Nonconvex Constrained Optimization Problems Research Articles

Joint passive beamforming and deployment design for active intelligent reflecting surface-assisted energy harvesting-cognitive radio sensor networks.

To prolong the network lifetime of energy harvesting-cognitive radio sensor networks (EH-CRSNs), this paper integrates active intelligent reflecting surface (IRS) to support both downlink EH and uplink data transmissions and formulates a constrained non-convex optimization problem that maximizes the net energy gain. To solve this problem, a joint passive beamforming and IRS deployment mechanism is proposed to determine the optimal deployment location of the active IRS and optimally configure the IRS reflection coefficient matrices. Specifically, the net energy gain maximization problem at a specified location is divided into two sub-problems according to its characteristics: maximizing the cumulative energy harvested by all CRSNs nodes in the downlink and minimizing the energy consumption of cluster heads in uplink data transmissions, with each sub-problem being solved independently. Furthermore, this paper explores how factors such as the number of active reflecting elements and the amplification power budget influence the optimal deployment location of the active IRS. The findings from this investigation can guide the design of active IRS-assisted EH-CRSNs under specified parameters. Simulation results confirm the aforementioned conclusions and highlight the benefits of the proposed joint mechanism in enhancing net energy gain over various benchmark mechanisms.

Comparison of a Hybrid Firefly–Particle Swarm Optimization Algorithm with Six Hybrid Firefly–Differential Evolution Algorithms and an Effective Cost-Saving Allocation Method for Ridesharing Recommendation Systems

The optimization and allocation of transport cost savings among stakeholders are two important issues that influence the satisfaction of information providers, drivers and passengers in ridesharing recommendation systems. For optimization issues, finding optimal solutions for nonconvex constrained discrete ridesharing optimization problems poses a challenge due to computational complexity. For the allocation of transport cost savings issues, the development of an effective method to allocate cost savings in ridesharing recommendation systems is an urgent need to improve the acceptability of ridesharing. The hybridization of different metaheuristic approaches has demonstrated its advantages in tackling the complexity of optimization problems. The principle of the hybridization of metaheuristic approaches is similar to a marriage of two people with the goal of having a happy ending. However, the effectiveness of hybrid metaheuristic algorithms is unknown a priori and depends on the problem to be solved. This is similar to a situation where no one knows whether a marriage will have a happy ending a priori. Whether the hybridization of the Firefly Algorithm (FA) with Particle Swarm Optimization (PSO) or Differential Evolution (DE) can work effectively in solving ridesharing optimization problems needs further study. Motivated by deficiencies in existing studies, this paper focuses on the effectiveness of hybrid metaheuristic algorithms for solving ridesharing problems based on the hybridization of FA with PSO or the hybridization of FA with DE. Another focus of this paper is to propose and study the effectiveness of a new method to allocate ridesharing cost savings to the stakeholders in ridesharing systems. The developed hybrid metaheuristic algorithms and the allocation method have been compared with examples of several application scenarios to illustrate their effectiveness. The results indicate that hybridizing FA with PSO creates a more efficient algorithm, whereas hybridizing FA with DE does not lead to a more efficient algorithm for the ridesharing recommendation problem. An interesting finding of this study is very similar to what happens in the real world: “Not all marriages have happy endings”.

Distributed Resilient Initialization-Free Jacobi Descent Algorithm for Constrained Optimization Against DoS Attacks

This paper investigates one type of distributed constrained optimization problem, e.g., the economic dispatch problem, in the presence of DoS attacks. Therein, multiple DoS attackers are collaborative to impede the communication transmission and change the communication topology at will. Consequently, the convergence and/or optimality of distributed algorithm may be compromised. To reduce the effect of this kind of DoS attacks, a distributed resilient initialization-free Jacobi descent algorithm is proposed. It is designed with three switched control protocols which enable the proposed algorithm reasonably employing the estimations to replace the missing information when attacks occur. Meanwhile, the proposed method is embedded with second order information, resulting in faster convergence speed. Moreover, theoretical analysis results are provided to show that the proposed algorithm can exponentially converge to the global optimal solution of the studied problem. Finally, simulation results tested in IEEE 30-bus system validate its effectiveness and flexibility. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The economic dispatch is a key issue in smart grid, which can be formulated as a kind of distributed constrained optimization problem. Since the distributed algorithms work under distributed sensor networks, they are easier to undergo DoS attacks. To address this issue, this paper presents a distributed resilient initialization-free Jacobi descent algorithm, which features strong robustness to resist DoS attacks and faster convergence. Meanwhile, the proposed method is shaped for common constrained optimization problem with better expansibility. We conduct the global convergence and optimality proofs, which benefits the practitioners to estimate the convergence performance, e.g., the convergence rate. Simulations further show the correctness and effectiveness of the proposed method. In future, we will pay more attention on the non-convex constrained optimization problem.

Over-the-Air Computation with Quantized CSI and Discrete Power Control Levels

In this paper, an Over-the-Air Computation (AirComp) scheme for fast data aggregation is considered. Multisource data are simultaneously transmitted by single-antenna mobile devices to a single-antenna fusion center (FC) through a wireless multiple-access channel. The optimal power levels at the devices and a postprocessing scaling function at the FC are jointly derived such that mean square error of the computation is minimized. Different than the existing approaches that rely on perfect channel state information (CSI) at the FC and assume that the devices’ optimal power levels can be selected from an infinite solution set, in the present paper, it is assumed that only quantized CSI is available at the FC and that the aforementioned optimal power levels lie in a finite discrete set of solutions. To derive the optimal power levels and FC’s scaling factor, a difficult nonconvex constrained optimization problem is formulated. An efficient and robust solution to quantization errors is developed via the deep reinforcement learning framework. Numerical results verify the good performance of the proposed approach while it exhibits a significant reduction in the required feedback.

Adaptive Beamforming with Sidelobe Level Control for Multiband Sparse Linear Array

Multiband antenna arrays have the capability of effectively working in multiple frequency bands and thus significantly simplify the antenna system. To further reduce the system overhead, this paper discusses the joint design of antenna selection and adaptive beamforming for multiband antenna arrays, where the sidelobe level is also controlled so as to alleviate the effect of unknown sporadic interference. Based on the maximum signal-to-interference-plus-noise ratio (SINR) criterion and sidelobe level constraints, the proposed multiband sparse array design is formulated into a nonconvex constrained nonlinear optimization problem with an l0,2-mixed norm regularization. This problem ensures that the same antenna positions are selected at all operating frequencies while the beamformer weights of each frequency are optimized independently. By exploiting the semi-definite relaxation and the reweighted l1,∞-norm approximation, the problem is converted into a series of convex subproblems and is then effectively solved by the proposed iterative reweighted method. Numerical results show that the proposed multiband sparse array significantly reduces the sidelobe levels in all operating frequencies while maintaining the maximum SINR, so it provides superior performance of interference suppression.

A hybrid quantum-behaved particle swarm optimization solution to non-convex economic load dispatch with multiple fuel types and valve-point effects

Economic dispatch problems (EDPs) can be reduced to non-convex constrained optimization problems, and most of the population-based algorithms are prone to have problems of premature and falling into local optimum when solving EDPs. Therefore, this paper proposes a hybrid quantum-behaved particle swarm optimization (HQPSO) algorithm to alleviate the above problems. In the HQPSO, the Solis and Wets local search method is used to enhance the local search ability of the QPSO so that the algorithm can find solutions that is close to optimal when the constraints are met, and two evolution operators are proposed and incorporated for the purpose of making a better balance between local search and global search abilities at the later search stage. The performance comparison is made among the HQPSO and the other ten population-based random search methods under two different experimental configurations and four different power systems in terms of solution quality, robustness, and convergence property. The experimental results show that the HQPSO improves the convergence properties of the QPSO and finally obtains the best total generation cost without violating any constraints. In addition, the HQPSO outperforms all the other algorithms on 7 cases of all 8 experimental cases in terms of global best position and mean position, which verifies the effectiveness of the algorithm.

Structured Static Output Feedback Mixed ${H}_2/{H}_\infty $ Control for Linear Control Systems

This note considers the design of static output feedback mixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${H}_2/{H}_\infty $</tex-math></inline-formula> controllers for linear control systems with certain equality and inequality constraints imposed directly on the feedback matrix. Based on the barrier method, we solve an auxiliary minimization problem to obtain an approximate solution to the original nonconvex constrained optimization problem. Necessary conditions for the optimal solution of the auxiliary minimization problem are derived using the Lagrange multiplier method. Subsequently, an iterative steepest descent algorithm is developed to find an approximate optimal solution. Finally, an example is provided to validate the proposed approach.

Perception-aware online trajectory generation for a prescribed manoeuvre of unmanned surface vehicle in cluttered unstructured environment

This paper proposes a perception-aware online trajectory generation system that facilitates prescribed manoeuvres of an unmanned surface vehicle (USV) in a dynamic unstructured environment. The proposed system is developed based on the principles of the inverse dynamics in the virtual domain (IDVD) method and an event-triggered receding horizon control (ETRHC) mechanism. This approach transforms the underlying nonconvex constrained optimization problem into a virtual space with a differentially flat dynamics and uses relatively few decision variables to prototype feasible quasi-optimal trajectories. The closed-loop configuration is provided by a computationally efficient ETRHC mechanism that uses situational awareness of operating environment to trigger trajectory replanning if/when required. This addresses the challenge of continuously updating a closed-loop trajectory which imposes unnecessary computational burden on a system with the limited onboard resources. To investigate the performance of the proposed trajectory generating system, a dynamic unstructured environment including variable and uncertain no-fly zone areas as well as variable current vector fields are modeled. Further, different operating conditions incorporating the uncertainties of environment and sudden failure on the USV propulsion system are introduced to examine the effectiveness, agility, and robustness of the proposed trajectory generating system. A comparative study with benchmark solutions generated by the hp-adaptive Radau pseudo-spectral method is conducted to provide a detailed statistical analysis of the proposed approach robustness, computational complexity, and effectiveness. The simulation results confirm the effectiveness of the proposed trajectory generator and ability to produce a solution for online realization.

Leveraging Power Grid Topology in Machine Learning Assisted Optimal Power Flow

Machine learning assisted optimal power flow (OPF) aims to reduce the computational complexity of these non-linear and non-convex constrained optimization problems by consigning expensive (online) optimization to offline training. The majority of work in this area typically employs fully connected neural networks (FCNN). However, recently convolutional (CNN) and graph (GNN) neural networks have also been investigated, in effort to exploit topological information within the power grid. Although promising results have been obtained, there lacks a systematic comparison between these architectures throughout literature. Accordingly, we introduce a concise framework for generalizing methods for machine learning assisted OPF and assess the performance of a variety of FCNN, CNN and GNN models for two fundamental approaches in this domain: regression (predicting optimal generator set-points) and classification (predicting the active set of constraints). For several synthetic power grids with interconnected utilities, we show that locality properties between feature and target variables are scarce and subsequently demonstrate marginal utility of applying CNN and GNN architectures compared to FCNN for a fixed grid topology. However, with variable topology (for instance, modeling transmission line contingency), GNN models are able to straightforwardly take the change of topological information into account and outperform both FCNN and CNN models.

Global Convergence of Policy Gradient Primal–Dual Methods for Risk-Constrained LQRs

While the techniques in optimal control theory are often model-based, the policy optimization (PO) approach directly optimizes the performance metric of interest. Even though it has been an essential approach for reinforcement learning problems, there is little theoretical understanding of its performance. In this paper, we focus on the risk-constrained linear quadratic regulator (RC-LQR) problem via the PO approach, which requires addressing a challenging non-convex constrained optimization problem. To solve it, we first build on our earlier result that an optimal policy has a time-invariant affine structure to show that the associated Lagrangian function is coercive, locally gradient dominated, and has local Lipschitz continuous gradient, based on which we establish strong duality. Then, we design policy gradient primal-dual methods with global convergence guarantees in both model-based and sample-based settings. Finally, we use samples of system trajectories in simulations to validate our methods.

Memory, evolutionary operator, and local search based improved Grey Wolf Optimizer with linear population size reduction technique

Optimization of multi-modal functions is challenging even for evolutionary and swarm-based algorithms as it requires an efficient exploration for finding the promising region of the search space, and effective exploitation to precisely find the global optimum. Grey Wolf Optimizer (GWO) is a recently developed metaheuristic algorithm that is inspired by nature with a relatively small number of parameters for tuning. However, GWO and most of its variants may suffer from the lack of population diversity, premature convergence, and the inability to preserve a good balance between exploratory and exploitative behaviors. To address these limitations, this work proposes a new variant of GWO incorporating memory, evolutionary operators, and a stochastic local search technique. It further integrates Linear Population Size Reduction (LPSR) technique. The proposed algorithm is comprehensively tested on 23 numerical benchmark functions, high dimensional benchmark functions, 13 engineering case studies, four data classifications, and three function approximation problems. The benchmark functions are mostly taken from the CEC 2005 and CEC 2010 special sessions, and they include rotated, shifted functions. The engineering case studies are from the CEC 2020 real-world non-convex constrained optimization problems. The performance of the proposed GWO is compared with popular metaheuristics, namely, particle swarm optimization (PSO), gravitational search algorithm (GSA), slap swarm algorithm (SSA), differential evolution (DE), self-adaptive differential evolution (SADE), basic GWO and its three recently improved variants. Statistical analysis and Friedman tests have been conducted to thoroughly compare their performance. The obtained results demonstrate that the proposed GWO outperforms the algorithms compared for the benchmark functions and engineering case studies tested.

PANTR: A Proximal Algorithm With Trust-Region Updates for Nonconvex Constrained Optimization

This work presents PANTR, an efficient solver for nonconvex constrained optimization problems, that is well-suited as an inner solver for an augmented Lagrangian method. The proposed scheme combines forward-backward iterations with solutions to trust-region subproblems: the former ensures global convergence, whereas the latter enables fast update directions. We discuss how the algorithm is able to exploit exact Hessian information of the smooth objective term through a linear Newton approximation, while benefiting from the structure of box-constraints or ℓ1-regularization. An open-source C++ implementation of PANTR is made available as part of the NLP solver library ALPAQA. Finally, the effectiveness of the proposed method is demonstrated in nonlinear model predictive control applications.

Asymptotic analysis of scalarization functions and applications

&lt;p style='text-indent:20px;'&gt;In this paper, we consider two common scalarization functions and their applications via asymptotic analysis. We mainly analyze the recession and asymptotic properties of translation invariant function and oriented distance function, and discuss their monotonicity and Lipschitz continuity in terms of recession functions. Finally, we apply these scalarization functions to the characterization of the nonemptiness and boundedness of the solution set for a general constrained nonconvex optimization problem.&lt;/p&gt;

Reinforcement learning-based modified cuckoo search algorithm for economic dispatch problems

One of the crucial tasks of social development is carbon neutrality, which is mainly due to the emission of greenhouse gases. However, thermal power, which produces plenty of harmful gases, is still the main component of electric energy. Therefore, Economic Dispatch (ED) is proposed to utilize energy resources more efficiently and reduce the cost of power generation. ED is a nonlinear and nonconvex-constrained optimization problem that is difficult to optimize. In this paper, we propose a Reinforcement Learning-based Modified Cuckoo Search algorithm (RLMCS) to solve ED problems. The proposed algorithm employs the concept of Reinforcement Learning (RL) and develops an RL-based method to process population obtained from the explorative phase. The RL-based method can dynamically enhance the population based on cumulative rewards and the current environmental state. Thus, the comprehensive search ability of RLMCS has been well improved. Moreover, some proven technologies, i.e., Gaussian random walk, quasi-opposition learning, and adaptive switch parameter, are introduced to further enhance the efficiency of RLMCS. The performance of the RLMCS is tested on standard ED problems (6 and 11 units) and ED problems with valve-point effects (10, 14, and 40 units). RLMCS is also compared with some well-established CS variants. The experimental results have demonstrated that RLMCS is more competitive and robust.

A υ-Constrained Matrix Adaptation Evolution Strategy With Broyden-Based Mutation for Constrained Optimization.

To solve the nonconvex constrained optimization problems (COPs) over continuous search spaces by using a population-based optimization algorithm, balancing between the feasible and infeasible solutions in the population plays an important role over different stages of the optimization process. To keep this balance, we propose a constraint handling technique, called the υ -level penalty function, which works by transforming a COP into an unconstrained one. Also, to improve the ability of the algorithm in handling several complex constraints, especially nonlinear inequality and equality constraints, we suggest a Broyden-based mutation that finds a feasible solution to replace an infeasible solution. By incorporating these techniques with the matrix adaptation evolution strategy (MA-ES), we develop a new constrained optimization algorithm. An extensive comparative analysis undertaken using a broad range of benchmark problems indicates that the proposed algorithm can outperform several state-of-the-art constrained evolutionary optimizers.

Quasi-oppositional turbulent water flow-based optimization for cascaded short term hydrothermal scheduling with valve-point effects and multiple fuels

In an interconnected power system, short-term hydrothermal scheduling (SHTS) portrays a paramount important task in the economic operation of electric power system. SHTS is discerned as a formidable non-convex constrained optimization problem considering the cascade nature of hydro plants, and valve-point loading (VPL) and multi-fuel sources of thermal plants. In the present study, a novel metaheuristic approach, a quasi-oppositional turbulent water flow-based optimization (QOTWFO) is proposed to solve the SHTS problem. Turbulent water flow-based optimization (TWFO) is evolved by the whirlpool prodigy formed in a turbulent flow of water. QOTWFO is an amended version of TWFO, where the quasi-opposition-based learning is pioneered for population initialization and iteration vaulting. It generates a quasi-oppositional solution which has the highest probability to elicit the global optimum solution than a randomly generated solution. Thus, the global searching ability and computational efficiency of the algorithm is enhanced. A new heuristic constraints handling mechanism is proposed to fulfil the equality constraints notably active power balance and dynamic water flow balance. The cogency of proposed QOTWFO is examined on standard benchmark functions, and multi-reservoir cascaded hydrothermal test systems with the cogitation of VPL effects and multi-fuel sources of thermal plants. Besides, Newton Raphson load flow approach is employed to determine power losses and line flows in a 16-bus, and 35 transmission lines hydrothermal power system. To model a realistic power system model, the maximum power transfer capability of each transmission line is taken into account. Simulation outcomes substantiate that, compared with state-of-the-art heuristic techniques, QOTWFO offers better feasible solutions as well as guarantees the robustness of the algorithm.

Joint Optimization on Power Allocation and Splitting for WSNs With SWIPT-Based Relay

How to prolong network lifetime has become an important issue in the design of large scale wireless sensor networks (WSNs). In this paper, a novel power saving scheme for conventional WSNs via simultaneous wireless information and power transfer (SWIPT) based relays is proposed. Unlike conventional relays based communications need extra energy supply for data forwarding, the relay applied in this paper works in an energy-free manner with the support of SWIPT. The power saving model with both power splitting (PS) and time switching (TS) based SWIPT are proposed. Then, we formulate the joint power allocation and splitting problems of SWIPT relay assisted WSNs as non-convex constrained optimization problems. Since the formulated problems are non-convex, semi-positive definite programming (SDP) algorithms are proposed to find the joint optimization on power allocation and splitting with low complexity. The simulation results show that the PS model has more advantages than the traditional direct communication model in long distance transmission. Under the same information receiving strategy, the PS model outperforms the TS model.

Provably training overparameterized neural network classifiers with non-convex constraints

Training a classifier under non-convex constraints has gotten increasing attention in the machine learning community thanks to its wide range of applications such as algorithmic fairness and class-imbalanced classification. However, several recent works addressing non-convex constraints have only focused on simple models such as logistic regression or support vector machines. Neural networks, one of the most popular models for classification nowadays, are precluded and lack theoretical guarantees. In this work, we show that overparameterized neural networks could achieve a near-optimal and near-feasible solution of non-convex constrained optimization problems via the project stochastic gradient descent. Our key ingredient is the no-regret analysis of online learning for neural networks in the overparameterization regime, which may be of independent interest in online learning applications.

Local saddles of relaxed averaged alternating reflections algorithms on phase retrieval

Phase retrieval can be expressed as a non-convex constrained optimization problem to identify one phase minimizer one a torus. Many iterative transform techniques have been proposed to identify the minimizer, e.g., relaxed averaged alternating reflections (RAAR) algorithms. In this paper, we present one optimization viewpoint on the RAAR algorithm. RAAR algorithm is one alternating direction method of multipliers with one penalty parameter. Pairing with multipliers (dual vectors), phase vectors on the primal space are lifted to higher dimensional vectors, RAAR algorithm is one continuation algorithm, which searches for local saddles in the primal-dual space. The dual iteration approximates one gradient ascent flow, which drives the corresponding local minimizers in a positive-definite Hessian region. Altering penalty parameters, the RAAR avoids the stagnation of these corresponding local minimizers in the primal space and thus screens out many stationary points corresponding to non-local minimizers.

Joint Transmit Waveform and Receive Filter Design for Dual-Function Radar-Communication Systems

In this paper, the problem of joint transmit waveform and receive filter design for dual-function radar-communication (DFRC) systems is studied. The considered system model involves a multiple antenna base station (BS) of a cellular system serving multiple single antenna users on the downlink. Furthermore, the BS simultaneously introduces sensing capabilities in the form of point-like target detection from the reflected return signals in a signal-dependent interference environment. A novel framework based on constrained optimization problems is proposed for the joint design of the transmit waveform and the radar receive filter such that different constraints related to the power amplifiers and the radar waveform are satisfied. In contrast to the existing approaches in the DFRC systems’ literature, the proposed approach does not require the knowledge of a predetermined radar beampattern in order to optimize the performance of the radar part through its approximation. Instead, a beampattern is generated by maximizing the radar receive signal-to-interference ratio (SINR) thus, enabling a more flexible design. Moreover, the radar receive filter processing and its optimization is considered for the first time on DFRC systems, enabling the effective exploitation of the available degrees of freedom in the radar receive array. Efficient algorithmic solutions with guaranteed convergence are developed for the defined constrained nonconvex optimization problems. The effectiveness of the proposed solutions is verified via numerical results.