Mathematical Optimization Techniques

From the beginning the ROXIE program was structured such that mathematical optimization techniques can be applied to the design of the superconducting magnets. With the concept of features it is possible to create the complex coil assemblies in 2 and 3 dimensions with only a small number of engineering data which can then be addressed as design variables of the optimization problem. In this chapter some background information on the application of mathematical optimization techniques is given. 1 Historical overview Mathematical optimization including numerical techniques such as linear and nonlinear programming, integer programming, network flow theory and dynamic optimization has its origin in operations research developed in world war II, e.g., Morse and Kimball 1950 [45]. Most of the real-world optimization problems involve multiple conflicting objectives which should be considered simultaneously, so-called vector-optimization problems. The solution process for vector-optimization problems is threefold, based on decision-making methods, methods to treat nonlinear constraints and optimization algorithms to minimize the objective function. Methods for decision-making, based on the optimality criterion by Pareto in 1896 [48], have been introduced and applied to a wide range of problems in economics by Marglin 1966 [42], Geoffrion 1968 [18] and Fandel 1972 [12]. The theory of nonlinear programming with constraints is based on the optimality criterion by Kuhn and Tucker, 1951 [37]. Methods for the treatment of nonlinear constraints have been developed by Zoutdendijk 1960 [70], Fiacco and McCormick 1968 [13] and Rockafellar 1973 [54] among others. Numerous optimization algorithms both using deterministic and stochastic elements have been developed in the sixties and covered in the books by Wilde 1964 [67], Rosenbrock 1966 [55], Himmelblau 1972 [25], Brent 1973 [5], and Schwefel 1977 [62]. Researchers tend to come back to genetic and evolutionary algorithms recently as they are suited for parallel processing, finding global optima, and are reported to be suitable for a large number of design variables Fogel 1994 [15], Holland 1992 [26]. Mathematical optimization techniques have been applied to computational electromagnetics already for decades. Halbach 1967 [23] introduced a method for optimizing coil arrangements and pole shapes of magnets by means of finite element (FE) field calculation. Armstrong, Fan, Simkin and Trowbridge 1982 [2] combined optimization algorithms with the volume integral method for the pole profile optimization of a H-magnet. Girdinio, Molfino, Molinari and Viviani 1983 [20] optimized a profile of an electrode. These attempts tended to be application-specific, however. Only since the late 80 th, have numerical field calculation packages for both 2d and 3d applications been placed in an optimization environment. Reasons for this delay have included constraints in computing power, problems with discontinuities and nondifferentiabilities in the objective function arising from FE meshes, accuracy of the field solution and software implementation problems. A small selection of papers can be found in the references. The variety of methods applied shows that no general method exists to solve nonlinear optimization problems in computational electromagnetics in the same way that the simplex algorithm exists to