Optimization penalty

Type of Site Initial Penalty Optimized Penalty... 

It might be possible to reduce the inventory significantly by changing reactor conversion and recycle inert concentration without a large cost penalty if the cost optimization profiles are fairly flat. 

The general constrained optimization problem can be considered as minimizing a function of n variables F(x), subject to a series of m constraints of the fomi C.(x) = 0. In the penalty fiinction method, additional temis of the fomi. (x), a.> 0, are fomially added to the original fiinction, thus... 

The perfomiance of the penalty fiinction algoritlun is heavily influenced by the value chosen for a.. The larger the value of o. the better the constraints are satisfied but the slower the rate of convergence. Optimizations with very high values of a, encounter severe convergence problems. However, the method is very general and... 

Fig. 10.16 Finding the optimal sequence alignment using dynamic programming with a scoring scheme in which a match scores 1, a mismatch scores —1 and the gap penalty is —2.

The penalty function approach adds a tenn of tire type k r — ro) to the function to be optimized. The variable r is constrained to be near the target value ro, and the force constant k describes how important the constraint is compared with the unconstrained optimization. By making k arbitrary large, tire constraint may be fulfilled to any given... 

A related problem in eomposites is the need to design optimal fiber orientations for a eomposite part given the set of stress vectors and levels to whieh the part will be subjected. These design eonsiderations would be useful in designing airframe eomponents sueh as parts for the tail, wing, or fuselage. A similar problem is assessment of the peiformanee penalties that might result from imperfections in manufacture. 

The second term in Eq. (1) is a penalty term representing constraints on the control field e(r) via a functional / and is extremely important in determining the outcome of the optimization. The most common penalty term, and that first introduced by Rabitz and co-workers [41], is... 

In early work in the optimal control theory design of laser helds to achieve desired transformations, the optimal control equations were solved directly, without constraints other than those imposed implicitly by the inclusion of a penalty term on the laser huence [see Eq. (1)]. This inevitably led to laser helds that suddenly increased from very small to large values near the start of the laser pulse. However, physically realistic laser helds should tum-on and -off smoothly. Therefore, during the optimization the held is not allowed to vary freely but is rather expressed in the form [60] ... 

An alternate method for introducing pulse restrictions has been introduced by one of us (AB) recently within an iterative scheme for solving the optimal control equations [70], The idea is that a new reference field e(t) is constructed based on the field from the previous iteration after the application of a filter function F to ensure the fulfilment of some predesigned temporal and spectral properties. Therefore, a penalty term of the form... 

The user supplied weighting constant, (>0), should have a large value during the early iterations of the Gauss-Newton method when the parameters are away from their optimal values. As the parameters approach the optimum, should be reduced so that the contribution of the penalty function is essentially negligible (so that no bias is introduced in the parameter estimates). 

Implicit estimation offers the opportunity to avoid the computationally demanding state estimation by formulating a suitable optimality criterion. The penalty one pays is that additional distributional assumptions must be made. Implicit formulation is based on residuals that are implicit functions of the state variables as opposed to the explicit estimation where the residuals are the errors in the state variables. The assumptions that are made are the following ... 

If the capital cost of new heat transfer area is expressed in the form of Equation 18.6, then this will lead to poor retrofit projects. The problem with Equation 18.6 is that the optimization is likely to spread the new heat transfer area in the network in many locations, without incurring a cost penalty associated with the many modifications that would result. To ensure that new heat transfer area is not spread around throughout the existing heat exchanger network, a capital cost correlation should be used that is of the form ... 

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