Kaj Madsen

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.

Combined lp and quasi-Newton methods for minimax optimization

We present an algorithm for minimax optimization that combines LP methods and quasi-Newton methods. The quasi-Newton algorithm is used only if an irregular solution is detected, in which case second-order derivative information is needed in order to obtain a fast final rate of convergence. We prove that the algorithm can converge only to a stationary point and that normally the final rate of convergence will be either quadratic or superlinear. The performance is illustrated through some numerical examples.

A Superlinearly Convergent Minimax Algorithm for Microwave Circuit Design

A new and highly efficient algorithm for nonlinear minimax optimization is presented. The algorithm, based on the work of Hald and Madsen, combines linear programming methods with quasi-Newton methods and has sure convergence properties. A critical review of the existing minimax algorithms is given, and the relation of the quasi-Newton iteration of the algorithm to the Powell method for nonlinear programming is discussed. To demonstrate the superiority of this algorithm over the, existing ones, the classical three-section transmission-line transformer problem is used. A novel approach to worst-case design of microwave circuits using the present algorithm is proposed. The robustness of the algorithm is proved by using it for practical design of contiguous and noncontiguous-band multiplexer, involving up to 75 design variables.

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.

Efficient optimization with integrated gradient approximations

A flexible and effective algorithm is proposed for efficient optimization with integrated gradient approximations. It combines the techniques of perturbations, the Broyden update, and the special iterations of Powell. Perturbations are used to provide an initial approximation as well as regular corrections. The approximate gradient is updated using C.G. Broydens formula (1965) in conjunction with the special iterations of M.J.D. Powell (1970). A modification to the Broyden update is introduced to exploit possible sparsity of the Jacobian. Utilizing this algorithm, powerful gradient-based nonlinear optimization tools for circuit CAD can be used without the effort of calculating exact derivatives Applications of practical significance are demonstrated. The examples include robust small-signal FET modeling using the l/sub 1/ techniques and simultaneous processing of multiple circuits, worst-case design of a microwave amplifier, and minimax optimization of a five-channel manifold multiplexer. Computational efficiency is greatly improved over estimating derivatives entirely by perturbations. >

Mean value forms in interval analysis

An interval extension of a function written in the centered form or the mean value form offers a second order approximation to the range of values of the function over an interval. However, the two forms differ with respect to inclusion monotonicity; the mean value form is inclusion monotone while the centered form is not. This is demonstrated in the following paper. Further, the mean value form is more generally applicable, and a mean value form for an integral operator is considered. It is shown that this form is also a second order inclusion monotone approximation.ZusammenfassungSowohl in der zentrierten Form wie auch in der Mittelwertform liefert die Intervall-Fortsetzung einer Funktion eine Näherung zweiter Ordnung für den Wertebereich der Funktion über einem Intervall. Die beiden Formen unterscheiden sich jedoch bezüglich der Einschließungsmonotonie: Die Mittelwertform ist einschließungsmonoton, die zentrierte Form dagegen i.a. nicht, wie in der Arbeit gezeigt wird. Ferner ist die Mittelwertform allgemeiner anwendbar; eine Mittelwertform für einen Integraloperator wird diskutiert, und es wird gezeigt, daß sie ebenfalls eine einschließungsmonotone Näherung zweiter Ordnung ist.

A nonlinear l_1 optimization algorithm for design, modeling, and diagnosis of networks

This paper presents a fast and highly efficient algorithm for nonlinear l_1 optimization and its applications to circuits employing the properties of the l_1 norm. The algorithm, based on the work of Hald and Madsen, is similar to a minimax algorithm originated by the same authors. It is a combination of a first-order method that approximates the solution by successive linear programming and a quasi-Newton method using approximate second-order information to solve a system of nonlinear equations resulting from the first-order necessary conditions for an optimum. The new l_1 algorithm is particularly useful in fault location methods using the l_1 norm. A new technique for isolating the most likely faulty elements, based on an exact penalty function, is presented. Another important application of the algorithm is the design of contiguous-band multiplexers consisting of multicavity filters distributed along a waveguide manifold which is illustrated by a 12-channel multiplexer design. We also present a formulation using the l_1 norm for model parameter identification problems in the presence of large isolated errors in measurements and illustrate it with a sixth-order filter.

A root-finding algorithm based on Newton's method

This note is a short description of a procedure searching for the zero of least modulus of a given polynomial. Further details may be found in [8].The method is based on Newtons formula, and the main problem is to find an approximation to the zero, close enough to make Newtons method converge. We solve this problem iteratively, making use of the information Newtons formula gives about the direction of steepest descent. In this way we obtain a sequence of points giving decreasing function values. When a certain condition is fulfilled, securing convergence of Newtons method, this is used directly.With very few modifications the procedure can be used to find a zero of an arbitrary analytic functionf:C →C.

Iterative methods for interval inclusion of fixed points

The paper discusses a technique for handling numerical, iterative processes that combines the efficiency of ordinary floating-point iterations with the accuracy control that may be obtained by iterations in interval arithmetic. As illustration the technique is used for the solution of fixed point problems in one and several variables.