Multi-Objective Design of Antennas using Surrogate Models by Slawomir Koziel, Adrian Bekasiewicz
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9in x 6in b2523-ch03 Multi-Objective Design of Antennas Using Surrogate Models EM-based objective functions and might be a problem particularly for gradient-based methods. Surrogate-assisted optimization is one of the most promising ways of handling computationally expensive EM-driven design problems. The key concept is to replace direct optimization of the highfidelity EM model by optimization of its cheap and analytically tractable representation referred to as a surrogate model. Introduction to surrogate-based methods is provided in Chapter 5.
9in x 6in b2523-ch03 Introduction to Numerical Optimization 39 1 0 0 0 y −1 −2 −2 y 1 y 1 −1 −1 0 −2 −2 1 −1 −1 0 −2 −2 1 x x (a) (b) (c) 1 0 0 0 y −1 0 1 1 0 1 −1 −1 −1 0 y 1 y 1 −2 −2 −1 x −2 −2 −1 0 −2 −2 1 −1 x x x (d) (e) (f) Fig. 4. Constrained optimization of the function f (x, y) = x2 + 4y 2 ; the equality constraint is h1 (x, y) = (x + 2)2 + (y + 2)2 − 2. 725]T . 26) for the increasing values of the penalty factor β1 . Constrained minimum marked using a circle. 1) as x(i+1) = arg min φ(i) (x) x µ(i) = arg min f (x) + x 2 M k=1 h2k (x) M − λk hk (x) .
30) hk (x(i) ) + ∇hk (x(i) )T h = 0, k = 1, . . , M. 31) and Here, x(i) is a current iteration point, whereas H is a symmetric, positive definite matrix (preferably an approximation of the Hessian of the Lagrangian of f , cf. 23)). The new approximation x(i+1) is subsequently obtained using line search. 3 Derivative-Free Optimization Despite strong theoretical foundations of gradient-based techniques and their good performance on smooth objective functions, they cannot be applied to a growing number of practical optimization problems.
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