Convex Optimization Algorithm Research Articles

A passive convex optimal control algorithm for teleoperating a redundant robotic arm in minimally invasive surgery

AbstractIn recent years, the remote center of motion (RCM) constraint has moved from purely mechanical to software implementation, enabling the use of serial manipulators in robotic‐assisted minimal invasive surgery. However, ensuring safety with software‐based RCM presents challenges. This article addresses this issue by introducing a novel control algorithm for a 7‐DOF redundant robotic arm, taking into account a software RCM while considering system passivity and kinematic constraints. The algorithm is both a control optimizer and a robot kinematics inversion, enabling precise and dexterous control for various surgical tasks and other control applications. By formulating a quadratic optimization problem under linear constraints, the algorithm guarantees the robotic arm's performance, safety, and stability under communication delay. Experimental validation demonstrates the effectiveness and accuracy of the control algorithm in executing a surgical training task (peg‐and‐ring) during bilateral teleoperation. The results highlight the successful implementation of the RCM constraint, ensuring patient safety and optimizing the manipulation capabilities of the robotic arm.

An optimization based limiter for enforcing positivity in a semi-implicit discontinuous Galerkin scheme for compressible Navier–Stokes equations

We consider an optimization-based limiter for enforcing positivity of internal energy in a semi-implicit scheme for solving gas dynamics equations. With Strang splitting, the compressible Navier–Stokes system is split into the compressible Euler equations, which are solved by the positivity-preserving Runge–Kutta discontinuous Galerkin (DG) method, and the parabolic subproblem, which is solved by Crank–Nicolson in time with interior penalty DG method. Such a scheme is at most second order accurate in time, high order accurate in space, conservative, and preserves positivity of density. To further enforce the positivity of internal energy, we impose an optimization-based limiter for the total energy variable to post-process DG polynomial cell averages. The optimization-based limiter can be efficiently implemented by the popular first order convex optimization algorithms such as the Douglas–Rachford splitting method by using nearly optimal algorithm parameters. Numerical tests suggest that the DG method with Qk basis and the optimization-based limiter is robust for demanding low-pressure problems such as high-speed flows.

Distributed event-triggered algorithm for convex optimization with coupled constraints

This paper develops a distributed primal–dual algorithm via an event-triggered mechanism to solve a class of convex optimization problems subject to local set constraints, coupled equality and inequality constraints. Different from some existing distributed algorithms with the diminishing step-sizes, our algorithm uses the constant step-sizes, and is shown to achieve an exact convergence to an optimal solution with an ergodic convergence rate of O(1/k) for general convex objective functions, where k>0 is the iteration number. Based on the event-triggered communication mechanism, the proposed algorithm can effectively reduce the communication cost without sacrificing the convergence rate. Finally, a numerical example is presented to verify the effectiveness of the proposed algorithm.

Online dual robot–human collaboration trajectory generation by convex optimization

For dynamic collision-free trajectory planning in dual-robot and human collaborative tasks, this paper develops an online dual-robot Mutual Collision Avoidance (MCA) scheme based on convex optimization. A novel convex optimization formulation model, named Disciplined Convex programming by Shifting reference paths (DCS), is proposed for solving the single-robot trajectory optimization problem. Furthermore, a new dual-robot trajectory convex optimization algorithm is presented for online adjustment of the dual-robot trajectories according to the collaborative task priority. The overall pipeline, named DCS-MCA, generates collision-free and time-optimal dual-robot trajectories, while prioritizing the task accessibility of the high-priority robot. Simulation experiments demonstrate that DCS exhibits comparable performance to the current state-of-the-art single-robot motion planner, while the DCS-MCA outperforms common algorithms by up to 30% in time optimality for dual-robot collaborative tasks. The feasibility and dynamic performance of the proposed approach are further validated in a real collaborative cell, illustrating its suitability for collaborative dual-robot tasks in moderately dynamic environments.

A dual scale spatio‐temporal control for complex distributed parameter systems

AbstractIt is very difficult to control the distributed parameter system (DPS) for consistent spatial performance. In this paper, a dual scale spatio‐temporal control is designed for the DPS to achieve a consistent performance across the entire workspace under unknown exogenous disturbances. First, the generalized “spatial observer” should be designed under spectral decomposition/synthesis, to act as the nominal model of the DPS, through which the spatial mismatch between the model and the process could be estimated. Then a distributed disturbance compensator can be constructed to suppress all the undesirable spatial disturbance through the inner loop, and the compensated system will become closer to the nominal model and easier to control. Third, a dual controller will be designed in the outer loop for the final consistent spatial performance. Since the spatial performance is mainly affected by the dominant dynamics on the slow scale, a convex optimization algorithm can be effectively established in terms of nonlinear matrix inequalities for the dual controller. In this way, the nonlinear optimization problem is solved by converting it into the problem of a linear matrix inequalities with constraints. Besides, the spatio‐temporal state variable of the controlled plant is demonstrated to converge in Hilbert space. The feasibility and effectiveness of the proposed control method are verified by temperature control of a catalytic rod, a benchmark in chemical reactors.

Secure transmission of IRS-UAV buffer-aided relaying system with delay constraint

This paper investigates the secure communication between legitimate users in the presence of eavesdroppers, where the Intelligent Reflective Surface-Unmanned Aerial Vehicle (IRS-UAV) and Buffer-Aided (BA) relaying techniques are utilized to enhance secrecy performance. By jointly optimizing the link selection strategy, the UAV position, and the reflection coefficient of the IRS, we aim to maximize the long-term average secrecy rate. Specifically, we propose a novel buffer in/out stabilization scheme based on the Lyapunov framework, which transforms the long-term average secrecy rate maximization problem into two per-slot drift-plus-penalty minimization problems with different link selection factors. The hybrid Particle Swarm Optimization-Artificial Fish Swarm Algorithm (PSO-AFSA) is adopted to optimize the UAV position, and the IRS reflection coefficient optimization problem is solved by iterative optimization in which auxiliary variables and standard convex optimization algorithms are introduced. Finally, the delay constraint is set to ensure the timeliness of information packets. Simulation results demonstrate that our proposed scheme outperforms the comparison schemes in terms of average secrecy rate. Specifically, the addition of BA improves the average secrecy rate by 1.37 bps/Hz, and the continued optimizations of IRS reflection coefficients and UAV positions improve the average secrecy rate by 2.46 bps/Hz and 3.75 bps/Hz, respectively.

Optimal control of a solar sail

SummaryOptimal control of a solar sail orbiting around a fixed center of mass is considered. The sail is modelled as a plane surface with two sides having similar optical properties. The control is assumed to be the attitude of the sail and is represented as an element of the projective plane. Mapping this plane into the three dimensional ambient space to compute the actual force generated by a given attitude results in a generally non‐convex set of admissible control values. A suitable convex relaxation is introduced to study the optimality system associated with maximising the change of the sail orbital parameters in a given direction along one orbit. Existence for both the relaxed and the original problems are deduced. Building on previous works, a refined analysis of the control structure is given, proving rigorously that it contains only switchings, and a global bound of the number of switchings is obtained. In order to compute effective solutions, a multiple shooting approach is retained that is well suited for systems with a sequence of switchings between various control subarcs. Three issues are addressed: initialization of shooting by means a tailored semi‐definite relaxation that takes advantage of good convergence properties of modern convex optimization algorithms; changes in the control structure that are accommodated by coupling shooting with differential continuation; implicit character of the Hamiltonian maximization when using Pontrjagin maximum principle that is taken care of by incorporating the associated equation for the dynamical feedback into the shooting procedure. The method is illustrated on an example from NASA for which the sail inclination is optimally changed over one orbital period.

An approximation algorithm for multiobjective mixed-integer convex optimization

In this article we introduce an algorithm that approximates the nondominated sets of multiobjective mixed-integer convex optimization problems. The algorithm constructs an inner and outer approximation of the front exploiting the convexity of the patches for problems with an arbitrary number of criteria. In the algorithm, the problem is decomposed into patches, which are multiobjective convex problems, by fixing the integer assignments. The patch problems are solved using (simplicial) Sandwiching. We identify parts of patches that are dominated by other patches and ensure that these patch parts are not refined further. We prove that the algorithm converges and show a bound on the reduction of the approximation error in the course of the algorithm. We illustrate the behaviour of our algorithm using some numerical examples and compare its performance to an algorithm from literature.

Hidden Convexity, Optimization, and Algorithms on Rotation Matrices

This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices [Formula: see text]. Such problems are nonconvex because of the constraint [Formula: see text]. Nonetheless, we show that certain linear images of [Formula: see text] are convex, opening up the possibility for convex optimization algorithms with provable guarantees for these problems. Our main technical contributions show that any two-dimensional image of [Formula: see text] is convex and that the projection of [Formula: see text] onto its strict upper triangular entries is convex. These results allow us to construct exact convex reformulations for constrained optimization problems over [Formula: see text] with a single constraint or with constraints defined by low-rank matrices. Both of these results are maximal in a formal sense. Funding: A. Ramachandran was supported by the H2020 program of the European Research Council [Grant 805241-QIP]. A. L. Wang was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant OCENW.GROOT.2019.015 (OPTIMAL)]. K. Shu was supported by the Georgia Institute of Technology (ACO-ARC fellowship).

Relaxed static output feedback control for discrete-time Takagi-Sugeno fuzzy systems: A switching sequence convex optimization algorithm

In this paper, a new switching sequence convex optimization (SSCO) algorithm is proposed for solving non-convex optimization problem with complex time-varying relaxation matrix structures that arises during output feedback design. Firstly, the introduced time-varying relaxation matrix combines the membership functions and the designed switching mechanism to adjust the positive and negative terms of the inequality constraints. As a result, relaxed controller design conditions with complex matrix structures are established. The proposed SSCO algorithm employs switching optimization variables and inner approximation strategy, which is able to compute non-convex optimization problems with complex matrix structures more flexibly and converge quickly. It is worth noting that the implementation of the SSCO algorithm requires a set of strictly feasible initial solutions. Therefore, an initialization iterative algorithm is proposed, which overcomes the difficulties of transforming the solving problem into a typical non-convex optimization problem and linearizing multiple different concave parts, by which a set of optimized feasible solutions are obtained. Finally, simulation examples are used to demonstrate the superiority of the design scheme proposed in this paper.

Convex optimization topology identification of distribution network under distributed conditions

Abstract The distributed phenomenon will occupy a very important proportion in the future distribution network. In order to ensure the stable operation of the power system, it is very important to study the topological adaptability identification of new distribution networks. In this paper, an adaptive topology identification regression matrix for a low-voltage distribution network is proposed, which is combined with a convex optimization algorithm to build a topology identification model for a low-voltage distribution network. This method can identify the topology of the distribution network accurately, which is helpful to improve the operation efficiency and safety of the power system.

Robust optimal power flow considering uncertainty in wind power probability distribution

This paper proposes an optimal power flow model that takes into account the uncertainty in the probability distribution of wind power. The model can schedule controllable generators under any possible distribution of wind power to ensure the safe and economic operation of the system. Firstly, considering the incompleteness of historical wind power data, the paper models the uncertainty of wind power using second-order moments of probability distribution and their fluctuation intervals. Subsequently, a robust optimal power flow model based on probability distribution model and joint chance constraints is established. The Lagrangian duality theorem is then employed to eliminate random variables from the optimization model, transforming the uncertainty model into a deterministic linear matrix inequality problem. Finally, a convex optimization algorithm is used to solve the deterministic problem, and the results are compared with traditional chance-constrained optimal power flow model. The feasibility and effectiveness of the proposed method are validated through case study simulations.

A powered descent trajectory planning method with quantitative consideration of safe distance to obstacle

High scientific value areas on celestial bodies such as the Moon and Mars are often located in hazardous terrains. To achieve safe landing exploration, a novel planning method is proposed, which can ensure that the planned trajectory maintains a user-specified distance from obstacles, thus reducing potential collision risk induced by factors such as the body size and model uncertainties. Firstly, the basic model of the trajectory optimization problem and its convexification version is given. The obstacles are modeled as polynomial functions, based on which the “if-then” obstacle avoidance logic explicitly considering the safe distance, is described as a sum of squares constraint. This constraint formulation applies to any obstacle described by a finite number of polynomials, independent of the specific expression of the polynomials (reflecting the shape of the obstacle). Subsequently, the convexification process for the obstacle avoidance constraint is given. Finally, the sequential sum of squares programming problem for the obstacle avoidance trajectory is established, which boils down to a series of semidefinite programming problems. Simulation results show that the closest distance between the planned trajectory and obstacles strictly satisfies the specified distance constraint, and the trajectory could avoid non-convex obstacles. With the promising convergence properties of underlying convex optimization algorithms, advanced autonomous obstacle avoidance guidance schemes are expected to be formed based on the proposed trajectory planning method.

Asynchronous event-triggered guaranteed cost control of uncertain 2-D Markov jump Roesser systems with actuator saturation

This paper investigates the asynchronous event-triggered guaranteed cost control problem for two-dimensional Markov jump Roesser systems with parameter uncertainties and actuator saturation. The hidden Markov model is constructed to characterize the asynchronous phenomenon induced by the mode mismatch between the system and the controller, and a hidden Markov model-based event-triggered mechanism is adopted in controller design to save communication resources. The convex hull representation is employed to process saturated inputs. By using the quadratic Lyapunov function methods and linear matrix inequality techniques, some sufficient criteria are developed to guarantee the asymptotic stability of the addressed system with a guaranteed cost under three different boundary conditions. Finally, a convex optimization algorithm with linear matrix inequality constraints is proposed to design the optimal asynchronous event-triggered guaranteed cost controller, and a numerical example is considered to demonstrate its validity.

An Approach to Maximize the Admitted Device-to-Device Pairs in MU-MIMO Cellular Networks

Due to the shortage of wireless resources and the emergence of a large number of users, determining how to guarantee the quality-of-service (QoS) requirements of users and make more users work in the same spectrum has become an urgent research topic. In this paper, we study a multi-user MIMO (MU-MIMO) cellular network system model in which cellular users (CUs) share the same spectrum resource with multiple device-to-device (D2D) pairs. To maximize the number of admitted D2D pairs sharing the same spectrum with the CUs, a joint power allocation and channel gain (JPACG) algorithm is proposed. The optimization problem is divided into two steps to be solved. First, the power allocation of CUs without D2D pairs admitted is solved. Then, the optimization problem is transformed into minimizing the interference to CUs when CUs are treated as primary users. The admittance order of D2D pairs is determined by the transmission power and channel gain. The proposed algorithm uses a convex optimization algorithm to solve the problem of power allocation joint interference channel gain in order to maximize the number of admitted D2D pairs under the constraints of the signal-to-interference-plus-noise ratio (SINR) threshold and maximum transmission power. In addition, the effect of the number of admitted D2D pairs on the total sum rate of all users is also analyzed. The simulation results show that the proposed JPACG algorithm can achieve better performance in admitting D2D pairs.

On approximating dependence function and its derivatives

Bivariate extreme value distributions can be used to model dependence of observations from random variables in extreme levels. There is no finite dimensional parametric family for these distributions, but they can be characterized by a certain one-dimensional function which is known as Pickands dependence function. In many applications the general approach is to estimate the dependence function with a non-parametric method and then conduct further analysis based on the estimate. Although this approach is flexible in the sense that it does not impose any special structure on the dependence function, its main drawback is that the estimate is not available in a closed form. This paper provides some theoretical results which can be used to find a closed form approximation for an exact or an estimate of a twice differentiable dependence function and its derivatives. We demonstrate the methodology by calculating approximations for symmetric and asymmetric logistic dependence functions and their second derivatives. We show that the theory can be even applied to approximating a non-parametric estimate of dependence function using a convex optimization algorithm. Other discussed applications include a procedure for testing whether an estimate of dependence function can be assumed to be symmetric and estimation of the concordance measures of a bivariate extreme value distribution. Finally, an Australian annual maximum temperature dataset is used to illustrate how the theory can be used to build semi-infinite and compact predictions regions.

A multi-modal vision-language pipeline strategy for contour quality assurance and adaptive optimization

Objective. Accurate delineation of organs-at-risk (OARs) is a critical step in radiotherapy. The deep learning generated segmentations usually need to be reviewed and corrected by oncologists manually, which is time-consuming and operator-dependent. Therefore, an automated quality assurance (QA) and adaptive optimization correction strategy was proposed to identify and optimize ‘incorrect’ auto-segmentations. Approach. A total of 586 CT images and labels from nine institutions were used. The OARs included the brainstem, parotid, and mandible. The deep learning generated contours were compared with the manual ground truth delineations. In this study, we proposed a novel contour quality assurance and adaptive optimization (CQA-AO) strategy, which consists of the following three main components: (1) the contour QA module classified the deep learning generated contours as either accepted or unaccepted; (2) the unacceptable contour categories analysis module provided the potential error reasons (five unacceptable category) and locations (attention heatmaps); (3) the adaptive correction of unacceptable contours module integrate vision-language representations and utilize convex optimization algorithms to achieve adaptive correction of ‘incorrect’ contours. Main results. In the contour QA tasks, the sensitivity (accuracy, precision) of CQA-AO strategy reached 0.940 (0.945, 0.948), 0.962 (0.937, 0.913), and 0.967 (0.962, 0.957) for brainstem, parotid and mandible, respectively. The unacceptable contour category analysis, the (FI,AccI,Fmicro,Fmacro) of CQA-AO strategy reached (0.901, 0.763, 0.862, 0.822), (0.855, 0.737, 0.837, 0.784), and (0.907, 0.762, 0.858, 0.821) for brainstem, parotid and mandible, respectively. After adaptive optimization correction, the DSC values of brainstem, parotid and mandible have been improved by 9.4%, 25.9%, and 13.5%, and Hausdorff distance values decreased by 62%, 70.6%, and 81.6%, respectively. Significance. The proposed CQA-AO strategy, which combines QA of contour and adaptive optimization correction for OARs contouring, demonstrated superior performance compare to conventional methods. This method can be implemented in the clinical contouring procedures and improve the efficiency of delineating and reviewing workflow.

Distributed Primal-Dual Proximal Algorithms for Convex Optimization Involving Three Composite Functions

Distributed Primal-Dual Proximal Algorithms for Convex Optimization Involving Three Composite Functions

Rocket landing guidance based on second-order Picard-Chebyshev-Newton type algorithm

This paper proposes a rocket substage vertical landing guidance method based on the second-order Picard-Chebyshev-Newton type algorithm. Firstly, the continuous-time dynamic equation is discretized based on the natural second-order Picard iteration formulation and the Chebyshev polynomial. Secondly, the landing problem that considers terminal constraints is transformed into a nonlinear least-squares problem with respect to the terminal constraint function and solved with the Gauss-Newton method. In addition, the projection method is introduced to the iteration process of the Gauss-Newton method to realize the inequality constraints of the thrust. Finally, the closed-loop strategy for rocket substage vertical landing guidance is proposed and the numerical simulations of the rocket vertical landing stage are carried out. The simulation results demonstrate that compared with the sequential convex optimization algorithm, the proposed algorithm has higher computational efficiency.

Distributed Newton Step Projection Algorithm for Online Convex Optimization Over Time-Varying Unbalanced Networks

Distributed Newton Step Projection Algorithm for Online Convex Optimization Over Time-Varying Unbalanced Networks