Convex Programming Method Research Articles

Receding horizon guidance and control using sequential convex programming for spacecraft 6-DOF close proximity

This paper presents a receding horizon implementation of sequential convex programming for the spacecraft six-degree-of-freedom close proximity to a non-cooperative target satellite. With linearized relative translational dynamics and newly derived discrete rotational equations in terms of modified Rodrigues parameters, nonlinear system dynamics of the original optimal guidance and control problem for proximity are converted into convex ones for sequentially planning. The nonconvex constraint on field-of-view of the visual sensor pointing with coupled attitude and relative position is then approximately relaxed as a convex standard second-order cone, and concave spherical and ellipsoidal obstacle regions respectively around the target's body and solar arrays are convexified by affine constraints in terms of tangent planes. The original nonlinear optimal guidance and control problem is accordingly transformed into a series of second-order cone programming sub-problems via iteratively successive convexification with the trust region constraints, and sequentially solved using disciplined convex programming method. A close-loop guidance and control using the proposed sequential convex programming scheme is then demonstrated by means of receding horizon to robustly drive the spacecraft maneuvering close to the target. Numerical simulations and results reveal that the proposed method provides rapid and reliable guidance and control performance for six-degree-of-freedom close proximity and shows the potential for on-board implementations in real-time applications.

Generalized Uniformly Optimal Methods for Nonlinear Programming

In this paper, we present a generic framework to extend existing uniformly optimal convex programming algorithms to solve more general nonlinear, possibly nonconvex, optimization problems. The basic idea is to incorporate a local search step (gradient descent or Quasi-Newton iteration) into these uniformly optimal convex programming methods, and then enforce a monotone decreasing property of the function values computed along the trajectory. Algorithms of these types will then achieve the best known complexity for nonconvex problems, and the optimal complexity for convex ones without requiring any problem parameters. As a consequence, we can have a unified treatment for a general class of nonlinear programming problems regardless of their convexity and smoothness level. In particular, we show that the accelerated gradient and level methods, both originally designed for solving convex optimization problems only, can be used for solving both convex and nonconvex problems uniformly. In a similar vein, we show that some well-studied techniques for nonlinear programming, e.g., Quasi-Newton iteration, can be embedded into optimal convex optimization algorithms to possibly further enhance their numerical performance. Our theoretical and algorithmic developments are complemented by some promising numerical results obtained for solving a few important nonconvex and nonlinear data analysis problems in the literature.

Resonance Theory Reboot.

What is now called "resonance theory" has a long and conflicted history. We first sketch the early roots of resonance theory, its heritage of diverse physics and chemistry conceptions, and its subsequent rise to reigning chemical bonding paradigm of the mid-20th century. We then outline the alternative "natural" pathway to localized Lewis- and resonance-structural conceptions that was initiated in the 1950s, given semi-empirical formulation in the 1970s, recast in ab initio form in the 1980s, and successfully generalized to multi-structural "natural resonance theory" (NRT) form in the 1990s. Although earlier numerical applications were often frustrated by the ineptness of then-available numerical solvers, the NRT variational problem was recently shown to be amenable to highly efficient convex programming methods that yield provably optimal resonance weightings at a small fraction of previous computational costs. Such convexity-based algorithms now allow a full "reboot" of NRT methodology for tackling a broad range of chemical applications, including the many familiar resonance phenomena of organic and biochemistry as well as the still broader range of resonance attraction effects in the inorganic domain. We illustrate these advances for prototype chemical applications, including (i) stable near-equilibrium species, where resonance mixing typically provides only small corrections to a dominant Lewis-structural picture, (ii) reactive transition-state species, where strong resonance mixing of reactant and product bonding patterns is inherent, (iii) coordinative and related supramolecular interactions of the inorganic domain, where sub-integer resonance bond orders are the essential origin of intermolecular attraction, and (iv) exotic long-bonding and metallic delocalization phenomena, where no single "parent" Lewis-structural pattern gains pre-eminent weighting in the overall resonance hybrid.

The method of conditional minimization based on the selection of the optimal value of the objective function

One of the convex programming methods that allows to reduce the solution of the original problem to the successive minimization of auxiliary functions on the whole space or on simple sets is the method of parametrization of the objective function. A method close to the one mentioned above is proposed in this paper. As in the known method, in order to find an iterative point, we solve the minimization problem of some auxiliary function constructed with cuts of the parametrized objective function and the constraint functions. The proposed method has some advantages over the parametrization method of the objective function due to a different principle of setting parameter values. The method convergence is substantiated. The properties associated with its implementation are discussed.

A penalty method for nonlinear programming

This paper presents a variant of logarithmic penalty methods for nonlinear convex programming. If the descent direction is obtained through a classical Newton-type method, the line search is done on a majorant function. Numerical tests show the efficiency of this approach versus classical line searches.

Fast Trajectory Optimization using Sequential Convex Programming with No-Fly Zone Constraints

Abstract This paper proposes a new trajectory optimization method for Unmanned Aerial Vehicle (UAV). The proposed L1-Penalized Sequential Convex Programming (LPSCP) method improves the initial infeasibility of the standard sequential convex programming. LPSCP method converges within 0.2 seconds which is more than one-tenth of the pseudospectral (PS) method. Therefore, LPSCP has the potential to enable UAV’s real-time autonomous air mission if implemented on-board. The UAV trajectory optimization problem is defined at the beginning of the paper and a convexification process is performed when there are several no-fly zones along the trajectory. Then the LPSCP method iteratively solves locally approximated subproblems in conic forms. Simulation results illustrate the proposed method satisfies the required constraints and has a computation time advantage over the conventional PS method.

Joint Spectrum and Power Allocation for NOMA Enhanced Relaying Networks

In this paper, we investigate the joint resource allocation for the non-orthogonal multiple access (NOMA)-enhanced relaying networks involving the subcarrier pair, subcarrier-user assignment, as well as power allocation. In the NOMA-enhanced relaying networks, the relay is capable of communicating with multiple users on one subcarrier using the NOMA technology. To maximize the system throughput, the joint resource allocation problem is formulated as a mixed-integer nonlinear programming problem, which is difficult to tackle in general. Furthermore, there is strong coupling between the subcarrier-user assignment and the power allocation due to the multi-user interference in the NOMA system. To reduce the complexity, we separate the joint resource allocation problem as two subproblems in terms of the subcarrier assignment (subcarrier pair and subcarrier-user assignment) and power allocation, respectively. In particular, we propose a subcarrier assignment scheme based on the simulated annealing algorithm to optimize the subcarrier pair and subcarrier-user assignment with the fixed power allocation. Then, the power allocation problem is transformed as a difference of convex functions programming problem, and the sequence convex programming method is adopted to solve it. The simulation results illustrate that the proposed algorithm can effectively improve the system throughput.

Modified hybrid decomposition of the augmented Lagrangian method with larger step size for three-block separable convex programming

The Jacobian decomposition and the Gauss–Seidel decomposition of augmented Lagrangian method (ALM) are two popular methods for separable convex programming. However, their convergence is not guaranteed for three-block separable convex programming. In this paper, we present a modified hybrid decomposition of ALM (MHD-ALM) for three-block separable convex programming, which first updates all variables by a hybrid decomposition of ALM, and then corrects the output by a correction step with constant step size alpha in(0,2-sqrt{2}) which is much less restricted than the step sizes in similar methods. Furthermore, we show that 2-sqrt{2} is the optimal upper bound of the constant step size α. The rationality of MHD-ALM is testified by theoretical analysis, including global convergence, ergodic convergence rate, nonergodic convergence rate, and refined ergodic convergence rate. MHD-ALM is applied to solve video background extraction problem, and numerical results indicate that it is numerically reliable and requires less computation.

Optimal Relaying Beamforming in Multiple Access Broadcast Channel (MABC) Bidirectional Cognitive Radio Networks in Presence of Interferers

In this paper, a general cognitive radio system consisting of a set of users with different level of spectrum access including two primary transceivers and several types of secondary users is considered. It is assumed that two secondary users operate based on an underlay model at the same frequency bandwidth and at the same time as the primary users based on a multiple access broadcast channel bidirectional beamforming scheme. Other secondary users provide a relaying service to the primary users in exchange for the opportunity to send their messages towards their own destinations for a fixed portion of the communication cycle. In addition, it is assumed that some interferers are active during the communication cycle and cause interference for the network. Furthermore, it is assumed that only partial channel state information (CSI) between interferers and other nodes in the network is available. We provide a robust optimization method against imperfection on the interferers’ CSI to maximize the joint primary and secondary signal-to-interference-plus-noise-ratio with the assumption of limited available power at the secondary relays. An amplify-and-forward relaying scheme is deployed at the secondary relays and the optimal beamforming is obtained using second order convex programming method. The simulation results show the performance of the proposed beamforming method against the existence of interferers, and demonstrate the effectiveness of our robust method against uncertainty in knowledge of interferers’ CSIs.

A New Backpressure Algorithm for Joint Rate Control and Routing With Vanishing Utility Optimality Gaps and Finite Queue Lengths

The backpressure algorithm has been widely used as a distributed solution to the problem of joint rate control and routing in multi-hop data networks. By controlling an algorithm parameter, the backpressure algorithm can achieve an arbitrarily small utility optimality gap. However, this in turn brings in a large queue length at each node and hence causes large network delay. This phenomenon is known as the fundamental utility-delay tradeoff. The best known utility-delay tradeoff for general networks is $[O(\epsilon ), O(1/\epsilon )]$ and is attained by a backpressure algorithm based on a drift-plus-penalty technique. This may suggest that to achieve an arbitrarily small utility optimality gap, backpressure-based algorithms must incur arbitrarily large queue lengths. However, this paper proposes a new backpressure algorithm that has a vanishing utility optimality gap, so utility converges to exact optimality as the algorithm keeps running, while queue lengths are bounded throughout by a finite constant. The technique uses backpressure and drift concepts with a new method for convex programming.

Relaxed inertial proximal Peaceman-Rachford splitting method for separable convex programming

The strictly contractive Peaceman-Rachford splitting method is one of effective methods for solving separable convex optimization problem, and the inertial proximal Peaceman-Rachford splitting method is one of its important variants. It is known that the convergence of the inertial proximal Peaceman-Rachford splitting method can be ensured if the relaxation factor in Lagrangian multiplier updates is underdetermined, which means that the steps for the Lagrangian multiplier updates are shrunk conservatively. Although small steps play an important role in ensuring convergence, they should be strongly avoided in practice. In this article, we propose a relaxed inertial proximal Peaceman-Rachford splitting method, which has a larger feasible set for the relaxation factor. Thus, our method provides the possibility to admit larger steps in the Lagrangian multiplier updates. We establish the global convergence of the proposed algorithm under the same conditions as the inertial proximal Peaceman-Rachford splitting method. Numerical experimental results on a sparse signal recovery problem in compressive sensing and a total variation based image denoising problem demonstrate the effectiveness of our method.

Energy-efficient trajectory planning for UAV-aided secure communication

Wireless communication with unmanned aerial vehicles (UAVs) has aroused great research interest recently. This paper is concerned with the UAV's trajectory planning problem for secrecy energy efficiency maximization (SEEM) in the UAV communication system. Specifically, we jointly consider the secrecy throughput and UAV's energy consumption in a three-node (fixed-wing UAV-aided source, destination, and eavesdropper) wiretap channel. By ignoring the energy consumption on radiation and signal processing, the system's secrecy energy efficiency is defined as the total secrecy rate normalized by the UAV's propulsion energy consumption within a given time horizon. Nonetheless, the SEEM problem is nonconvex and thus is intractable to solve. As a compromise, we propose an iterative algorithm based on sequential convex programming (SCP) and Dinkel-bach's method to seek a suboptimal solution for SEEM. The algorithm only needs to solve convex problems, and thus is computationally efficient to implement. Additionally, we prove that the proposed algorithm has Karush-Kuhn-Tucker (KKT) point convergence guarantee. Lastly, simulation results demonstrate the efficacy of our proposed algorithm in improving the secrecy energy efficiency for the UAV communication system.

Compressive Sensing Matrix Design for Fast Encoding and Decoding via Sparse FFT

Compressive sensing (CS) is proposed for signal sampling below the Nyquist rate based on the assumption that the signal is sparse in some transformed domain. Most sensing matrices (e.g., Gaussian random matrix) in CS, however, usually suffer from unfriendly hardware implementation, high computation cost, and huge memory storage. In this letter, we propose a deterministic sensing matrix for collecting measurements fed into sparse fast Fourier transform (sFFT) as the decoder. Compared with the conventional paradigm with Gaussian random matrix at encoder and convex programming or greedy method at decoders, sFFT can reconstruct sparse signals with very low computation cost under the comparable number of measurements. But, the limitation is that the signal must be sparse in the frequency domain. We further show how to relax this limitation into any domains with the transformation matrix or dictionary being circulant. Experimental and theoretical results validate that the proposed method achieves fast sensing, fast recovery, and low memory cost.

Autonomous Entry Guidance for Hypersonic Vehicles by Convex Optimization

Advanced entry guidance systems could potentially enable future vehicles to generate feasible/optimal flight trajectories onboard for the latest mission requirements and track the new trajectories under uncertainties. To this end, an autonomous entry guidance algorithm is proposed for hypersonic vehicles using convex optimization methods. Based on previous sequential convex programming methods for entry trajectory optimization, a different successive procedure is developed for the online generation and tracking of reference trajectories from the current location to a desired target. In each guidance cycle, a new reference trajectory is generated by solving a second-order cone programming problem, and an optimal feedback guidance law is designed using a quadratically constrained quadratic programming method to track this reference trajectory. Because these two problems are convex, they are expected to be solved by state-of-the-art interior-point methods very efficiently. To demonstrate the effectiveness and performance of the proposed entry guidance algorithm, an offline convex entry guidance method is also designed for comparison based on a predetermined reference trajectory. Numerical simulations are presented for both offline and online approaches, which show that the proposed autonomous entry guidance algorithm is robust and efficient for potential future onboard guidance applications.

Lifetime optimization framework for a hybrid renewable energy system based on receding horizon optimization

In this work, a novel convex sequence framework for real-time receding horizon operation optimization of a hybrid renewable energy system integrated with optimal sizing is presented to increase the penetration rate of renewable energy in supplying the demand. The proposed framework optimizes the entire lifetime cost of a system consisting of two main steps which are 1) design & installation and 2) operation as two sequence modules. This framework is applied to a hybrid renewable energy system which includes PV, wind turbine, batteries and a diesel generator. In the operation optimization, receding horizon strategy is used to optimize the operation schedule. Mixed integer convex programming method is applied in order to achieve the optimal operation. The hybrid renewable energy system is installed to actualize the design optimization outputs and to measure the required data for real-time operation optimization. The results show the proposed framework can be applied to facilitate the reliable real-time operation using real optimal input data for taking better advantage of the renewable energy resources. The effect of length of the horizon on optimal scheduling is also investigated. The results indicate that increasing of prediction horizon length enhances the economic performance and increases the share of renewable energy in the hybrid renewable energy system (from 68.5% to 81.4%).

Admission Control Algorithms for QoS-Constrained Multicell MISO Downlink Systems

The problem of admission control in a multicell downlink multiple-input single-output system is considered. The objective is to maximize the number of admitted users subject to the signal-to-interference-plus-noise ratio constraint for each admitted user and a transmit power constraint at each base station. We cast the admission control problem as an $\ell _{0}$ minimization problem. This problem is known to be combinatorial NP-hard. Hence, we have to rely on suboptimal algorithms to solve it. We first approximate the $\ell _{0}$ minimization problem via a non-combinatorial one. Then, we propose centralized and distributed algorithms to solve the non-combinatorial problem. To develop the centralized algorithm, we use the sequential convex programming method. The distributed algorithm is derived by using the alternating direction method of multipliers in conjunction with sequential convex programming. We show numerically that the proposed admission control algorithms achieve a near-to-optimal performance. Next, we extend the admission control problem to provide fairness, where a long term fairness among the users is guaranteed. We focus on proportional and max–min fairness and propose dynamic control algorithms via Lyapunov optimization. It is shown numerically that the proposed fair admission control algorithms guarantee fairness among the users.

A prediction-correction primal-dual hybrid gradient method for convex programming with linear constraints

A prediction-correction primal-dual hybrid gradient method for convex programming with linear constraints

A Regularized Semi-Smooth Newton Method with Projection Steps for Composite Convex Programs

The goal of this paper is to study approaches to bridge the gap between first-order and second-order type methods for composite convex programs. Our key observations are: (1) Many well-known operator splitting methods, such as forward–backward splitting and Douglas–Rachford splitting, actually define a fixed-point mapping; (2) The optimal solutions of the composite convex program and the solutions of a system of nonlinear equations derived from the fixed-point mapping are equivalent. Solving this kind of system of nonlinear equations enables us to develop second-order type methods. These nonlinear equations may be non-differentiable, but they are often semi-smooth and their generalized Jacobian matrix is positive semidefinite due to monotonicity. By combining with a regularization approach and a known hyperplane projection technique, we propose an adaptive semi-smooth Newton method and establish its convergence to global optimality. Preliminary numerical results on $$\ell _1$$ -minimization problems demonstrate that our second-order type algorithms are able to achieve superlinear or quadratic convergence.

Complexity of first-order inexact Lagrangian and penalty methods for conic convex programming

In this paper we present a complete iteration complexity analysis of inexact first-order Lagrangian and penalty methods for solving cone-constrained convex problems that have or may not have optimal Lagrange multipliers that close the duality gap. We first assume the existence of optimal Lagrange multipliers and study primal–dual first-order methods based on inexact information and augmented Lagrangian smoothing or Nesterov-type smoothing. For inexact (fast) gradient augmented Lagrangian methods, we derive an overall computational complexity of projections onto a simple primal set in order to attain an ε-optimal solution of the conic convex problem. For the inexact fast gradient method combined with Nesterov-type smoothing, we derive computational complexity projections onto the same set. Then, we assume that optimal Lagrange multipliers might not exist for the cone-constrained convex problem, and analyse the fast gradient method for solving penalty reformulations of the problem. For the fast gradient method combined with penalty framework, we also derive an overall computational complexity of projections onto a simple primal set to attain an ε-optimal solution for the original problem.

On the convergence rate of the augmented Lagrangian-based parallel splitting method

The augmented Lagrangian method (ALM) is a well-regarded algorithm for solving convex optimization problems with linear constraints. Recently, in He et al. [On full Jacobian decomposition of the augmented Lagrangian method for separable convex programming, SIAM J. Optim. 25(4) (2015), pp. 2274–2312], it has been demonstrated that a straightforward Jacobian decomposition of ALM is not necessarily convergent when the objective function is the sum of functions without coupled variables. Then, Wang et al. [A note on augmented Lagrangian-based parallel splitting method, Optim. Lett. 9 (2015), pp. 1199–1212] proved the global convergence of the augmented Lagrangian-based parallel splitting method under the assumption that all objective functions are strongly convex. In this paper, we extend these results and derive the worst-case convergence rate of this method under both ergodic and non-ergodic conditions, where t represents the number of iterations. Furthermore, we show that the convergence rate can be improved from to , and finally, we also demonstrate that this method can achieve global linear convergence, when the involved functions satisfy some additional conditions.