Hans Schjær-Jacobsen

Linearly constrained minimax optimization

We present an algorithm for nonlinear minimax optimization subject to linear equality and inequality constraints which requires first order partial derivatives. The algorithm is based on successive linear approximations to the functions defining the problem. The resulting linear subproblems are solved in the minimax sense subject to the linear constraints. This ensures a feasible-point algorithm. Further, we introduce local bounds on the solutions of the linear subproblems, the bounds being adjusted automatically, depending on the quality of the linear approximations. It is proved that the algorithm will always converge to the set of stationary points of the problem, a stationary point being defined in terms of the generalized gradients of the minimax objective function. It is further proved that, under mild regularity conditions, the algorithm is identical to a quadratically convergent Newton iteration in its final stages. We demonstrate the performance of the algorithm by solving a number of numerical examples with up to 50 variables, 163 functions, and 25 constraints. We have also implemented a version of the algorithm which is particularly suited for the solution of restricted approximation problems.

Algorithms for worst-case tolerance optimization

New algorithms are presented for the solution of optimum tolerance assignment problems. The problems considered are defined mathematically as a worst-case problem (WCP), a fixed tolerance problem (FTP), and a variable tolerance problem (VTP). The basic optimization problem without tolerances is denoted the zero tolerance problem (ZTP). For solution of the WCP we suggest application of interval arithmetic and also alternative methods. For solution of the FTP an algorithm is suggested which is conceptually similar to algorithms previously developed by the authors for the ZTP. Finally, the VTP is solved by a double-iterative algorithm in which the inner iteration is performed by the FTP- algorithm. The application of the algorithm is demonstrated by means of relatively simple numerical examples. Basic properties, such as convergence properties, are displayed based on the examples.

Automated minimax design of networks

A new gradient algorithm for the solution of nonlinear minimax problems has been developed. The algorithm is well suited for automated minimax design of networks and it is very simple to use. It compares favorably with recent minimax and least p th algorithms. General convergence problems related to minimax design of networks are discussed. Finally, minimax design of equalization networks for reflectiontype microwave amplifiers is carried out by means of the proposed algorithm.

Synthesis of nonuniformly spaced arrays using a general nonlinear minimax optimisation method

Antenna patterns can be synthesized using a new nonlinear minimax optimization method with sure convergence properties. Not requiring derivatives, the proposed method is general and easy to use so that it might be applied to a wide variety of nonlinear synthesis problems for which analytical solutions are not known. To test the algorithm a group of test problems for which exact analytical solutions are known has been considered, namely, optimization of Dolph-Chebyshev arrays by spacing variation. The method is further applied to find the element positions in nonuniformly spaced linear arrays with uniform excitation that produce minimized (equal) sidelobe levels, and comparisons are made with conventional Dolph-Chebyshev arrays.

Singularities in minimax optimization of networks

A theoretical treatment of singularities in nonlinear minimax optimization problems, which allows for a classification in regular and singular problems, is presented. A theorem for determining a singularity that is present in a given problem is formulated. A group of problems often used in the literature to test nonlinear minimax algorithms, i.e., minimax design of multisection quarter-wave transformers, is shown to exhibit singularities and the reason for this is pointed out. Based on the theoretical results presented an algorithm for nonlinear minimax optimization is developed. The new algorithm maintains the quadratic convergence property of a recent algorithm by Madsen et al. when applied to regular problems and it is demonstrated to significantly improve the final convergence on singular problems.

Coupling Effects in Multibeam Reflector Antennas

New results and conceptual design principles concerning multiple beam reflector antennas in which coupling effects are of major influence on antenna performance are presented. Special attention is paid to a certain class of feeds consisting of minimum scattering antennas (MSAs). The mutual impedance between two identical MSAs is expressed as an integral over their power patterns including complex angles [1]. For a given element spacing power patterns that minimize mutual coupling may be found. However, the patterns which lead to minimum coupling may not be optimum when the illumination of the reflector is considered. This is accounted for by constraining the efficiency for a given f/D-ratio. As an example of circularly polarized feed antennas detailed numerical analysis of coupling between arbitrarily oriented crossed dipoles is presented. In evaluating the scattered field from crossed dipoles the feeding network is included since a crossed dipole is never an MSA. The secondary field pattern is calculated by a two-dimensional Romberg-integration of the complex current distribution on the reflector found by the physical optics approximation and contour plots of gain and polarization loss are discussed. Also the scattered near field in the focal region is calculated in order to evaluate coupling between feed elements due to the presence of the reflector. Finally examples for coverage of specific areas on earth from a synchronous orbit satellite are presented.