Real-Time Convex Optimization in Signal Processing

Abstract

Abstract-Convex optimization has been used in signal processing for a long time, to choose coefficients for use in fast (linear) algorithms, such as in filter or array design; more recently, it has been used to carry out (nonlinear) processing on the signal itself. Examples of the latter case include total variation de-noising, compressed sensing, fault detection, and image classification. In both scenarios, the optimization is carried out on time scales of seconds or minutes, and without strict time constraints. Convex optimization has traditionally been considered computationally expensive, so its use has been limited to applications where plenty of time is available. Such restrictions are no longer justified. The combination of dramatically increased computational power, modern algorithms, and new coding approaches has delivered an enormous speed increase, which makes it possible to solve modest-sized convex optimization problems on microsecond or millisecond time scales, and with strict deadlines. This enables real-time convex optimization in signal processing.

Real-time implementations of sparse linear prediction for speech processing

“…Specialized software to solve real-time convex optimization exists but it has a few limitations. For example, to implement small/sparse problems it is possible to use automatic tools for C code generation of a primal-dual interior-point method for a specific problem family which results in fast implementations [14,19]. The sparse LPC problem is small but X is in general not sparse and even m, n ≈ 40 will result in the number of coefficients in the problem to grow above the suggested limit of 4000 coefficients that the system can handle (as reported on cvxgen.com, software page of [19]).…”

“…However, modern algorithms along with technology advances in processing power, have dramatically reduced solution times. This introduces the possibility of embedding convex optimization directly in signal processing algorithms that run online, with strict real-time constraints [14,15].…”

Some applications of polynomial optimization in operations research and real-time decision making

“…Hence, a real-time implementation of this approach on a hardware platform is plausible. Such a hardware implementation can benefit from already-existing SOCP solvers that are specifically designed to run on embedded systems [29], [28]. In particular, [17] presents an approach for generating stand-alone C code for an SOCP solver that can run very efficiently and with low memory footprint.…”

Data-driven and adaptive statistical residual evaluation for fault detection with an automotive application

“…State-of-theart algorithms often exploits code-generation, where solvers are customized to a specific problem class. One such example is CVXGEN [42], which enables real-time, i.e., solving time scales in microseconds or milliseconds with strict deadlines, solving of modest-sized quadratic optimization problems [41].…”