Nonconvex Programming Research Articles

Synergizing shared micromobility and public transit towards an equitable multimodal transportation network

This paper assesses the equity impacts of shared micromobility and investigates regulatory policies that improve transport equity and promote synergy between public transit and shared micromobility. We consider a multimodal transportation network, where a micromobility platform deploys docking stations and operates a fleet of micromobility vehicles to provide shared micromobility services and a public transit agency offers transit services over a transportation network. A market equilibrium model is developed to capture the intimate interactions among access and egress times of shared micromobility services, waiting times of transit services, the spatial distribution of docking stations, passenger demand, platform pricing and fleet sizing, vehicle repositioning and the micromobility platform profit. The platform decision problem is cast as a high-dimensional non-convex program. A solution method is proposed to efficiently compute the solution through problem reformulation and dimensionality reduction. Based on the proposed framework, we evaluate spatial equity in transport accessibility using the Gini index, and find that although shared micromobility improves overall transport accessibility, the benefits are not fairly distributed across different geographic zones, which leads to enlarged spatial inequity gaps after introducing shared micromobility. To promote transport equity, we investigate three policy directions: (a) to impose a vehicle density floor on shared micromobility; (b) to offer a subsidy on shared micromobility rides for first/last-mile connections; and (c) to promote collaboration between public transit and shared micromobility. We show that different regulatory policies have advantages and limitations. The minimum vehicle density requirement can simultaneously improve spatial equity and passengers’ surplus, but has limited equity improvements. In contrast, the subsidy on bundled services could significantly mitigate spatial inequity, but it hurts passengers and the platform profit. Compared to the other two policies, the transit-micromobility collaboration can lead to higher equity improvement, higher passenger surplus, while offering a guarantee on the platform profit, which turns out to be the most cost-effective approach. These insights are validated through realistic numerical studies for San Francisco.

Online Stochastic DCA With Applications to Principal Component Analysis.

Stochastic algorithms are well-known for their performance in the era of big data. In this article, we study nonsmooth stochastic Difference-of-Convex functions (DC) programs-the major class of nonconvex stochastic optimization, which have a variety of applications in divers domains, in particular, machine learning. We propose new online stochastic algorithms based on the state-of-the-art DC Algorithm (DCA)-a powerful approach in nonconvex programming framework, in the online context of streaming data continuously generated by some (unknown) source distribution. The new schemes use the stochastic approximations (SAs) principle: deterministic quantities of the standard DCA are replaced by their noisy estimators constructed using newly arriving samples. The convergence analysis of the proposed algorithms is studied intensively with the help of tools from modern convex analysis and martingale theory. Finally, we study several aspects of the proposed algorithms on an important problem in machine learning: the expected problem in principal component analysis (PCA).

Scalable Computation of Robust Control Invariant Sets of Nonlinear Systems

Ensuring robust constraint satisfaction for an infinite time horizon is a challenging, yet crucial task when deploying safety-critical systems. We address this issue by synthesizing robust control invariant sets of perturbed nonlinear sampled-data systems. This task can be encoded as a non-convex program that we approximate by a tailored, computationally efficient successive convexification algorithm. Based on the zonotopic representation of invariant sets, we obtain an updated candidate for the invariant set and the invariance-enforcing controller by solving a single convex program. To obtain a possibly large region of safe operation, our algorithm is designed so that the sequence of candidate invariant sets is volume-wise monotonically increasing. We demonstrate the efficacy and scalability of our approach using a broad range of nonlinear control systems from the literature with up to 20 dimensions.

A 0–1 linear relaxation and stochastic approximation method for real-time pricing in a smart grid with non-convex constraints

In this article, real-time pricing (RTP) for a smart grid with complicated non-convex and/or non-smooth constraints is explored. First, piecewise linear functions are adopted to approximate any kinds of utility function and RTP is formulated into bilevel non-convex programming. Then, a 0–1 relaxation method is used to linearize sparse constraints and piecewise linear utility functions, to obtain mixed integer linear programming. Finally, a distributed simultaneous perturbation stochastic approximation (DSPSA) method is designed to solve the bilevel non-convex programming. The RTP strategy is generalizable because sparse constraints and non-differentiable and/or non-concave utility functions are widespread in practice, and the piecewise linear functions approximation method is applicable to all kinds of utility function, not just to non-smooth and/or non-concave utility functions. In addition, the 0–1 linear relaxation and stochastic approximation method make the bilevel non-convex programming easily solvable by DSPSA, which converges rapidly while maintaining the high accuracy of solutions.

A new iterative mixed integer linear programming based real time energy efficient management of AC-DC distribution networks

In the smart grid era, AC-DC distribution networks are evolving for integrating massive amounts of distributed energy resources, especially battery associated solar power generation units, into distribution networks in most energy efficient ways. Performing real time energy management for such networks is challenging due to presence of different power electronic converters, grid-edge devices and voltage controlling equipment. Existing studies either focused on developing offline day ahead methods, or shortsighted greedy strategies by neglecting the offline beneficial attributes in real time optimization frameworks. On the contrary, this article proposes a novel real time AC-DC distribution network energy management portfolio by merging load control and conservation voltage reduction techniques to achieve energy efficient operation of the network by regulating the operation of the controllable resources. Initially, the overall optimization problem is developed as a mixed integer non-convex programming by modelling the offline benefits as time coupled stochastic expressions in real time optimization platform. The non-convex expressions are later simplified by adopting different linearization techniques and the entire energy management framework is revised as a simple mixed integer linear programming (MILP). The original problem is decoupled for each time step by simplifying the time coupled expressions using virtual queues and Lyapunov functions. Additionally, a new iterative MILP algorithm is proposed to solve the revised energy management problem with less computation complexity and time. Unlike the previous solution methods, the proposed algorithm initializes the variables by their median values and handles the infeasibilities by adding tangential cutting plane constraints and penalty variable minimization objective. Demonstrating on 33-bus AC-DC distribution network, the superiority of the proposed real time energy management process and iterative MILP algorithm is established after comparing with standard energy management portfolios and existing solution methods.© 2017 Elsevier Inc. All rights reserved.

Cooperative Localization With the Fusion of GNSS and Relative Range Information in Vehicular Ad-Hoc Networks

Accurate vehicle position is crucial to vehicular Ad-Hoc network (VANET) applications. The wildly used global navigation satellite systems (GNSS) have limited localization accuracy, especially when GNSS signals are blocked or contaminated with reflected signals. Cooperative localization based on multi-vehicle information fusion will be the core component in VANET in the near future. This paper strives to enhance the localization accuracy using the fusion of GNSS and relative range information in static scenes. The formulation following the Maximum-Likelihood strategy is a quadratic non-convex programming, which is NP-hard. An iterative localization algorithm is proposed, where an approximate quadratic programming with linear constraints is solved in each iteration. Additional inspection steps and correction step are designed to ensure the convergence and feasibility of the proposed algorithm. The theoretical analysis shows that the limit of the proposed algorithm must be a global optimal solution of the formulated problem and the mean squared error can reach the Cramér-Rao lower bound. In terms of localization accuracy, the proposed method outperforms the existing linearized weighted least-squares method and semidefinite relaxation method, while exhibiting moderate computational complexity.

A Hierarchical Framework for Passenger Inflow Control in Metro System With Reinforcement Learning

Since mobility needs grow rapidly, the metro system of modern cities is suffering from the oversaturated situation in the rush hour, which makes the metro system vulnerable and inefficient. Thus, passenger inflow control is proposed and implemented. This paper investigates the station-based passenger inflow control problem with the objective of maximizing overall transport efficiency and fairness. To solve the formulated nonlinear and nonconvex programming model, a novel framework integrating unsupervised subgoal discovery and hierarchical reinforcement learning, named SD-HIC, is proposed. With a series of subgoals, the complicated and long-horizon original task is decomposed and can be solved by reinforcement learning responsively and far-sightedly. A real-world case with operational data in the Guangzhou metro is presented to demonstrate the performance of the proposed model and framework. According to the results, the overall transport utility is improved by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$28.87\%$</tex-math> </inline-formula> compared with the benchmark inflow control strategy that is frequently used in daily operations. Through solution algorithm studies and critical parameter analyses, the performance of the proposed passenger inflow control framework is further verified. Notably, the revealed subgoals are also distinguishable and interpretable, which is helpful for the operation staff in practice.

Dislocation hyperbolic augmented Lagrangian algorithm for nonconvex optimization

The dislocation hyperbolic augmented Lagrangian algorithm (DHALA) solves the nonconvex programming problem considering an update rule for its penalty parameter and considering a condition to ensure the complementarity condition. in this work, we ensure that the sequence generated by DHALA converges to a Karush-Kuhn-Tucker (KKT) point, and we present computational experiments to demonstrate the performance of our proposed algorithm.

Eco-Driving for Metro Trains: A Computationally Efficient Approach Using Convex Programming

Eco-driving for trains has traditionally focused on minimizing mechanical energy consumption at wheels, while completely ignoring traction chain losses that are rather significant. This paper presents a computationally efficient approach to minimize the total electrical energy consumption from traction substations (TS). After a nonlinear and non-convex program is formulated in time domain, a nonlinear and non-convex program is formulated in space domain to overcome the drawbacks of the model in time domain. By convex modeling steps, the non-convex program in space domain is reformulated as a convex program that can be efficiently solved. To further reduce computational effort, a real-time iteration sequential quadratic programming (SQP) algorithm is proposed to solve the convex program in a model predictive control framework. Numerical results indicate that the proposed SQP method yields a near-optimal solution with high computational efficiency. Compared to a traditional mechanical energy consumption model, a TS-to-traction energy efficiency can be improved.

Efficient Day-Ahead Scheduling of PV-STATCOMs in Medium-Voltage Distribution Networks Using a Second-Order Cone Relaxation

This paper utilizes convex optimization to implement a day-ahead scheduling strategy for operating a photovoltaic distribution static compensator (PV-STATCOM) in medium-voltage distribution networks. The nonlinear non-convex programming model of the day-ahead scheduling strategy is transformed into a convex optimization model using the second-order cone programming approach in the complex domain. The main goal of efficiently operating PV-STATCOMs in distribution networks is to dynamically compensate for the active and reactive power generated by renewable energy resources such as photovoltaic plants. This is achieved by controlling power electronic converters, usually voltage source converters, to manage reactive power with lagging or leading power factors. Numerical simulations were conducted to analyze the effects of different power factors on the IEEE 33- and 69-bus systems. The simulations considered operations with a unity power factor (active power injection only), a zero power factor (reactive power injection only), and a variable power factor (active and reactive power injections). The results demonstrated the benefits of dynamic, active and reactive power compensation in reducing grid power losses, voltage profile deviations, and energy purchasing costs at the substation terminals. These simulations were conducted using the CVX tool and the Gurobi solver in the MATLAB programming environment.

Ellipsotopes: Uniting Ellipsoids and Zonotopes for Reachability Analysis and Fault Detection

Ellipsoids are a common representation for reachability analysis, because they can be transformed efficiently under affine maps, and they allow conservative approximation of Minkowski sums, which let one incorporate uncertainty and linearization error in a dynamical system by expanding the size of the reachable set. Zonotopes, a type of symmetric, convex polytope, are similarly frequently used, because they allow efficient numerical implementations of affine maps and exact Minkowski sums. Both of these representations also enable efficient, convex collision detection for fault detection or formal verification tasks, wherein one checks if the reachable set of a system collides (i.e., intersects) with an unsafe set. However, both representations often result in conservative representations for reachable sets of arbitrary systems, and neither is closed under intersection. Recently, representations such as constrained zonotopes and constrained polynomial zonotopes have been shown to overcome some of these conservativeness challenges, and are closed under intersection. However, constrained zonotopes can not represent shapes with smooth boundaries such as ellipsoids, and constrained polynomial zonotopes can require solving a non-convex program for collision checking or fault detection. This paper introduces <i>ellipsotopes</i>, a set representation that is closed under affine maps, Minkowski sums, and intersections. Ellipsotopes combine the advantages of ellipsoids and zonotopes while ensuring convex collision checking. The utility of this representation is demonstrated on several examples.

Initial point for enhancing DC Programming and DCA in Physical Layer Security

Abstract— Physical layer security (PLS) is to enable confidential data transmission through wireless communication systems in the presence of illegitimate receivers, without basing on higher-layer encryption. The goal of PLS is to make use of the properties of the physical to enable critical aspects of secure communication. PLS's additional strength is that it is based on information-theoretic security, which assumes no limit on the opponent's computational power and is thus inherently quantum-resistant. In general, PLS problems are non-convex programming and thus hard to deal with. One approach to handling these problems is based on DC programming and DCA, in which finding a good initial point of DCA scheme is still a complicated, challenging and open problem. This paper proposal some good initial points for the PLS problem DCA-DF scheme on Decode-and-Forward relaying wireless networks.

Optimizing fairness in cellular networks with mobile drone relays

Aiding the ground cellular network with aerial base stations carried by drones has experienced an intensive raise of interest in the past years. Reconfigurable air-to-ground channels enable aerial stations to enhance users’ access links by means of seeking good line-of-sight connectivity while hovering in the air. In this article, we propose an analytical framework for the 3D placement of a fleet of coordinated drone relays. This framework optimizes network performance in terms of user throughput fairness, expressed through the α-fairness metric. The optimization problem is formulated as a mixed-integer non-convex program, which is intractable. Hence, we propose an extremal-optimization-based algorithm, Parallelized Alpha-fair Drone Deployment (PADD), which solves the problem online, in low-degree polynomial time. We evaluate our proposal by means of numerical simulations over the real topology of a dense city. We discuss the advantages of integrating drone relay stations in current networks and test several resource scheduling approaches in both static and dynamic scenarios, including with progressively larger and denser crowds.

Fast Quantum State Reconstruction via Accelerated Non-Convex Programming

We propose a new quantum state reconstruction method that combines ideas from compressed sensing, non-convex optimization, and acceleration methods. The algorithm, called Momentum-Inspired Factored Gradient Descent (MiFGD), extends the applicability of quantum tomography for larger systems. Despite being a non-convex method, MiFGD converges provably close to the true density matrix at an accelerated linear rate asymptotically in the absence of experimental and statistical noise, under common assumptions. With this manuscript, we present the method, prove its convergence property and provide the Frobenius norm bound guarantees with respect to the true density matrix. From a practical point of view, we benchmark the algorithm performance with respect to other existing methods, in both synthetic and real (noisy) experiments, performed on the IBM’s quantum processing unit. We find that the proposed algorithm performs orders of magnitude faster than the state-of-the-art approaches, with similar or better accuracy. In both synthetic and real experiments, we observed accurate and robust reconstruction, despite the presence of experimental and statistical noise in the tomographic data. Finally, we provide a ready-to-use code for state tomography of multi-qubit systems.

Joint Mode Selection and Resource Allocation for D2D and Femtocell Users in Dense Heterogeneous Networks with Full Frequency Reuse

We consider the problem of joint mode selection and resource allocation for D2D and femtocell users in a three-tier dense heterogeneous network in which all the users can reuse the spectrum. The goal is to maximize their total weighted sum rate, subject to minimum rate requirements and maximum tolerable interference on the cellular system. Since the spectrum can be fully reused, the co-tier and cross-tier interferences among the users result in a mixed integer non-linear, non-convex program that is difficult to solve directly. Using insights into the structure of the interference, we derive a close, conservative approximation of the joint problem that has a convex relaxation that is tight. That enables good solutions to the joint problem to be obtained using a customized iterative algorithm employing the Lagrange dual decomposition method. Since the proposed iterative scheme is semi-distributed, the signaling overhead is mitigated. To further reduce the computational complexity, a low-complexity (primal) decomposition-based method is also introduced, in which we select the transmission mode heuristically based on the traffic level in the network, and then sequentially perform admission control, power control, and sub-channel allocation for the femtocell users and D2D users. Our simulation results indicate that the proposed iterative and heuristic algorithms, on average, achieve around 94% and 82% of the sum rate of the optimal Branch & Bound method, respectively, and do so at much lower computational costs.

Global optimization for non-convex programs via convex proximal point method

&lt;p style='text-indent:20px;'&gt;In this study, a convex proximal point algorithm (CPPA) is considered for solving constrained non-convex problems, and new theoretical results are proposed. It is proved that every cluster point of CPPA is a stationary point, and the initial point of CPPA is key to global optimization. Several sufficient conditions for the initial point selection are provided for CPPA to find the global minimum. Motivated by these results, numerical experiments were conducted on non-convex quadratic programming problems with convex quadratic constraints. The performance of CPPAs was compared, with the initial point randomly selected or obtained through the Lagrangian dual problem. The numerical results demonstrate that the quality of the CPPA with the computed Lagrangian dual initial point is much better than that with the random initial point, in terms of the objective function value.&lt;/p&gt;

Joint Multi-Domain Resource Allocation and Trajectory Optimization in UAV-Assisted Maritime IoT Networks

The integration of Maritime Internet of Things (M-IoT) technology and unmanned aerial/surface vehicles (UAVs/USVs) has been emerging as a promising navigational information technique in intelligent ocean systems. In this article, we consider the UAV-assisted M-IoT network where USVs offload computation-intensive maritime tasks via non-orthogonal multiple access (NOMA) to the UAV equipped with the mobile-edge computing (MEC) server subject to the UAV mobility. To improve the energy efficiency of offloading transmission and workload computation, we focus on minimizing the total energy consumption by jointly optimizing the USVs’ offloaded workload, transmit power, computation resource allocation, as well as the UAV trajectory subject to the USVs’ latency requirements. Despite the nature of mixed discrete and non-convex programming of the formulated problem, we exploit the vertical decomposition and propose a two-layered algorithm for solving it efficiently. Specifically, the top-layered algorithm is proposed to solve the problem of optimizing the UAV trajectory based on the idea of deep reinforcement learning (DRL), and the underlying algorithm is proposed to optimize the underlying multidomain resource allocation problem based on the idea of the Lagrangian multiplier method. Numerical results are provided to validate the effectiveness of our proposed algorithms as well as the performance advantage of NOMA-enabled computation offloading in terms of overall energy consumption.

A GEOMETRIC PROGRAMMING APPROACH TO CONVEX MULTI-OBJECTIVE PROGRAMMING PROBLEMS

Over the past few years, convex optimization has played a vital role in the study of complex engineering problems in different fields. Geometric programming is one of the available techniques particularly used for solving nonconvex programming problems. But in this article, a suitable attempt has been made to solve a real-life model on convex multi-objective using geometric programming technique with help of the E-constraint method and result is compared with the solutions obtained by fuzzy technique. Finally, a conclusion is presented by analyzing the solutions to a numerical problem.

Deep Reinforcement Learning Based Resource Allocation in Multi-UAV-Aided MEC Networks

Resource allocation for mobile edge computing (MEC) in unmanned aerial vehicle (UAV) networks has been a popular research issue. Different from existing works, this paper considers a multi-UAV-aided uplink communication scenario and investigates a resource allocation problem of minimizing the total system latency and the energy consumption, subject to constraints on transmit power of mobile users (MUs), system latency caused by transmission and computation. The problem is confirmed to be a challenging time-series mixed-integer non-convex programming problem, and we propose a joint UAV Movement control, MU Association and MU Power control (UMAP) algorithm to solve it effectively, where three sub-problems are optimized iteratively. Specifically, UAV movement and MU association are optimized utilizing deep reinforcement learning (DRL) to decrease the energy consumption and system latency. Next, a closed-form solution of the MU transmit power is derived. Finally, simulation results show that the UMAP algorithm can significantly decrease the system latency and energy consumption and increase the coverage rate compared with benchmark algorithms.

A two-level distributed algorithm for nonconvex constrained optimization

This paper aims to develop distributed algorithms for nonconvex optimization problems with complicated constraints associated with a network. The network can be a physical one, such as an electric power network, where the constraints are nonlinear power flow equations, or an abstract one that represents constraint couplings between decision variables of different agents. Despite the recent development of distributed algorithms for nonconvex programs, highly complicated constraints still pose a significant challenge in theory and practice. We first identify some difficulties with the existing algorithms based on the alternating direction method of multipliers (ADMM) for dealing with such problems. We then propose a reformulation that enables us to design a two-level algorithm, which embeds a specially structured three-block ADMM at the inner level in an augmented Lagrangian method framework. Furthermore, we prove the global and local convergence as well as iteration complexity of this new scheme for general nonconvex constrained programs, and show that our analysis can be extended to handle more complicated multi-block inner-level problems. Finally, we demonstrate with computation that the new scheme provides convergent and parallelizable algorithms for various nonconvex applications, and is able to complement the performance of the state-of-the-art distributed algorithms in practice by achieving either faster convergence in optimality gap or in feasibility or both.