Gradient search

With many variables and constraints, linear and nonlinear programming may be applicable, as well as various numerical gradient search methods. Maximum principle and dynamic programming are laborious and have had only limited applications in this area. The various mathematical techniques are explained and illustrated, for instance, by Edgar and Himmelblau Optimization of Chemical Processes, McGraw-Hill, 1988). 

Optimal design of ammonia synthesis by differential equation solution and a numerical gradient search... 

As a comparison, the results from a back-propagation training routine with a gradient search for PTE is... 

Figure 22 Root-mean-squared error progression plot for Fletcher nonlinear optimization and back-propagation algorithms during training.-----Fletcher routine —A—Gradient search.

The gradient search methods require derivatives of the objective functions whereas the direct methods are derivative-free. The derivatives may be available analytically or otherwise they are approximated in some way. It is assumed that the objective function has continuous second derivatives, whether or not these are explicitly available. Gradient methods are still efficient if there are some discontinuities in the derivatives. On the other hand, direct search techniques, which use function values, are more efficient for highly discontinuous functions. 

Direct search methods use only function evaluations. They search for the minimum of an objective function without calculating derivatives analytically or numerically. Direct methods are based upon heuristic rules which make no a priori assumptions about the objective function. They tend to have much poorer convergence rates than gradient methods when applied to smooth functions. Several authors claim that direct search methods are not as efficient and robust as the indirect or gradient search methods (Bard, 1974 Edgar and Himmelblau, 1988 Scales, 1986). However, in many instances direct search methods have proved to be robust and reliable particularly for systems that exhibit local minima or have complex nonlinear constraints (Wang and Luus, 1978). 

Starting with an arbitrary set of poles, Hild used Brogan s method ( ) to determine the matrices K and P of Figure 6. The integrations of Equation 19 were performed in closed form on the linearized equations and a gradient search was conducted in "pole space" to minimize P. All poles were restricted to negative, real, and distinct values. 

By direct gradient search, iterated 16 times from a starting temperature of 50°F, the total annual cost is found to have a local minimum at x = 5.94 in. and t = 6.29°F, where the cost is 53,400/yr. The reader can verify, however, that an ambient system (80°F) without any recondensation only costs 52,000/yr, a saving of 3%. 

Gradient search using created response surface (first pass). 

Two types of nonlinear optimizers—the sectionalized linear program and the gradient search—have been successfully implemented in advanced computer control schemes. 

Recently, a semiclassical formulation of the optimal control theory has been derived [23, 24] by combining the conjugate gradient search method... 

Global Optimization Procedure and Conjugate Gradient Search Method... 

To find the optimal field, we employ the simplest form of global optimization procedure with the iterational conjugate gradient search method [7,8]. At each iteration of this algorithm, the correction to the optimized laser field is determined from the following equations ... 

The Estimates, Derivatives and Search parameters can be changed to optimize the solution process. The Search parameter specifies which gradient search method to use the Newton method requires more memory but fewer iterations, the Conjugate method requires less memory but more iterations. The Derivatives parameter specifies how the gradients for the search are calculated the Central derivatives method requires more calculations but may be helpful if the Solver reports that it is unable to find a solution. The Estimates parameter determines the method by which new estimates of the coefficients are obtained from previous values the Quadratic method may improve results if the system is highly nonlinear. 

Once the construction of an appropriately constrained cost function is complete, its minimization in the absolute sense becomes the target. Gradient search methods are patently unsuitable for the minimization of... 

At W/F = 0, which is equal to the initial rate (r0), the above derivative reduces to CD. The best values of C and D were obtained by a gradient search using a digital PDP-9 computer. Mezaki and Kittrell (6) have demonstrated that this expression is a convenient and relatively nondiscriminating way to extrapolate to zero conversion. The range of partial pressures (torr) of reactants used were hydrogen sulfide, 4-22, and sulfur dioxide, 2-22. 

A strategy suggested by Chang (1982) is based on the assumption that the perturbed trajectory is in the vicinity of the nominal trajectory and uses one-step iteration in a gradient search technique to improve the perturbed performance index. This approach assumes that the moments of the CSD (the states of the lumped model resulting from the method of moments) can be measured or estimated. 

We have briefly sketched six important topics of MCDM problems. Many more topics, such as interactive methods including ad ted gradient search method, surrogate-worth trade-off methods (Hetimes and Hall 1974), the Zionts-WaUenius method (Zionts and Wallenius 1976), the paired com-... 

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