*** Matlab codes for optimization under unitary matrix constraint Author of the codes: Traian Emanuel Abrudan (joint work with Jan Eriksson and Visa Koivunen) SMARAD CoE, Department of Signal Processing and Acoustics, Aalto Univestity, Espoo, Finland ** General description This set of codes may be used to optimize a smooth (differentiable) cost function J (W) over the Lie group of n × n unitary matrices. The codes we provide can be used to either minimize or maximize an arbitrary smooth cost function under unitary matrix constraint. The title and description of this software/code correspond with the situation when the software metadata was imported to ACRIS. The most recent version of metadata is available in the original repository. • The following gradient-based optimization algorithms are implemented (see Sections 3.2.1 and 3.2.2, and Tables 3.2 and 3.3 in [1]) – SD/SA: Steepest Descent/Ascent , – CG-PR: Conjugate Gradient (Polak-Ribière formula) – CG-FR: Conjugate Gradient (Fletcher-Reeves formula) • The following line search methods are implemented and used together with the gradient algorithms (see Section 3.2.3, and Table 3.4 in [1]): – Armijo step method, – polynomial approximation-based method – DFT approximation-based method All combinations of Riemannian gradient methods and line search methods proposed in [1] are tested. In practical applications, just one of these algorithms (SD/SA, CG- PR/CG-FR) together with one of the line search methods (Armijo, polynomial of DFT- based) is sufficient to solve the problem at hand. The algorithm/method to be used may be chosen based on experimental testing. The difference in performance may depend on the cost function. In general, based on our experience, first we recommend CG-PR with the polynomial-based line search method, and second, SD/SA with the DFT-based line search method. As a test example, the Brockett cost function (see [1] for details). NOTE: This set of codes was designed such that the Riemannian optimization scripts are separated from the cost function specific scripts. Therefore, these codes can also be easily used to optimize other smooth cost functions, simply by changing the cost function- specific parameters/scripts. The cost function specific parts are: the cost function evalu- ation, the gradient computation and the order of the cost function in the coefficients of W (details are provided in the next section). For a much more detailed description of the methods, codes and their operation, see README.pdf file. ** References [1] T. Abrudan. “Advanced Optimization Algorithms for Sensor Arrays and Multi- antenna Communications”, (PhD. thesis) Department of Signal Processing and Acous- tics, Aalto University, Finland, 21 Nov. 2008. PDF available at: http://lib.tkk.fi/Diss/2008/isbn9789512296071/ ------- BibTeX citation @PhdThesis{Abr08_PhD_thesis, author = {T. Abrudan}, title = {Advanced Optimization Algorithms for Sensor Arrays and Multi-antenna Communications}, school = {Department of Signal Processing and Acoustics, Aalto University, Finland}, year = {2008}, month = {21 Nov.}, url = {http://lib.tkk.fi/Diss/2008/isbn9789512296071/}, }

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