electromagnetics problems. The antenna array has historically been one of the
most popular optimization targets in electromagnetics, so we continue that
tradition as well.
The fi rst optimum antenna array distribution is the binomial distribution
proposed by Stone [1]. As is now well known, the amplitude weights of the
elements in the array correspond to the binomial coeffi cients, and the resulting
array factor has no sidelobes. In a later paper, Dolph mapped the Chebyshev
polynomial onto the array factor polynomial to get all the sidelobes at an equal
level [2]. The resulting array factor polynomial coeffi cients represent the
Dolph–Chebyshev amplitude distribution. This amplitude taper is optimum in
that specifying the maximum sidelobe level results in the smallest beamwidth,
or specifying the beamwidth results in the lowest possible maximum sidelobe
level. Taylor developed a method to optimize the sidelobe levels and beam-
width of a line source [3]. Elliot extended Taylor’s work to new horizons,
including Taylor-based tapers with asymmetric sidelobe levels, arbitrary side-
lobe level designs, and null-free patterns [4]. It should be noted that Elliot’s
methods result in complex array weights, requiring both an amplitude and
phase variation across the array aperture. Since the Taylor taper optimized
continuous line sources, Villeneuve extended the technique to discrete arrays
[5]. Bayliss used a method similar to Taylor’s amplitude taper but applied to
a monopulse difference pattern [6]. The fi rst optimized phase taper was devel-
oped for the endfi re array. Hansen and Woodyard showed that the array
directivity is increased through a simple formula for phase shifts [7].
Iterative numerical methods became popular for fi nding optimum array
tapers beginning in the 1970s. Analytical methods for linear array synthesis
were well developed. Numerical methods were used to iteratively shape the
mainbeam while constraining sidelobe levels for planar arrays [8–10]. The
Fletcher–Powell method [11] was applied to optimizing the footprint pattern
of a satellite planar array antenna. An iterative method has been proposed to
optimize the directivity of an array via phase tapering [12] and a steepest-
descent algorithm used to optimize array sidelobe levels [13]. Considerable
interest in the design of nonuniformly spaced arrays began in the late 1950s
and early 1960s. Numerical optimization attracted attention because analytical
synthesis methods could not be found. A spotty sampling of some of the tech-
niques employed include linear programming [14], dynamic programming
[15], and steepest descent [16]. Many statistical methods have been used as
well [17].
1.1 OPTIMIZING A FUNCTION OF ONE VARIABLE
Most practical optimization problems have many variables. It’s usually best
to learn to walk before learning to run, so this section starts with optimizing
one variable; then the next section covers multiple variable optimization.
After describing a couple of single-variable functions to be optimized, several
2 INTRODUCTION TO OPTIMIZATION IN ELECTROMAGNETICS