Lipschitz-Based Precoding & Equalization in Reconfigurable Intelligent Surface Assisted MIMO Broadcast Channels
This work addresses central mathematical issues in modern optimization efforts of wireless communications systems via Intelligent Reflecting Surfaces and offers bounds and algorithms designed to optimize these systems and work around the induced mathematical difficulties.
Chapters 1-3 set up the system environment and derive the optimization problem inherent to the industry incentive of maximizing rates of data transmission. This central maximization problem is decomposed into an alternating maximization and minimization procedure, denoted by P6 and P8, respectively.
Chapter 4 focuses on the objective function of P8 and in utilizing analytical approximations which function as favorable surrogates to our original objective. Such approximations require additional information provided by the Hessian matrix, as well as require an approximation for the L-smooth Lipschitz constant of the objective. In the subsequent sections of Chapter 4, I find and demonstrate the structure of the Hessian matrix, and then construct and prove a lemma deriving an upper bound for the L-smooth constant of the objective function. I then offer and prove a corollary which provides a tighter approximation for this Lipschitz constant, under a slightly stricter condition. I then construct three algorithms with some added tuning considerations designed to achieve the highest possible performance metrics.
Chapter 5, however, shifts focus away from the objective function of P8 and focuses instead on the set of non-convex constraints and in reformulating P8 into a more tractable form. I offer two problem reformulations and prove that any solution to the reformulated problem is also a solution to P8. I then offer explicit lower bounds on the penalties required by each reformulation. For each, I construct an accelerated and a non-accelerated gradient-descent based algorithm, which are meant to serve as standardized comparisons to my fifth, non-standard, algorithm. For this fifth algorithm, I create a customized step-size searching procedure which is designed to work symbiotically with the unit-modulus set of constraints. Numerical simulations demonstrate that the tuned, Lipschitz-based algorithm from Chapter 4, and the customized projected gradient descent algorithm from Chapter 5 outperform all other algorithms derived in this work.
In my concluding remarks, I use the insight gained from all previous chapters to deduce a lowest-complexity algorithm with equally high-performance metrics with the top performing algorithm.
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