Slawomir Koziel

Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization

During the last five years, several methods have been proposed for handling nonlinear constraints using evolutionary algorithms (EAs) for numerical optimization problems. Recent survey papers classify these methods into four categories: preservation of feasibility, penalty functions, searching for feasibility, and other hybrids. In this paper we investigate a new approach for solving constrained numerical optimization problems which incorporates a homomorphous mapping between n-dimensional cube and a feasible search space. This approach constitutes an example of the fifth decoder-based category of constraint handling techniques. We demonstrate the power of this new approach on several test cases and discuss its further potential.

A Space-Mapping Framework for Engineering Optimization—Theory and Implementation

This paper presents a comprehensive approach to engineering design optimization exploiting space mapping (SM). The algorithms employ input SM and a new generalization of implicit SM to minimize the misalignment between the coarse and fine models of the optimized object over a region of interest. Output SM ensures the matching of responses and first-order derivatives between the mapped coarse model and the fine model at the current iteration point in the optimization process. We provide theoretical results that show the importance of the explicit use of sensitivity information to the convergence properties of our family of algorithms. Our algorithm is demonstrated on the optimization of a microstrip bandpass filter, a bandpass filter with double-coupled resonators, and a seven-section impedance transformer. We describe the novel user-oriented software package SMF that implements the new family of SM optimization algorithms

Surrogate-Based Methods

Objective functions that appear in engineering practice may come from measurements of physical systems and, more often, from computer simulations. In many cases, optimization of such objectives in a straightforward way, i.e., by applying optimization routines directly to these functions, is impractical. One reason is that simulation-based objective functions are often analytically intractable (discontinuous, non-differentiable, and inherently noisy). Also, sensitivity information is usually unavailable, or too expensive to compute. Another, and in many cases even more important, reason is the high computational cost of measurement/simulations. Simulation times of several hours, days or even weeks per objective function evaluation are not uncommon in contemporary engineering, despite the increase of available computing power. Feasible handling of these unmanageable functions can be accomplished using surrogate models: the optimization of the original objective is replaced by iterative re-optimization and updating of the analytically tractable and computationally cheap surrogate. This chapter briefly describes the basics of surrogate-based optimization, various ways of creating surrogate models, as well as several examples of surrogate-based optimization techniques.

Computational Optimization, Methods and Algorithms

Computational optimization is an important paradigm with a wide range of applications. This book reviews and discusses the latest developments concerning optimization and modelling with a focus on methods and algorithms for computational optimization.

Demystifying Surrogate Modeling for Circuits and Systems

In this article, grey-box and black-box surrogate modeling are described, with some key findings. The important point is that surrogate modeling has a solid mathematical basis leading to what has become a dramatic increase in our ability to develop engineering models and to engineer systems. In Section 2, a systematic approach to constructing surrogate models is provided. Each step is explained using published methods. Section 3 presents surrogate modeling examples from the domain of circuits and systems.

Shape-Preserving Response Prediction for Microwave Design Optimization

A shape-preserving response prediction methodology for microwave design optimization is introduced. The presented technique allows us to estimate the response of the microwave structure being optimized (fine model) using a computationally cheap representation of the structure (coarse model). The change of the coarse model response is described by the translation vectors corresponding to certain (finite) number of characteristic points of the response. These translation vectors are subsequently used to predict the response change of the fine model. The presented method has very good generalization capability and it is not based on any extractable parameters, which makes it easy to implement. Applications for microwave design optimization are discussed. The robustness of the proposed approach is demonstrated by extensive comparison with space mapping, which is one of the most efficient optimization approaches in microwave engineering so far.

Accelerated Microwave Design Optimization With Tuning Space Mapping

We introduce a tuning space-mapping technology for microwave design optimization. The general tuning space-mapping algorithm is formulated, which is based on a so-called tuning model, as well as on a calibration process that translates the adjustment of the tuning model parameters into relevant updates of the design variables. The tuning model is developed in a fast circuit-theory based simulator and typically includes the fine model data at the current design in the form of the properly formatted scattering parameter values. It also contains a set of tuning parameters, which are used to optimize the model so that it satisfies the design specification. The calibration process may involve analytical formulas that establish the dependence of the design variables on the tuning parameters. If the formulas are not known, the calibration process can be performed using an auxiliary space-mapping surrogate model. Although the tuning space mapping can be considered to be a specialized case of the standard space-mapping approach, it can offer even better performance because it enables engineers to exploit their experience within the context of efficient space mapping. Our approach is demonstrated using several microwave design optimization problems.

Space Mapping With Adaptive Response Correction for Microwave Design Optimization

Output space mapping is a technique introduced to enhance the robustness of the space-mapping optimization process in case the space-mapped coarse model cannot provide sufficient matching with the fine model. The technique often works very well; however, in some cases it fails. Especially in the microwave area where the typical model response (e.g., |S 21|) is a highly nonlinear function of the free parameter (e.g., frequency), the output space-mapping correction term may actually increase the mismatch between the surrogate and fine models for points other than the one at which the term was calculated, as in the surrogate model optimization process. In this paper, an adaptive response correction scheme is presented to work in conjunction with space-mapping optimization algorithms. This technique is designed to alleviate the difficulties of the standard output space mapping by adaptive adjustment of the response correction term according to the changes of the space-mapped coarse model response. Examples indicate the robustness of our approach.

Multi-Objective Design of Antennas Using Variable-Fidelity Simulations and Surrogate Models

A computationally-efficient procedure for multi-objective design of antenna structures is presented. Our approach exploits the multi-objective evolutionary algorithm (MOEA) working with a fast antenna surrogate model obtained with kriging interpolation of coarse-discretization simulation data. Response correction techniques are subsequently applied to refine the designs obtained by MOEA. Our methodology allows us to obtain-at a low computational cost-a set of designs corresponding to various trade-offs between the antenna size and the refection coefficient. Two illustration examples are considered: (i) an UWB monocone with two objectives being reduction of the antenna size and minimization of the antenna reflection coefficient in the bandwidth of interest, and (ii) a planar Yagi antenna with the objectives being an increase of the end-fire gain and minimization of the reflection coefficient, both in the bandwidth of interest.

Efficient Multi-Objective Simulation-Driven Antenna Design Using Co-Kriging

A methodology for fast multi-objective antenna optimization is presented. Our approach is based on response surface approximation (RSA) modeling and variable-fidelity electromagnetic (EM) simulations. In the design process, a computationally cheap RSA surrogate model constructed from sampled coarse-discretization EM antenna simulations is optimized using a multi-objective evolutionary algorithm. The initially determined Pareto optimal set representing the best possible trade-offs between conflicting design objectives is then iteratively refined. In each iteration, a limited number of high-fidelity EM model responses are incorporated into the RSA model using co-kriging. The enhanced RSA model is subsequently re-optimized to yield the refined Pareto set. Combination of low- and high-fidelity simulations as well as co-kriging results in the low overall optimization cost. The proposed approach is validated using two UWB antenna examples.