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Organizations and businesses strive toward excellence, and solutions to problems are based mostly on judgment and experience. However, increased competition and consumer demands require that the solutions be optimum and not just feasible. Theory leads to algorithms. Algorithms need to be translated into computer codes. Engineering problems need to be modeled. Optimum solutions are obtained using theory and computers, and then interpreted. Revised and expanded in its third edition, this textbook integrates theory, modeling, development of numerical methods, and problem solving, thus preparing students to apply optimization to real-world problems. This text covers a broad variety of optimization problems using: unconstrained, constrained, gradient, and non-gradient techniques; duality concepts; multi-objective optimization; linear, integer, geometric, and dynamic programming with applications; and finite element-based optimization. It is ideal for advanced undergraduate or graduate courses in optimization design and for practicing engineers.
Multivariable minimization can be approached using gradient and Hessian information, or using the function evaluations only. We have discussed the gradient- or derivative-based methods in earlier chapters. We present here several algorithms that do not involve derivatives. We refer to these methods as direct methods. These methods are referred to in the literature as zero order methods or minimization methods without derivatives. Direct methods are generally robust. A degree of randomness can be introduced in order to achieve global optimization. Direct methods lend themselves valuable when gradient information is not readily available or when the evaluation of the gradient is cumbersome and prone to errors. We present here the algorithms of cyclic coordinates, method of Hooke and Jeeves , method of Rosenbrock , simplex method of Nelder and Mead , Powell's  method of conjugate directions. The concepts of simulated annealing, genetic, and differential evolution algorithms are also discussed. Box's complex method for constrained problems is also included. All these algorithms are implemented in complete computer programs.
Cyclic Coordinate Search
In this method, the search is conducted along each of the coordinate directions for finding the minimum. If ei is the unit vector along the coordinate direction i, we determine the value αi minimizing f(α) = f(x + αei), where αi is a real number. A move is made to the new point x + αiei at the end of the search along the direction i.