IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 11, NOVEMBER 1997 1689 Designing Broad-Band Patch Antennas Using the Sequential Quadratic Programming Method Zhifang Li, Panos Y. Papalambros, and John L. Volakis, Fellow, IEEE Abstract— The utility of numerical codes is greatly enhanced if they can be used in design, a situation that typically in- volves iterative optimization algorithms. An attractive way is to use gradient-based algorithms developed for solving nonlinear programming (NLP) problems. In this letter, we examine the performance of a general sequential quadratic programming (SQP) optimization algorithm for designing patch antennas in conjunction with a finite-element boundary-integral code. Index Terms— Microstrip antennas. I. INTRODUCTION A NTENNA design involves the selection of the physical antenna parameters to achieve optimal gain, pattern per- formance, VSWR, bandwidth, and so on, subject to specified constraints. Over the past ten years, sophisticated computer codes have been developed for antenna analysis [1]–[3] based on a variety of popular methods. By and large, these codes have not been extended to include design capabilities pri- marily because of their complexity and nonlinearity with respect to the physical properties of the antenna (material constants, dimensions, feed location, and type, etc.). Some design algorithms have been proposed but these are applicable to specialized antenna shapes and do not address the general antenna optimization problem [4]. Recently, genetic algorithms (GA’s) have been examined for array design and absorber optimization [5]–[7]. However, GA’s, although robust, require large number of function evalu- ations to complete the optimization study. Also, GA’s are more suitable for discrete variable problems. In contrast, antenna simulations rely on complex computationally intensive codes, which generate continuous functions. It may, therefore, be impractical to generate a sufficiently large sample space for carrying out an optimization study using GA’s. An alternative optimization algorithm is the sequential quadratic programming (SQP) method, suitable for continuous nonlinear objective functions such as the input impedance, gain, pattern shape, etc. with both equality and inequality constraints. Convergence is typically achieved in a few iterations and, therefore, their interface with rigorous (but expensive) numerical antenna analysis codes is much more practical. SQP and other similar algorithms are routinely used for large structural design problems involving finite-element Manuscript received April 18, 1997; revised July 10, 1997. Z. Li and J. L. Volakis are with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 USA. P. Y. Papalambros is with the Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI 48109 USA. Publisher Item Identifier S 0018-926X(97)07992-1. analysis [8] and, thus, we can benefit from the extensive experience available in other disciplines. In this letter, we examine the performance of a general SQP code [9] for designing patch antennas in conjunction with a finite-element boundary-integral code [10]. Both are rigorous general-purpose codes. The main point of the paper is to exam- ine the suitability of SQP for antenna parameter optimization to achieve the design objectives subject to constraints. We will illustrate the performance of the optimizer using a few illustrative examples from simple to more complex. II. SEQUENTIAL QUADRATIC PROGRAMMING ALGORITHM DESCRIPTION SQP is a gradient-based class of methods that became prominent in the late 1970’s [11]. They are considered the most efficient general-purpose nonlinear programming algorithms today. The basic principle of sequential approximations is to replace the given nolinear problem by a sequence of quadratic subproblems that are easier to solve. Consider the equality constrained problem (1) where is the design variable vector, is the objective function, and is the vector of equality constraints. Using a Lagrange–Newton method (see, for example, [11]), at the th iteration, we have (2) where and is the vector of Lagrange multipliers. Solving the above equations iteratively, we obtain the iterates and which should eventually approach and , the optimal values. We observe that the above equation can be viewed as the first-order optimality (Karush–Kuhn–Tucker) conditions for the quadratic model (3) where . Solving the quadratic pro- gramming subproblem (3) gives the same and as solving (2) and thus the two formulations are equivalent. The values of and can be obtained from solving a sequence 0018–926X/97$10.00  1997 IEEE