Real-time Convex Optimization Research Articles

Exploiting Block Structures of KKT Matrices for Efficient Solution of Convex Optimization Problems

Convex optimization solvers are widely used in the embedded systems that require sophisticated optimization algorithms including model predictive control (MPC). In this paper, we aim to reduce the online solve time of such convex optimization solvers so as to reduce the total runtime of the algorithm and make it suitable for real-time convex optimization. We exploit the property of the Karush–Kuhn–Tucker (KKT) matrix involved in the solution of the problem that only some parts of the matrix change during the solution iterations of the algorithm. Our results show that the proposed method can effectively reduce the runtime of the solvers.

Distributed Optimization for Robot Networks: From Real-Time Convex Optimization to Game-Theoretic Self-Organization

Recent advances in sensing, communication, and computing technologies have enabled the use of multirobot systems for practical applications such as surveillance, area mapping, and search and rescue. For such systems, a major challenge is to design decision rules that are real-time-implementable, require local information only, and guarantee some desired global performance. Distributed optimization provides a framework for designing such local decision-making rules for multirobot systems. In this article, we present a collection of selected results for distributed optimization for robot networks. We will focus on two special classes of problems: 1) real-time path planning for multirobot systems and 2) self-organization in multirobot systems using game-theoretic approaches. For multirobot path planning, we will present some recent approaches that are based on approximately solving distributed optimization problems over continuous and discrete domains of actions. The main idea underlying these approaches is that a variety of path planning problems can be formulated as convex optimization and submodular minimization problems over continuous and discrete action spaces, respectively. To generate local update rules that are efficiently implementable in real time, these approaches rely on approximate solutions to the global problems that can still guarantee some level of desired global performance. For game-theoretic self-organization, we will present a sampling of results for area coverage and real-time target assignment. In these results, the problems are formulated as games, and online updating rules are designed to enable teams of robots to achieve the collective objective in a distributed manner.

Time-optimized 4D phase contrast MRI with real-time convex optimization of gradient waveforms and fast excitation methods.

To shorten 4D flow acquisitions by shortening TRs with fast RF pulses and gradient waveforms. Real-time convex optimization is used to generate these gradients waveforms on the scanner. RF and slab-select waveforms were shortened with a minimum phase SLR excitation and the time-optimal variable-rate selective excitation method. Real-time convex optimization was used to shorten bipolar and spoiler gradients by finding the shortest gradient waveforms that satisfied constraints on scan parameters, gradient hardware, M0 , M1 , and peripheral nerve stimulation. Waveforms were calculated and TE and/or TR values were compared for a range of scan parameters and compared to a conventional 4D flow sequence. The method was tested in flow phantoms, and in the aorta and neurovasculature of volunteers (N = 10). Additionally, eddy current error was measured in a large phantom. TEs and TRs were shortened by 21-32% and 20-34%, respectively, compared to the conventional sequence over a range of scan parameters. Bland-Altman analysis of 2 flow phantom configurations showed flow rate bias of 0.3 mL/s and limits of agreement (LOA) of [-6.9, 7.5] mL/s for a cardiac phantom and a bias of -0.1 mL/s with LOA = [-0.4, 0.2] mL/s for a neuro phantom. Similar agreement was also seen for flow measurements in volunteers (bias = -1.0 and -0.1 mL/s, LOA = [-34.9, 33.0] and [-0.7, 0.6] mL/s). Measured eddy currents were 39% larger with the CVX + mpVERSE method. The real-time optimized 4D flow gradients and fast slab-selection excitation methods produced up to 34% faster TRs with excellent flow measurement agreement compared to a conventional 4D flow sequence.

Real-Time Convex Optimization in Signal Processing

This article shows the potential for convex optimization methods to be much more widely used in signal processing. In particular, automatic code generation makes it easier to create convex optimization solvers that are made much faster by being designed for a specific problem family. The disciplined convex programming framework that has been shown useful in transforming problems to a standard form may be extended to create solvers themselves. Much work remains to be done in exploring the capabilities and limitations of automatic code generation. As computing power increases, and as automatic code generation improves, the authors expect convex optimization solvers to be found more and more often in real-time signal processing applications.

A hybrid steepest descent method for constrained convex optimization

This paper describes a hybrid steepest descent method to decrease over time any given convex cost function while keeping the optimization variables in any given convex set. The method takes advantage of the properties of hybrid systems to avoid the computation of projections or of a dual optimum. The convergence to a global optimum is analyzed using Lyapunov stability arguments. A discretized implementation and simulation results are presented and analyzed. This method is of practical interest to integrate real-time convex optimization into embedded controllers thanks to its implementation as a dynamical system, its simplicity, and its low computation cost.

Control of a climbing robot using real-time convex optimization

This paper presents a controller for a free-climbing robot called Cartesian Force Convex control. This controller computes in real-time control inputs that will move a robot safely along a nominal plan by using a convex optimization technique based on an LP solver to compute torques to joint motors. For the LP solution, a cost function is chosen that increases robustness by weighting tracking performance of the robot’s body against the margins in the constraint variables (e.g. proximity of the contact forces to their friction cone limits). The constraints are formulated to characterize the climbing problem and include both static and dynamic limits. The effectiveness of this controller is demonstrated both with simulations and hardware experiments.