### Table 2.- Schematic classification of stochastic optimization methods. Seminal references and selected examples of their application in process engineering are also given.

"... In PAGE 12: ...riori assumptions or pre-processing work. . There are at least four different classes of approaches which were apparently generated independently. A schematic classification, together with key seminal references and selected examples of their application in process engineering are given in Table2 . Some more details about each type follows: g183g32 Adaptive stochastic methods were developed in the domains of electrical and control engineering and applied mathematics (e.... In PAGE 13: ...stable configuration as slow cooling of a metal takes place). Apart from those methods presented in Table2 , during recent years a number of other (so called) meta-heuristics have been presented, mostly based on other biological or physical phenomena, and with combinatorial optimization as their original domain of application. Examples of these more recent methods are Taboo Search (TS), Ant Colony Optimization (ACO) (Dorigo, Maniezzo amp; Colorni, 1996; Bonabeau, Dorigo amp; Theraulaz, 2000; Jayaraman, Kulkarni amp; Shelokar, 2000) and particle swarm methods (Bonabeau, Dorigo amp; Theraulaz, 1999).... ..."

### Table 5. Full state and output feedback control laws with sensor numbers.

1998

"... In PAGE 20: ... The ideal case is to eliminate the acoustic sensors entirely and use the model with coupling along with structural data to reconstruct the acoustic state. For the simulations presented here, three sensor con gurations were considered as sum- marized in Table5 . In all cases, the number of sensors measuring the potential was taken to be N = 0 in (2.... In PAGE 22: ... In comparing the rms values and time plots of the three compensators, it is noted that the performance of Compensator I with measurements of pressure, displacement and velocity is only 1-2 dB better than that of Compensators II and III. Recall from Table5 that Compensator III employs only 5 velocity sensors for the actual state reconstruction. The pre- computed gains and coupled model provide the remaining information required for accurate state estimation and control computation.... In PAGE 25: ... Hence while signi cant attenuation is achieved throughout most of the cavity, optimization issues concerning patch number and orientation should be investigated to attain global attenuation. Similar results obtained with Compensators I and III described in Table5 are plotted in Figure 9. The small patch having radius R=12 was employed as an actuator and rms sound pressure levels along axis 2 are reported in the gure.... In PAGE 27: ... The example we consider in this section reinforces the tenet held by many acousticians that this strategy is not e ective in general and should be used only for certain exogenous frequencies (see, for example, [13, 20]). It also illustrates the bene ts of utilizing a compensator for the coupled system which employs only structural sensors (see Compensator III of Table5 ) rather than a purely structural controller. For the structural acoustic system in this work, a purely structural controller would be designed for the discretized plate model quot; KP 0 0 MP # quot; _ #(t) #(t) # = quot; 0 KP ?KP ?CP # quot; #(t) _ #(t) # + quot; 0^ B # u(t) + quot; 0 ^ g(t) # + ^ D (t) where again, #(t) contains the generalized Fourier coe cients for displacement and MP ; KP and CP are the mass, sti ness and damping matrices for the plate (see Section 2).... ..."

Cited by 2

### Table 1. outlines a typical EA. A population of individuals is randomly initialized and then evolved from generation to generation by repeated applications of evaluation, selection, mutation and recombination. In the selection step, the algorithm selects the parents of the next generation. The population is subjected to environmental pressure , which means the selection of the fittest individuals. The most important automated selection methods are Stochastic Uniform Sampling, Tournament Selection, Fitness Ranking Selection and Fitness Proportional Selection. After the selection of the individuals, the new individuals of the next generation (also called offspring) are created by recombination and mutation.

"... In PAGE 1: ... The key of EA is that the fitness also determines how successful the individual will be at propagating its genes (its code) to subsequent generations. Table1 . A typical evolutionary algorithm procedure EA; { Initialize population; Evaluate all individuals; while (not terminate) do { Select individuals; Create offspring from selected individuals using Recombination and Mutation; Evaluate offspring; Replace some old individuals by some offspring;} } In practical system identification, process optimization or controller design it is often desirable to simultaneously handle several objectives and constraints.... ..."

### Table 10: Parameters for Stochastic Experiments

"... In PAGE 32: ... In these experiments, we do not consider LAMA because its control structure is not suited to stochastic problems. The data parameters in Table10 are used to construct the stochastic experiment data sets and are similar to the #0Crst set of deterministic experiments. The main exception is the single period time windows on the tasks that enables an optimal posterior bound to be calculated.... ..."

### Table 2. Discrete Time Stochastic Volatility Models Estimated from Stock Prices: Optimized Value of the Criterion for the Non- linear Nonparametric Score. Model

1995

"... In PAGE 15: ... Their work is summarized in two tables reproduced here as Tables 1 and 2. ||||||||||||{Table 1 about here ||||||||||||{ ||||||||||||{ Table2 about here ||||||||||||{ Table 1 shows the optimized values of the EMM objective function scaled to follow the chi- squared distribution, as described in Section 2. From the top panel of the table it is seen that the standard stochastic volatility model with Gaussian errors... In PAGE 18: ...Table2 displays the objective function surface for versions of the stochastic volatility model against the Nonlinear Nonparametric Score. The standard model is overwhelmingly rejected.... ..."

### Table 1. Discrete Time Stochastic Volatility Models Es- timated from Stock Prices: Optimized Value of the Cri- terion for the Semiparametric ARCH Score. Model

1995

"... In PAGE 15: ... Their work is summarized in two tables reproduced here as Tables 1 and 2. ||||||||||||{ Table1 about here ||||||||||||{ ||||||||||||{Table 2 about here ||||||||||||{ Table 1 shows the optimized values of the EMM objective function scaled to follow the chi- squared distribution, as described in Section 2. From the top panel of the table it is seen that the standard stochastic volatility model with Gaussian errors... In PAGE 15: ... Their work is summarized in two tables reproduced here as Tables 1 and 2. ||||||||||||{Table 1 about here ||||||||||||{ ||||||||||||{Table 2 about here ||||||||||||{ Table1 shows the optimized values of the EMM objective function scaled to follow the chi- squared distribution, as described in Section 2. From the top panel of the table it is seen that the standard stochastic volatility model with Gaussian errors... In PAGE 16: ... The objective function is so at for values of the degrees of freedom parameter 2 (10; 20) that the optimizer gets stuck and makes no progress when it sees as free parameter along with the rest. Thus, in the second panel of Table1 the value of the objective function is successively xed at = 10; 15; 20; 25. Spline yt ? y = c1(yt?1 ? y) + c2(yt?2 ? y) + exp(wt)ryTz(zt) Tz(zt) = bz0 + bz1zt + bz2z2 t + bz3I+(zt)z2 t wt ? w = a1(wt?1 ? w) + a2(wt?2 ? w) + rw~ zt The idea is to allow a deviation from the Gaussian speci cation by transforming zt through a di erentiable quadratic spline that has one knot at zero.... ..."

### Table 2: Non-convex quarticly constrained optimization problem for hierarchy and policy discovery in bounded stochastic recursive controllers.

in Abstract

"... In PAGE 5: ... 3.3 Algorithms Since the problem in Table2 has non-convex (quartic) constraints in Eq. 5 and 6, it is difficult to solve.... In PAGE 5: ... 5 and 6, it is difficult to solve. We consider three approaches inspired from the techniques for non-hierarchical controllers: Non-convex optimization: Use a general non-linear solver, such as SNOPT, to directly tackle the optimization problem in Table2 . This is the most convenient approach, however a globally optimal solution may not be found due to the non-convex nature of the problem.... In PAGE 7: ... 4 Experiments We report on some preliminary experiments with three toy problems (paint, shuttle and maze) from the POMDP repository3. We used the SNOPT package to directly solve the non-convex optimization problem in Table2 and bounded hierarchical policy iteration (BHPI) to solve it iteratively. Table 3 reports the running time and the value of the hierarchical policies found.... ..."

### Table 2: Non-convex quarticly constrained optimization problem for hierarchy and policy discovery in bounded stochastic recursive controllers.

in Abstract

"... In PAGE 5: ... 3.3 Algorithms Since the problem in Table2 has non-convex (quartic) constraints in Eq. 5 and 6, it is difficult to solve.... In PAGE 5: ... 5 and 6, it is difficult to solve. We consider three approaches inspired from the techniques for non-hierarchical controllers: Non-convex optimization: Use a general non-linear solver, such as SNOPT, to directly tackle the optimization problem in Table2 . This is the most convenient approach, however a globally optimal solution may not be found due to the non-convex nature of the problem.... In PAGE 7: ... 4 Experiments We report on some preliminary experiments with three toy problems (paint, shuttle and maze) from the POMDP repository3. We used the SNOPT package to directly solve the non-convex optimization problem in Table2 and bounded hierarchical policy iteration (BHPI) to solve it iteratively. Table 3 reports the running time and the value of the hierarchical policies found.... ..."

### Table 2: Non-convex quarticly constrained optimization problem for hierarchy and policy discovery in bounded stochastic recursive controllers.

"... In PAGE 5: ... 3.3 Algorithms Since the problem in Table2 has non-convex (quartic) constraints in Eq. 5 and 6, it is difficult to solve.... In PAGE 5: ... 5 and 6, it is difficult to solve. We consider three approaches inspired from the techniques for non-hierarchical controllers: Non-convex optimization: Use a general non-linear solver, such as SNOPT, to directly tackle the optimization problem in Table2 . This is the most convenient approach, however a globally optimal solution may not be found due to the non-convex nature of the problem.... In PAGE 7: ... 4 Experiments We report on some preliminary experiments with three toy problems (paint, shuttle and maze) from the POMDP repository3. We used the SNOPT package to directly solve the non-convexoptimization problem in Table2 and bounded hierarchical policy iteration (BHPI) to solve it iteratively. Table 3 reports the running time and the value of the hierarchical policies found.... ..."