Stochastic optimization

Hashemi and Epstein (1982) linearized the set of ordinary differential equations (ODEs) resulting from the application of the method of moments on an MSMPR crystallizer model and used singular value decomposition to define controllability and observability indices. These indices aid in selecting measurements and manipulated and control variables. Myerson et al. (1987) suggested the manipulation of the feed flow rate and the crystallizer temperature according to a nonlinear optimal stochastic control scheme with a nonlinear Kalman filter for state estimation. 

In the areas of process identification (Ljung, 1987), optimal stochastic controller design (Box and Jenkins, 1976) and controller performance assessment (Harris, 1989), it is often desirable or necessary to determine a model... 

Note The segmentation operation yields a near-optimal estimate x that may be used as initialization point for an optimization algoritlim that has to find out the global minimum of the criterion /(.). Because of its nonlinear nature, we prefer to minimize it by using a stochastic optimization algorithm (a version of the Simulated Annealing algorithm [3]). 

Let u be a vector valued stochastic variable with dimension D x 1 and with covariance matrix Ru of size D x D. The key idea is to linearly transform all observation vectors, u , to new variables, z = W Uy, and then solve the optimization problem (1) where we replace u, by z . We choose the transformation so that the covariance matrix of z is diagonal and (more importantly) none if its eigenvalues are too close to zero. (Loosely speaking, the eigenvalues close to zero are those that are responsible for the large variance of the OLS-solution). In order to liiid the desired transformation, a singular value decomposition of /f is performed yielding... 

Wlren the door is open, the optimal net flux into the store isgiven by equation (C2.14.7). It may be that the stochastically gated diffusion treated by Szabo et aJ [47], see also [48] is a good representation of typical biological storage reactions (C2.14.8). 

Application of a Stochastic Path Integral Approach to the Computations of an Optimal Path and Ensembles of Trajectories ... 

A Stochastic Approach to Combinatorial Optimization and Neural Computing. John Wiley Sons, New York, 1989. 

Grimble, M.J. and Johnson, M.A. (1988) Optimal Control and Stochastic Estimation Theory and Application, Vols 1 and 2, John Wiley Sons, Chichester, UK. 

Whatever model is used to describe an operations research problem, be it a differential equation, a mathematical program, or a stochastic process, there is a natural tendency to seek a maximum or a minimum with a certain purpose in mind. Thus, one often finds optimization problems imbedded in the models of operations research. 

Most drug-like molecules adopt a number of conformations through rotations about bonds and/or inversions about atomic centers, giving the molecules a number of different three-dimensional (3D) shapes. To obtain different energy minimized structures using a force field, a conformational search technique must be combined with the local geometry optimization described in the previous section. Many such methods have been formulated, and they can be broadly classified as either systematic or stochastic algorithms. 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

The problem of multivariable optimization is illustrated in Figure 3.4. Search methods used for multivariable optimization can be classified as deterministic and stochastic. 

Stochastic search methods. In all of the optimization methods discussed so far, the algorithm searches the objec-... 

It is also worth noting that the stochastic optimization methods described previously are readily adapted to the inclusion of constraints. For example, in simulated annealing, if a move suggested at random takes the solution outside of the feasible region, then the algorithm can be constrained to prevent this by simply setting the probability of that move to 0. 

Stochastic optimization methods described previously, such as simulated annealing, can also be used to solve the general nonlinear programming problem. These have the advantage that the search is sometimes allowed to move uphill in a minimization problem, rather than always searching for a downhill move. Or, in a maximization problem, the search is sometimes allowed to move downhill, rather than always searching for an uphill move. In this way, the technique is less vulnerable to the problems associated with local optima. 

Having evaluated the system performance for each setting of the six variables, the variables are optimized simultaneously in a multidimensional optimization, using for example SQP, to maximize or minimize an objective function evaluated at each setting of the variables. However, in practice, many models tend to be nonlinear and hence a stochastic method can be more effective. 

One of the approaches that can be used in design is to carry out structural and parameter optimization of a superstructure. The structural optimization required can be carried out using mixed integer linear programming in the case of a linear problem or mixed integer nonlinear programming in the case of a nonlinear problem. Stochastic optimization can also be very effective for structural optimization problems. 

Marcoulaki E.C and Kokossis A.C (1999) Scoping and Screening Complex Reaction Networks Using Stochastic Optimization, AIChE J, 45 1977. 

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