Andersen, Martin Skovgaard

Borries, Oscar

Eltved, Anders

Numerical optimization is an essential part of the design process when shaping a contoured beam reflector antenna to produce a required coverage. The minimax formulation that has been used for decades generally results in a nonconvex problem, and since the evaluation of a design is costly, local optimization is typically used in practice. An inherent challenge with the minimax formulation is that it often leads to multiple local minima As a result, the quality of the final design obtained via local optimization depends strongly on the initial design.To address the sensitivity to the initial design, we investigate the use of initialization based on a different problem formulation. In particular, we first solve a one-sided nonlinear least-squares problem to improve an initial design, and the resulting design is then used as initialization of a minimax algorithm. We use this strategy to shape a reflector antenna starting from different initial shapes, and our results show that the initialization strategy improves the coverage with five out of six initial guesses

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Downloaded from orbit.dtu.dk on: Aug 23, 2018Improved Shaping of Reflector Antennas using a New Minimax Initialization StrategyEltved, Anders; Andersen, Martin Skovgaard; Bowies, OscarPublished in:Proceedings of 2018 International Applied Computational Electromagnetics Society SymposiumLink to article, DOI:10.23919/ROPACES.2018.8364326Publication date:2018Document VersionPeer reviewed versionLink back to DTU OrbitCitation (APA):Eltved, A., Andersen, M. S., & Bowies, O. (2018). Improved Shaping of Reflector Antennas using a NewMinimax Initialization Strategy. In Proceedings of 2018 International Applied Computational ElectromagneticsSociety Symposium IEEE. DOI: 10.23919/ROPACES.2018.8364326Improved Shaping of Reflector Antennasusing a New Minimax Initialization StrategyAnders Eltved* and Martin S. AndersenDTU Compute, Technical University of DenmarkKongens Lyngby, Denmark*Email: aelt@dtu.dkOscar BorriesTICRACopenhagen, DenmarkAbstract—Numerical optimization is an essential part of thedesign process when shaping a contoured beam reflector antennato produce a required coverage. The minimax formulation thathas been used for decades generally results in a nonconvexproblem, and since the evaluation of a design is costly, localoptimization is typically used in practice. An inherent challengewith the minimax formulation is that it often leads to multiplelocal minima. As a result, the quality of the final design obtainedvia local optimization depends strongly on the initial design.To address the sensitivity to the initial design, we investigatethe use of initialization based on a different problem formulation.In particular, we first solve a one-sided nonlinear least-squaresproblem to improve an initial design, and the resulting designis then used as initialization of a minimax algorithm. We usethis strategy to shape a reflector antenna starting from differentinitial shapes, and our results show that the initialization strategyimproves the coverage with five out of six initial guesses.I. INTRODUCTIONFor many modern communications satellites, the designof contoured beam reflector antennas involves the use ofoptimization to shape the reflector to produce a desired cov-erage. The performance of a reflector system is simulatedusing Physical Optics (PO) which is sometimes augmentedby Physical Theory of Diffraction (PTD) to account for theeffects from the reflector edges. Using these techniques, thegain, cross-polar performance, and other relevant quantitiescan be evaluated for a specific design.To optimize the design, it is common to consider theminimax problem where the objective is to maximize theminimum gain in the coverage, or equivalently, maximizingthe guaranteed gain in the coverage. The minimax problemhas been studied extensively [1], and good local optimizationalgorithms exist for solving the problem [2], [3]. However,with local optimization algorithms, the solution depends on aninitial guess which is usually provided by the user. In practice,the antenna designer comes up with an initial design which isthen improved by the optimization algorithm.Unfortunately, the minimax formulation can have manylocal minima, and hence local optimization algorithms can besensitive to the initial guess; if the antenna designer provides apoor initial design, the final design after optimization may notbe improved much. Multi-start methods (doing multiple localoptimizations from various starting points) and global opti-mization methods can improve this situation, but evaluatingthe performance of a design, which is done in each iteration ofthe optimization, can take hours in some applications, makingthese approaches too computationally expensive. In practice,the user often decides on a maximum number of functionevaluations and the optimization is terminated if this numberis exceeded even if the usual stopping criteria are not satisfied.In this study, we present an alternative optimization prob-lem to solve — the one-sided nonlinear least-squares (one-sided LS) problem — and an algorithm for finding a (local)minimum. Given a maximum number of function evaluations,kmax, to be used in the optimization, we propose the followingstrategy: (i) the initial design is first improved by optimizingthe one-sided LS formulation using at most kmax/2 iterations(one iteration corresponds to one function evaluation), and(ii) the resulting design is then optimized using the minimaxalgorithm with a cap on the number of iterations such that thetotal number of function evaluations is at most kmax.II. METHODIn the following, we will assume that x ∈ Rn is a vectorof n parameters that define the design of the reflector antenna(e.g., the elements of x can represent spline coefficients of ashaped surface, the position or excitation of feeds and arrays,etc.). For each of the m points in the coverage we define aresidual function:ri(x) = wi(gi − fi(x)), (1)where fi(x) is a function that measures the performance of thecurrent design, gi is a performance goal, and wi is a weightthat determines the importance of meeting the performancegoal at the ith point. To ensure that a design can be realized,it is often necessary to put constraints on x (e.g., restrictingthe curvature of the reflector to avoid having to bend the metalsurface of the reflector too much, or restricting the position ofa feed such that it does not coincide with other components onthe structure). Here we consider linear constraints representedby the set C = {x ∈ Rn | Cx+ d ≥ 0}, where C ∈ Rp×n,d ∈ Rp, p is the number of constraints, and the inequality iselement-wise. The minimax problem can now be formulatedas:minimizex∈Cmaxiri(x), (2)and the one-sided LS problem is given by:minimizex∈Cm∑i=1max {0, ri(x)}2 , (3)Notice that all positive residuals are penalized in the one-sided LS formulation; this is sensible since a positive residualimplies that the goal is not attained at the given point. Asmentioned in the introduction, there are efficient algorithmsfor optimizing (2) locally, and in the next section, we outlinea method for local optimization of (3).III. ALGORITHMTo optimize (3), we consider the equivalent problem:minimizex∈C,u∈Rm‖u‖2subject to ui ≥ ri(x), ui ≥ 0, i = 1, . . . ,m,(4)where u is a vector of auxiliary variables. This problem has aquadratic objective function, but due to the possibly nonconvexconstraints ui ≥ ri(x), it is still intractable to find a globalminimum. Thus, we use a trust-region method [4] to searchfor a local minimum. At a current iterate, xk, we linearizethe residual functions in (4), and find a candidate for thenext iterate, xk+1 = xk + ∆x, by solving the trust-regionsubproblem:minimize∆x∈Rn,u∈Rm‖u‖2subject to C∆x+ dk ≥ 0u ≥ rk + Jk∆xu ≥ 0, δk ≥ ‖∆x‖∞ ,(5)where δk > 0 is the trust-region radius, rk = r(xk), and:dk = Cxk + d, Jk =[∇r1(xk) · · · ∇rm(xk)]T . (6)The problem (5) is a convex quadratic problem with linearconstraints (since the∞-norm is linearly representable) whichwe solve with an interior-point method [5].IV. RESULTSTo illustrate the effect of the initialization strategy, weconsider a case where the objective is to shape a reflector forillumination of the continental United States (the US excludingAlaska and Hawaii). We have n = 783 variables (splinecoefficients of a reflector surface) and optimize the coverageover m = 810 points starting from six different initial shapes.The initial shapes were created by de-focusing the reflectorsuch that the initial guesses become progressively worse: guess1 is expected to produce the best results and initial guess 6is expected to produce the worst results. We did not imposeconstraints on x (i.e., p = 0). An example of the coverageobtained with the minimax algorithm initialized with initialguess 2 is shown in Fig. 1.In our experiments, we limited the number of functionevaluations (kmax) to 500, and for each of the six initialguesses we did two experiments. In the first experiment, weFig. 1. Contours of coverage optimized using the minimax algorithm. Thepeak gain of around 33 dB is at the cross in Florida. The Nadir direction ofthe satellite is shown in the black circle.TABLE I. VALUE OF MAXIMAL RESIDUAL IN DB AFTER 500 ITERATIONS.THE 6 INITIAL GUESSES BECOME PROGRESSIVELY WORSE: 1 IS A GOOD(GREEN) INITIAL GUESS AND 6 IS A POOR (RED) INITIAL GUESS.Initial Guess 1 2 3 4 5 6Minimax only -0.074 0.090 0.826 2.019 4.149 9.360OLS+minimax -0.074 -0.004 0.417 1.952 4.103 7.690Difference 0.000 0.094 0.409 0.067 0.046 1.67applied the one-sided LS algorithm for 250 iterations (or untila stopping criteria is met), followed by the minimax algorithmfor the remaining iterations. In the second experiment, we usedonly the minimax algorithm for 500 iterations. Our resultsare summarized in Table I; recall that smaller residuals arebetter. The results clearly show that the initialization strategycan improve the minimum gain in the coverage: the gain isimproved for initial guesses 2-6, and the result remains thesame for initial guess 1. Considering the maximal residual —it measures how far we are from reaching the goal value — theabsolute improvement is largest for initial guess 6 (1.67 dB),where there is much room for improvement, and the relativeimprovement is largest for initial guess 3 (49.5%).V. CONCLUSIONThe use of one-sided least-squares for initializing the mini-max algorithm can lead to improved results: in our numericalexperiments, the initialization strategy led to an improveddesign with five out of six initial guesses.REFERENCES[1] V. Demyanov and V. Molozemov, Introduction to Minimax. Wiley: NewYork, 1974.[2] J. Hald, “Two stage algorithms for linearly constrained minimax opti-mization,” Ph.D. dissertation, Technical University of Denmark, 1984.[3] K. Madsen and H. Schjær-Jacobsen, “Linearly constrained minimaxoptimization,” Mathematical Programming, vol. 14, no. 1, 1978.[4] A. Conn, N. Gould, and P. Toint, Trust Region Methods. Society forIndustrial and Applied Mathematics, 2000.[5] J. Nocedal and S. Wright, “Quadratic programming,” in NumericalOptimization, 2nd ed. Springer-Verlag New York, 2006, ch. 16.

Improved Shaping of Reflector Antennas using a New Minimax Initialization Strategy

Anders Eltved

Martin S. Andersen

Oscar Borries

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Improved shaping of reflector antennas using a new minimax initialization strategy

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Improved Shaping of Reflector Antennas using a New Minimax Initialization Strategy

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