Non-convex Quadratically Constrained Quadratic Programming: Hidden Convexity, Scalable Approximation and Applications

University of Minnesota Ph.D. dissertation. September 2017. Major: Electrical Engineering. Advisor: Nicholas Sidiropoulos. 1 computer file (PDF); viii, 85 pages.Quadratically Constrained Quadratic Programming (QCQP) constitutes a class of computationally hard optimization problems that have a broad spectrum of applications in wireless communications, networking, signal processing, power systems, and other areas. The QCQP problem is known to be NP–hard in its general form; only in certain special cases can it be solved to global optimality in polynomial-time. Such cases are said to be convex in a hidden way, and the task of identifying them remains an active area of research. Meanwhile, relatively few methods are known to be effective for general QCQP problems. The prevailing approach of Semidefinite Relaxation (SDR) is computationally expensive, and often fails to work for general non-convex QCQP problems. Other methods based on Successive Convex Approximation (SCA) require initialization from a feasible point, which is NP-hard to compute in general. This dissertation focuses on both of the above mentioned aspects of non-convex QCQP. In the first part of this work, we consider the special case of QCQP with Toeplitz-Hermitian quadratic forms and establish that it possesses hidden convexity, which makes it possible to obtain globally optimal solutions in polynomial-time. The second part of this dissertation introduces a framework for efficiently computing feasible solutions of general quadratic feasibility problems. While an approximation framework known as Feasible Point Pursuit-Successive Convex Approximation (FPP-SCA) was recently proposed for this task, with considerable empirical success, it remains unsuitable for application on large-scale problems. This work is primarily focused on speeding and scaling up these approximation schemes to enable dealing with large-scale problems. For this purpose, we reformulate the feasibility criteria employed by FPP-SCA for minimizing constraint violations in the form of non-smooth, non-convex penalty functions. We demonstrate that by employing judicious approximation of the penalty functions, we obtain problem formulations which are well suited for the application of first-order methods (FOMs). The appeal of using FOMs lies in the fact that they are capable of efficiently exploiting various forms of problem structure while being computationally lightweight. This endows our approximation algorithms the ability to scale well with problem dimension. Specific applications in wireless communications and power grid system optimization considered to illustrate the efficacy of our FOM based approximation schemes. Our experimental results reveal the surprising effectiveness of FOMs for this class of hard optimization problems

Adaptive Detection and Estimation in the Presence of Useful Signal and Interference Mismatches

This paper considers adaptive detection and estimation in the presence of useful signal and interference mismatches. We assume a homogeneous environment where the random disturbance components from the primary and secondary data share the same covariance matrix. Moreover, the data under test contains a deterministic interference vector in addition to the possible useful signal. We focus on the situation where an energy fraction of both the useful signal and the deterministic interference may lie outside their nominal subspaces (conical uncertainty model). Under these conditions, we devise a procedure for the computation of the joint maximum likelihood (ML) estimators of the useful signal and interference vectors, resorting to a suitable rank-one decomposition of a semidefinite program (SDP) problem optimal solution. Hence, we use the aforementioned estimators for the synthesis of adaptive receivers based on different generalized likelihood ratio test (GLRT). criteria. At the analysis stage, we assess the performance of the new detectors in comparison with some decision rules, available in open literature