Nonconvex Programming Problems Research Articles

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.

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.

Environment Warped Gait Trajectory Optimization for Complex Terrains

Contact-aware gait trajectory optimization is a challenging non-convex programming problem, especially for complex terrain shapes, where prominent numerical algorithms can fail to find a solution or fall into local minima. To alleviate this issue, we propose an environment warping technique that changes the coordinates for decision variables. Given a terrain of some general shape, our method first generates a locally injective, as-conformal-as-possible function that maps the ambient space around the terrain to a warped space. We then formulate the trajectory optimization in the warped space by remapping all the decision variables. Our method frees the numerical optimizer from tuning the trajectories to fit changing terrain shapes, leading to better numerical conditioning and fewer local minima. Numerical results show that our method outperforms direct trajectory optimization in terms of both success rates and quality of solutions.

Reconfigurable Intelligent Surface With Energy Harvesting Assisted Cooperative Ambient Backscatter Communications

The performance of cooperative ambient backscatter communications (CABC) can be enhanced by employing reconfigurable intelligent surface (RIS) to assist backscatter transmitters. Since the RIS power consumption is a non-negligible issue, we consider a RIS assisted CABC system where the RIS with energy harvesting circuit can not only reflect signal but also harvest wireless energy. We study a transmission design problem to minimize the RIS power consumption with the quality of service constraints for both active and backscatter transmissions. The optimization problem is a mixed-integer non-convex programming problem which is NP-hard. To tackle it, an algorithm is proposed by employing the block coordinate descent, semidefinite relaxation and alternating direction method of multipliers techniques. Simulation results demonstrate the effectiveness of the proposed algorithm.

ESN Reinforcement Learning for Spectrum and Flight Control in THz-Enabled Drone Networks

Terahertz (THz)-band communications have been envisioned as a key technology to support ultra-high-data-rate applications in 5G-beyond (or 6G) wireless networks. Compared to the microwave and mmWave bands, the main challenges with the THz band are in its i) large path loss hence limited network coverage and ii) visible-light-like propagation characteristics hence poor support of mobility in blockage-rich environments. This paper studies quantitatively the applicability of THz-band communications in blockage-rich mobile environments, focusing on a new network scenario called FlyTera. In FlyTera, a set of hotspots mounted on flying drones collaboratively provide data streaming services to ground users, in the microwave, mmWave and THz bands. We first provide a mathematical formulation of the FlyTera control problem, where the objective is to maximize the network spectral efficiency by jointly controlling the flight of the drone hotspots, their association to the ground users, and the spectrum bands used by the users. To solve the resulting problem, which is shown to be a mixed integer nonlinear nonconvex programming (MINLP) problem, we design distributed solution algorithms based on a combination of echo state learning and reinforcement learning. An extensive simulation campaign is then conducted with SimBAG, a newly developed Simulator of Broadband Aerial-Ground wireless networks. It is shown that no single spectrum band can meet the requirements of high data rate and wide coverage simultaneously. Moreover, from the network-level point of view, THz-band communications can significantly benefit from the mobility of the flying drones, and on average 4 - 6 times higher (rather than lower) throughput can be achieved in mobile than in static environments.

Over-the-Air Computation via Reconfigurable Intelligent Surface

Over-the-air computation (AirComp) is a disruptive technique for fast wireless data aggregation in Internet of Things (IoT) networks via exploiting the waveform superposition property of multiple-access channels. However, the performance of AirComp is bottlenecked by the worst channel condition among all links between the IoT devices and the access point. In this paper, a reconfigurable intelligent surface (RIS) assisted AirComp system is proposed to boost the received signal power and thus mitigate the performance bottleneck by reconfiguring the propagation channels. With an objective to minimize the AirComp distortion, we propose a joint design of AirComp transceivers and RIS phase-shifts, which however turns out to be a highly intractable non-convex programming problem. To this end, we develop a novel alternating minimization framework in conjunction with the successive convex approximation technique, which is proved to converge monotonically. To reduce the computational complexity, we transform the subproblem in each alternation as a smooth convex-concave saddle point problem, which is then tackled by proposing a Mirror-Prox method that only involves a sequence of closed-form updates. Simulations show that the computation time of the proposed algorithm can be two orders of magnitude smaller than that of the state-of-the-art algorithms, while achieving a similar distortion performance.

A successive convex optimization method for bilinear matrix inequality problems and its application to static output‐feedback control

AbstractThis article explores a successive convex optimization method for solving a class of nonconvex programming problems subject to bilinear matrix inequality (BMI) constraints. In particular, many control issues can be boiled down to BMI problems, which are typically NP‐hard. To get out of the predicament, we propose a more relaxed feasible set to approximate the original one, based on which a local optimization algorithm is developed and its convergence is analyzed. As a case of application, we consider static output‐feedback control for uncertain systems with disturbances in restricted frequency intervals. In order to strengthen the disturbance‐rejection capability over the given frequency range, we establish sufficient and necessary analysis conditions via the generalized Kalman‐Yakubovich‐Popov lemma, under which the homogeneous polynomially parameter‐dependent technique is adopted to facilitate the design. Finally, several examples are given to demonstrate the efficiency of the results.

Optimality conditions for nonconvex problems over nearly convex feasible sets

In this paper, we study the optimization problem (P) of minimizing a convex function over a constraint set with nonconvex constraint functions. We do this by given new characterizations of Robinson’s constraint qualification, which reduces to the combination of generalized Slater’s condition and generalized sharpened nondegeneracy condition for nonconvex programming problems with nearly convex feasible sets at a reference point. Next, using a version of the strong CHIP, we present a constraint qualification which is necessary for optimality of the problem (P). Finally, using new characterizations of Robinson’s constraint qualification, we give necessary and sufficient conditions for optimality of the problem (P).

A risk-sensitive task offloading strategy for edge computing in industrial Internet of Things

Edge computing has become one of the key enablers for ultra-reliable and low-latency communications in the industrial Internet of Things in the fifth generation communication systems and is also a promising technology in the future sixth generation communication systems. In this work, we consider the application of edge computing to smart factories for mission critical task offloading through wireless links. In such scenarios, although high end-to-end delays from the generation to completion of tasks happen with low probability, they may incur severe casualties and property loss and should be seriously treated. Inspired by the risk management theory widely used in finance, we adopt the Conditional Value at Risk to capture the tail of the delay distribution. An upper bound of the Conditional Value at Risk is derived through analysis of the queues both at the devices and the edge computing servers. We aim to find out the optimal offloading policy taking into consideration both the average and the worst-case delay performance of the system. Given that the formulated optimization problem is a non-convex mixed integer nonlinear programming problem, a decomposition into subproblems is performed and a two-stage heuristic algorithm is proposed. The simulation results validate our analysis and indicate that the proposed algorithm can reduce the risk in both the queueing and end-to-end delay.

A New Global Optimization Scheme for Quadratic Programs with Low-Rank Nonconvexity

We consider the classical convex constrained nonconvex quadratic programming problem where the Hessian matrix of the objective to be minimized has r negative eigenvalues, denoted by (QPr). Based on a biconvex programming reformulation in a slightly higher dimension, we propose a novel branch-and-bound algorithm to solve (QP1) and show that it returns an [Formula: see text]-approximate solution of (QP1) in at most [Formula: see text] iterations. We further extend the new algorithm to solve the general (QPr) with r > 1. Computational comparison shows the efficiency of our proposed global optimization method for small r. Finally, we extend the explicit relaxation approach for (QP1) to (QPr) with r > 1. Summary of Contribution: Nonconvex quadratic program (QP) is a classical optimization problem in operations research. This paper aims at globally solving the QP where the Hessian matrix of the objective to be minimized has r negative eigenvalues. It is known to be nondeterministic polynomial-time hard even when r = 1. This paper presents a novel algorithm to globally solve the QP for r = 1 and then extends to general r. Numerical results demonstrate the superiority of the proposed algorithm in comparison with state-of-the-art algorithms/software for small r.

Задачи устойчивости круговых арок с неудерживающими связями

The work deals with the problems of stability of arches and arch systems supported by inextensible threads that cannot withstand compressive forces. These questions relate to contact problems of elastic structures with an unknown region of active interaction of their elements, and the study is reduced to the search and analysis of bifurcation points ofsolutions to variational problems in the presence of inequality constraints on the sought functions. The paper considers the problem of stability of an arch supported by threads, one end of which is attached to some point of the arch, the other to the projection of this point on the chord connecting its ends. In the numerical solution, the problem is reducedto finding a load parameter in which the nonconvex quadratic programming problem with constraints on variables in the form of inequalities has a non-unique solution. Numerical calculations have shown that even with a small number of reinforcing threads, the critical load approximately doubles.

Secure and Energy-Efficient Precoding for MIMO Two-Way Untrusted Relay Systems

This paper focuses on the multiple-input-multiple-output (MIMO) two-way relay system with an untrusted relay and investigates its secure and energy-efficient precoding design issue based on the physical layer security technology. We first provide theoretical modeling for the index of secrecy energy efficiency (SEE) and formulate the optimal precoding design for SEE maximization (SEEM) as a high-dimensional non-convex programming problem. By exploring the techniques like fractional programming, alternate optimization and semi-definite programming, we then develop a hierarchical theoretical framework to solve the SEEM problem and thus to identify the optimal precoding designs for the source and relay. Furthermore, we demonstrate the proposed theoretical framework is also applicable to the problem of precoding design for secrecy sum rate maximization. Finally, with the help of generalized singular value decomposition, we propose a sub-optimal relay precoding design scheme with significantly lower computational complexity. Extensive numerical results provided in the paper indicate that the proposed schemes can remarkably improve the SEE performance in MIMO two-way untrusted relay systems.

Synthesis of Irregular Phased Arrays Subject to Constraint on Directivity via Convex Optimization

The synthesis of irregular phased arrays subject to constraint on directivity is fulfilled by convex optimization. In particular, based on the membership between the antenna elements and subarrays, the dictionary matrix, the resulting sparse excitation vector, and the sparse binary vector are introduced to describe the array pattern and exact tiling of the aperture. Therefore, the synthesis of irregular phased arrays without overlaps and holes satisfying given maximum sidelobe level (SLL) and minimum directivity can be summarized as a mixed integer nonconvex programming problem, where both the subarray tiling configurations and the associated complex excitations are optimized simultaneously. To avoid the nonconvexities, some mathematical transformations in terms of directivity are implemented and the binary sparse vector is relaxed to be a real sparse vector by the minimization of the weighted l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm. Consequently, the nonconvex problem is reduced to iterative convex optimization problems, where several convex optimization problems are included and can be efficiently solved using convex optimization solver. The excellent performances of the proposed method are verified by comparing with previously reported methods through several numerical examples.

Optimal placement of phase‐reconfiguration devices in low‐voltage distribution network with residential PV generation

As residential PV generation penetrates in the low-voltage distribution network (LVDN), the unbalance issue may be intensified due to the asymmetry of generations/loads in different phases. Phase-reconfiguration device (PRD), which can reconfigure connected phases of residential customers, provides an effective method to address this issue. Noting that although the benefit brought by PRDs can vary if they are placed at different locations in the network, little literature has been reported on this topic. To bridge the research gap, this paper presents a novel method to optimally place PRDs in an LVDN aiming at minimizing the power unbalance running through the distribution transformer. The problem considers both installation, operational constraints, and boils down to a challenging mixed-integer non-convex programming problem, which is then reformulated as an efficient solvable mixed-integer linear programming problem. Moreover, operational constraints are relaxed with slack variables penalized in the objective function, which makes sure a feasible solution is always available without or with minimal operational violations. Case studies based on a modified IEEE system and a practical system in Australia demonstrate that an efficient strategy can be provided to address the unbalance issue while improving the network's power supply qualities.

Existence and sensitivity analysis for nonconvex cubic optimization problems

In this paper, we propose necessary/sufficient conditions for existence of minimizers of nonconvex cubic programming problems via an Eaves type theorem. Directional differentiability of the optimal value function under total perturbation are characterized. Among our proposed assumptions, there are some weaker than the existing assumptions (applied to cubic programming).

Optimal correction of the absolute value equations

In this paper, we study the optimum correction of the absolute value equations through making minimal changes in the coefficient matrix and the right hand side vector and using spectral norm. This problem can be formulated as a non-differentiable, non-convex and unconstrained fractional quadratic programming problem. The regularized least squares is applied for stabilizing the solution of the fractional problem. The regularized problem is reduced to a unimodal single variable minimization problem and to solve it a bisection algorithm is proposed. The main difficulty of the algorithm is a complicated constraint optimization problem, for which two novel methods are suggested. We also present optimality conditions and bounds for the norm of the optimal solutions. Numerical experiments are given to demonstrate the effectiveness of suggested methods.

A Frank–Wolfe-Type Theorem for Cubic Programs and Solvability for Quadratic Variational Inequalities

In this paper, we present a Frank–Wolfe-type theorem for nonconvex cubic programming problems. This result is a direct extension of the previous ones by Andronov et al. (Izvestija Akadem. Nauk SSSR, Tekhnicheskaja Kibernetika 4:194–197, 1982) and Flores-Bazan et al. (Math. Program. 145:263–290, 2014). Under suitable conditions, we characterize the compactness of the solution set of cubic programming problems. Sufficient conditions for the existence of solutions of quadratic variational inequalities are proposed. We also provide several numerical examples, which not only illustrate the obtained results but also show that the existing results cannot apply.

A Deterministic Method for Solving the Sum of Linear Ratios Problem

Since the sum of linear ratios problem (SLRP) has many applications in real life, for globally solving it, an efficient branch and bound algorithm is presented in this paper. By utilizing the characteristic of the problem (SLRP), we propose a convex separation technique and a two-part linearization technique, which can be used to generate a sequence of linear programming relaxation of the initial nonconvex programming problem. For improving the convergence speed of this algorithm, a deleting rule is presented. The convergence of this algorithm is established, and some experiments are reported to show the feasibility and efficiency of the proposed algorithm.

Load Balancing in Low-Voltage Distribution Network via Phase Reconfiguration: An Efficient Sensitivity-Based Approach

Operational performance in the low-voltage distribution network (LVDN) can be undermined by its inherent unbalances, which may become worse as the penetration of rooftop solar continuously increases. To address this issue, load balancing via phase-reconfiguration devices (PRDs), which can change phase positions of residential customers as required, provides a cost-efficient option. However, most reported approaches to control PRDs require that demands of all residential customers are available, which are not viable for many LVDNs without smart meters or advanced metering infrastructure (AMI) installed. To bridging the gap in this field, this paper proposes a novel method to control PRDs purely based on measurable data from PRDs, and its controller. Based on limited information, sensitivity analysis in the network with PRDs is studied, followed by the optimization model that comprehensively considers operational requirements in the network. Moreover, slack variables are introduced to the model, and penalized in the objective function to assure either a strategy that is secure or with minimized violations can always be provided. The model is a challenging mixed-integer non-convex programming (MINCP) problem, which is reformulated as an efficient solvable mixed-integer second-order cone programming (MISOCP) based on exact reformulations or accurate linear approximations. Simulations based on two modified IEEE systems, and a real system in Australia demonstrate that an efficient strategy can be provided to mitigate unbalances in the network.