Derivatives optimization, discontinuities

There are different ways of implementing the cut-off approximation. The simplest is to neglect all contributions if the distance is larger than the cut-off. This is in general not a very good method as the energy function becomes discontinuous. Derivatives of the energy function also become discontinuous, which causes problems in optimization... 

In carrying out analytical or numerical optimization you will find it preferable and more convenient to work with continuous functions of one or more variables than with functions containing discontinuities. Functions having continuous derivatives are also preferred. Case A in Figure 4.1 shows a discontinuous function. Is case B also discontinuous ... 

A discontinuity in a function may or may not cause difficulty in optimization. In case A in Figure 4.1, the maximum occurs reasonably far from the discontinuity which may or may not be encountered in the search for the optimum. In case B, if a method of optimization that does not use derivatives is employed, then the kink in /(x) is probably unimportant, but methods employing derivatives might fail, because the derivative becomes undefined at the discontinuity and has different signs on each side of it. Hence a search technique approaches the optimum, but then oscillates about it rather than converges to it. 

As mentioned earlier, nonlinear objective functions are sometimes nonsmooth due to the presence of functions like abs, min, max, or if-then-else statements, which can cause derivatives, or the function itself, to be discontinuous at some points. Unconstrained optimization methods that do not use derivatives are often able to solve nonsmooth NLP problems, whereas methods that use derivatives can fail. Methods employing derivatives can get stuck at a point of discontinuity, but the function-value-only methods are less affected. For smooth functions, however, methods that use derivatives are both more accurate and faster, and their advantage grows as the number of decision variables increases. Hence, we now turn our attention to unconstrained optimization methods that use only first partial derivatives of the objective function. 

In practice, even though the use of a cut-off introduces only small disparities in the energy, the discontinuity of these disparities can cause problems for optimizers. A more stable approach is to use a switching function which multiplies the van der Waals interaction and causes it (and possibly its first and second derivatives) to go smoothly to zero at some cut-off distance. This function must, of course, be equal to 1 at short distances. 

Nonsmooth or nondifferentiable optimization plays an important role in large-scale programming and addresses mathematical programming problems in which the functions involved have discontinuous first derivatives. Thus, classical methods that rely on gradient information fail to solve these problems, and alternative nonstandard approaches must be used. These alternative methods include subgradient methods and bundle methods. The interested reader is referred to Shor (1985), Zowe (1985), and Fletcher (1987, pp. 357 14). 

Belonging to the class defined in Section 5.2.1 (p. 126), the optimal control u could be discontinuous in the interval [0, tf]. Therefore, we need to show that even when u is discontinuous, it minimizes the Hamiltonian. We will derive this result in the following three steps ... 

The optimal control problem represents one of the most difficult optimization problems as it involves determination of optimal variables, which are vectors. There are three methods to solve these problems, namely, calculus of variation, which results in second-order differential equations, maximum principle, which adds adjoint variables and adjoint equations, and dynamic programming, which involves partial differential equations. For details of these methods, please refer to [23]. If we can discretize the whole system or use the model as a black box, then we can use NLP techniques. However, this results in discontinuous profiles. Since we need to manipulate the techno-socio-economic poHcy, we can consider the intermediate and integrated model for this purpose as it includes economics in the sustainabiHty models. As stated earlier, when we study the increase in per capita consumption, the system becomes unsustainable. Here we present the derivation of techno-socio-economic poHcies using optimal control appHed to the two models. 

With the feasible path approach the optimization algorithm automatically performs case studies by variing input data. There are several drawbacks the process equations (32c) have to be solved every time the performance index is evaluated. Efficient gradient-based optimization techniques can only be used with great difficulties because derivatives can only be evaluated by perturbing the entire flowsheet with respect to the decision variables. This is very time consuming. Second, process units are often described by discrete and discontinuous relations or by functions that may be nondifferentiable at certain points. To overcome these problems quadratic module models can be... 

The algorithm was originally developed for use when the controller scan interval (ts) is signihcant compared to the process dynamics. This makes it suitable for use if the PV is discontinuous, such as that from some types of on-stream analysers. Analysers are a major source of deadtime. They may located well downstream of the MV and their sample systems and analytical sequence can introduce a delay. An optimally tuned PID controller would then have a great deal of derivative action. However this wUl produce the spiking shown in Figure 7.6. 

Because of the existence of first-derivative discontinuities when internal boundary conditions are specified, flows past multi-element airfoils in the aerospace industry, as cases in point, are simulated using singly connected computational domains such as that shown in Figure 9-12 which displace the source of the discontinuity to a computational boundary. Whereas the modeling by Sharpe and Anderson (1991) of wells as internal fixed points produces undesired discontinuities, aerospace methods produce meshes where all metrics and derivatives are continuous. Sharpe and Anderson also embed their elliptic operators in first-order, time-like systems. The complete process yields shocks in some instances, perhaps because the embedded system possesses nonlinear hyperbolic properties. Jameson (1975) has shown how various transient diffusive systems can be derived to host relaxation-based techniques these methods are further optimized for computational speed. 

Geometry optimization problems for molecules in the context of standard all-atom force fields in computational chemistry are typically of the multivariate, continuous, and nonlinear type. They can be formulated as constrained (as in adiabatic relaxation) or unconstrained. Discontinuities in the derivatives may be a problem in certain formulations involving truncation, such as of the nonbonded terms (see Section 7). 

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