  A memristor crossbar, which is constructed with memristor devices, has the
unique ability to change and memorize the state of each of its memristor
elements. It also has other highly desirable features such as high density, low
power operation and excellent scalability. Hence the memristor crossbar
technology can potentially be utilized for developing low-complexity and
high-scalability solution frameworks for solving a large class of convex
optimization problems, which involve extensive matrix operations and have
critical applications in multiple disciplines. This paper, as the first attempt
towards this direction, proposes a novel memristor crossbar-based framework for
solving two important convex optimization problems, i.e., second-order cone
programming (SOCP) and homogeneous quadratically constrained quadratic
programming (QCQP) problems. In this paper, the alternating direction method of
multipliers (ADMM) is adopted. It splits the SOCP and homogeneous QCQP problems
into sub-problems that involve the solution of linear systems, which could be
effectively solved using the memristor crossbar in O(1) time complexity. The
proposed algorithm is an iterative procedure that iterates a constant number of
times. Therefore, algorithms to solve SOCP and homogeneous QCQP problems have
pseudo-O(N) complexity, which is a significant reduction compared to the
state-of-the-art software solvers (O(N^3.5) - O(N^4)).
