any system or process is subjected to large changes, it appears nonlinear. In the present context, this means that the relation between X and Y becomes nonlinear. Also the relations between the X variables may become nonlinear, as may the relations between the Y variables. Even so, the X and Y matrices can always be approximated by the bilinear model (equations 6b and 3a). Hence, nonlinear situations can be modeled by non-linear PLS models, where the nonlinearities are expressed as nonlinear relations between the X scores and the Y scores Ua- These nonlinearities can be modeled as polynomial nonlinearities (quadratic, cubic, etc.), spline functions, or other nonlinear forms (e.g., bi-exponential). 

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