Convex Optimization Algorithm Research Articles

Note on the Equivalence of Orthogonalizing EM and Proximal Gradient Descent

Xiong et al. (2016) develop a method called orthogonalizing EM (OEM) to solve penalized regression problems for tall data. While OEM is developed in the context of the EM algorithm, we show that it is, in fact, an instance of proximal gradient descent, a popular first-order convex optimization algorithm.

Comparative analysis of accelerated gradient algorithms for convex optimization: high and super resolution ODE approach

We investigate convex differentiable optimization and explore the temporal discretization of damped inertial dynamics driven by the gradient of the objective function. This leads to three accelerated gradient algorithms: Nesterov Accelerated Gradient (NAG), Ravine Accelerated Gradient (RAG), and (IGAHD). The latter was introduced by discretizing inertial dynamics with Hessian-driven damping to attenuate inherent oscillations in inertial methods. By analysing the high-resolution ODEs of order p = 0,1,2 for these algorithms, we gain insights into their similarities and differences. All three algorithms share the same low-resolution ODE of order 0, which is the dynamic proposed as a continuous surrogate for (NAG). To differentiate Nesterov from Ravine, we refine the comparison and demonstrate distinct high-resolution ODEs of order 2 in h (termed super-resolution). The corresponding Taylor expansions in h reveal matching terms of order 1 but differing terms of order 2. To the best of our knowledge, this result is completely new and emphasizes the need to avoid confusion between the Ravine and Nesterov methods in the literature. We present numerical experiments to illustrate our theoretical results. Performance profiles, measuring the number of iterations, indicate that (IGAHD) outperforms both (NAG) and (RAG) methods. (RAG) exhibits a slight advantage over (NAG) in terms of the average number of iterations. When considering CPU-time, both (RAG) and (NAG) outperform (IGAHD). All three algorithms exhibit similar behaviour when evaluating based on gradient norms.

Adaptive proximal algorithms for convex optimization under local Lipschitz continuity of the gradient

Adaptive proximal algorithms for convex optimization under local Lipschitz continuity of the gradient

DDQN-based online computation offloading and application caching for dynamic edge computing service management

Multi-access Edge Computing (MEC) reduces task service latency and energy consumption by offloading computing tasks to MEC servers. However, constrained by the limited bandwidth and computing resources, MEC servers often cannot parallelly process all computing tasks. Simultaneously, the high dynamism of service popularity necessitates MEC servers to dynamically update cached applications, under ensuring compliance with storage resource constraints and the system cache updating cost budget for service providers. In response to the above two issues, this paper firstly formulates computation offloading and application caching as a dual-timescale decision optimization problem, aiming to minimize the average service latency for users by obtaining optimal offloading decision, cache decision, transmission bandwidth, and computing resource. Then, we propose a Deep Reinforcement Learning (DRL)-based two-stage online computation offloading and application caching (DTSO2C) algorithm, effectively stabilizing application cache update costs and enhancing Quality of Service (QoS) for users. Furthermore, we utilize convex optimization algorithms to derive the optimal communication bandwidth and computing resource allocation strategy, further reducing the average service latency for users. Simulation results demonstrate that the DTSO2C algorithm outperforms the compared algorithms, achieving an average reduction in service latency of 66.2%, with an average cache update cost of only 0.15 USD per time frame.

Optimal operation of three-phase unbalanced active distribution system based on semidefinite relaxation and convex-concave procedure

The three-phase distribution optimal power flow (D-OPF) problem has attracted much attention in recent years due to practical requirements for managing distributed energy resources and multi-energy customers in unbalanced active distribution systems. To deal with the strong nonconvexity of the three-phase OPF problem, this paper develops the semidefinite programming (SDP) formulation of three-phase OPF problems considering the mutual coupling in multi-phase networks. Instead of directly dropping the rank-one constraints, a sequential convex optimization algorithm based on the convex-concave procedure (CCP) is proposed for recovering a feasible and optimal solution of three-phase power flow. The proposed iterative algorithm not only has high computational efficiency but also guarantees the optimality when the semidefinite relaxation or second-order cone relaxation method is inexact. Case studies on various IEEE testing distribution systems verify that the proposed algorithm recovers the feasible and optimal power flow solution. It also exhibits a relatively low number of iterations and short solving time in large-scale IEEE testing systems.

Energy Efficiency Maximization for Multi-UAV-IRS-Assisted Marine Vehicle Systems

Mobile edge computing is envisioned as a prospective technology for supporting time-sensitive and computation-intensive applications in marine vehicle systems. However, the offloading performance is highly impacted by the poor wireless channel. Recently, an Unmanned Aerial Vehicle (UAV) equipped with an Intelligent Reflecting Surface (IRS), i.e., UIRS, has drawn attention due to its capability to control wireless signals so as to improve the data rate. In this paper, we consider a multi-UIRS-assisted marine vehicle system where UIRSs are deployed to assist in the computation offloading of Unmanned Surface Vehicles (USVs). To improve energy efficiency, the optimization problem of the association relationships, computation resources of USVs, multi-UIRS phase shifts, and multi-UIRS trajectories is formulated. To solve the mixed-integer nonlinear programming problem, we decompose it into two layers and propose an integrated convex optimization and deep reinforcement learning algorithm to attain the near-optimal solution. Specifically, the inner layer solves the discrete variables by using the convex optimization based on Dinkelbach and relaxation methods, and the outer layer optimizes the continuous variables based on the Multi-Agent Twin Delayed Deep Deterministic Policy Gradient (MATD3). The numerical results demonstrate that the proposed algorithm can effectively improve the energy efficiency of the multi-UIRS-assisted marine vehicle system in comparison with the benchmarks.

Nonsmooth projection-free optimization with functional constraints

This paper presents a subgradient-based algorithm for constrained nonsmooth convex optimization that does not require projections onto the feasible set. While the well-established Frank–Wolfe algorithm and its variants already avoid projections, they are primarily designed for smooth objective functions. In contrast, our proposed algorithm can handle nonsmooth problems with general convex functional inequality constraints. It achieves an ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document}-suboptimal solution in O(ϵ-2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(\\epsilon ^{-2})$$\\end{document} iterations, with each iteration requiring only a single (potentially inexact) Linear Minimization Oracle call and a (possibly inexact) subgradient computation. This performance is consistent with existing lower bounds. Similar performance is observed when deterministic subgradients are replaced with stochastic subgradients. In the special case where there are no functional inequality constraints, our algorithm competes favorably with a recent nonsmooth projection-free method designed for constraint-free problems. Our approach utilizes a simple separation scheme in conjunction with a new Lagrange multiplier update rule.

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.

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.

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.