Nonlinear Nonconvex Optimization Problem Research Articles

The deep neural network solver for B-spline approximation

This paper introduces a novel unsupervised deep learning approach to address the knot placement problem in the field of B-spline approximation, called deep neural network solvers (DNN-Solvers). Given discrete points, the DNN acts as a solver for calculating knot positions in the case of a fixed knot number. The input can be any initial knots and the output provides the desirable knots. The loss function is based on the approximation error. The DNN-Solver converts the lower-dimensional knot placement problem, characterized as a nonconvex nonlinear optimization problem, into a search for suitable network parameters within a high-dimensional space. Owing to the over-parameterization nature, DNN-Solvers are less likely to be trapped in local minima and robust against initial knots. Moreover, the unsupervised learning paradigm of DNN-Solvers liberates us from constructing high-quality synthetic datasets with labels. Numerical experiments demonstrate that DNN-Solvers are excellent in both approximation results and efficiency under the premise of an appropriate number of knots.

An evolution strategy approach for the balanced minimum evolution problem.

The Balanced Minimum Evolution (BME) is a powerful distance based phylogenetic estimation model introduced by Desper and Gascuel and nowadays implemented in popular tools for phylogenetic analyses. It was proven to be computationally less demanding than more sophisticated estimation methods, e.g. maximum likelihood or Bayesian inference while preserving the statistical consistency and the ability to run with almost any kind of data for which a dissimilarity measure is available. BME can be stated in terms of a nonlinear non-convex combinatorial optimization problem, usually referred to as the Balanced Minimum Evolution Problem (BMEP). Currently, the state-of-the-art among approximate methods for the BMEP is represented by FastME (version 2.0), a software which implements several deterministic phylogenetic construction heuristics combined with a local search on specific neighbourhoods derived by classical topological tree rearrangements. These combinations, however, may not guarantee convergence to close-to-optimal solutions to the problem due to the lack of solution space exploration, a phenomenon which is exacerbated when tackling molecular datasets characterized by a large number of taxa. To overcome such convergence issues, in this article, we propose a novel metaheuristic, named PhyloES, which exploits the combination of an exploration phase based on Evolution Strategies, a special type of evolutionary algorithm, with a refinement phase based on two local search algorithms. Extensive computational experiments show that PhyloES consistently outperforms FastME, especially when tackling larger datasets, providing solutions characterized by a shorter tree length but also significantly different from the topological perspective. The software and the data are available at https://github.com/andygaspar/PHYLOES.

An Efficient Path Planning Method for the Unmanned Aerial Vehicle in Highway Inspection Scenarios

Unmanned aerial vehicles (UAVs) have received widespread attention due to their flexible deployment characteristics. Automated airports equipped with UAVs are expected to become important equipment for improving quality and reducing costs in many inspection scenarios. This paper focuses on the automated inspection business of UAVs dispatched by automated airports in highway scenarios. On the basis of considering the shape of highway curves, inspection targets, and the energy consumption characteristics of UAVs, planning the flight parameters of UAVs is of great significance for ensuring the effectiveness of the inspection process. This paper first sets the inspection path points for the UAV based on highway curves, and then proposes an efficient heuristic method for the nonlinear non-convex parameter optimization problem, through which the parameters of the UAV’s inspection altitude, hovering altitude, and flight speed are planned. Simulation and analysis show that the proposed method possesses good parameter planning efficiency. By combining several existing trajectory planning methods, e.g., the traversal method, the deep Q-network based method, and the genetic method, it can be concluded that the proposed method in this paper has better overall planning performance including planning efficiency and inspection effectiveness.

A comparison of the homotopy method with linearisation approaches for a non-linear optimization problem of operations in a reservoir cascade

AbstractReservoir systems are often operated for multiple purposes. This can result conflicting operational goals. To efficiently control these systems and to satisfy the different interests as good as possible, mathematical optimization models can be used to support operational decisions. Common approaches for reservoir optimization apply linear optimization techniques. However, real-world systems often require non-linear functions to describe the relation between water level and volume in a reservoir or to account for the hydropower equation. When the non-linear equations form a non-convex optimization problem, the problem is not necessarily solved to a global optimum. Piecewise-linear or linear formulations of the non-linear equations are a common way to address non-linear non-convex optimization problems. In this paper, the novel homotopy method is compared with two established approaches—the piecewise-linear and the linear approximation—to account for non-linear components in the optimization problem. The analysis is carried out for a cascade of three reservoirs under two scenarios—a flood scenario and a load balance scenario. The optimization software is the open source package RTC-Tools 2.4. Compared to the piecewise-linear and the linear approach the homotopy method shows a better accuracy for the analysed cases, because the method solves the flow equations within the optimization in a non-simplified form. Different to the piecewise-linear and the linear approach, however, the homotopy method does not guarantee a global optimum. The solution is still path-stable, which is a basic pre-requisite for its application in an operational context of hydropwer scheduling. Compared to the piecewise-linear approach, the homotopy method is easier to implement under the condition that the software supports the method.

Algorithms for a risk-averse Stackelberg game with multiple adversaries

We consider a Stackelberg game that arises in a security domain (SSG), where a defender can simultaneously protect m out of n targets from an adversary that observes the defense strategy before deciding on an utility maximizing attack. Given the high stakes in security settings, it is reasonable that the defender in this game is risk averse with respect to the attacker’s decisions.Here we focus on developing efficient solution algorithms for a specific SSG, where the defender uses an entropic risk measure to model risk aversion to the attacker’s strategies, and where multiple attackers select targets following logit quantal response equilibrium models. This problem can be formulated as a nonconvex nonlinear optimization problem. We propose two solution methods: (1) approximate the problem through convex mixed integer nonlinear programs (MINR) and (2) a general purpose methodology (CELL) to optimize nonconvex and nonseparable fractional problems through mixed integer linear programming approximations. Both methods provide arbitrarily good incumbents and lower bounds on SSG. We present cutting plane methods to solve these problems for large instances. Our computational experiments illustrate the advantages of introducing risk aversion into the defender’s behavior and show that MINR dominates CELL, producing in 2 h solutions that are within 2% of optimal on average.

Uncertainty-Informed Operation Coordination in a Water-Energy Nexus

The widespread deployment of smart heterogeneous technologies and the growing complexity in our modern society calls for effective coordination of the interdependent lifeline networks. In particular, operation coordination of electric power and water infrastructures is urgently needed as the water system is one of the most energy-intensive networks, an interruption in which may quickly evolve into a dramatic societal concern. The closely-intertwined ecosystem of water and power infrastructures is commonly known as water-energy nexus. This paper develops a novel analytic for uncertainty-aware day-ahead operation optimization of the interconnected power and water systems (PaWS). Joint probabilistic constraint (JPC) programming is employed to capture the uncertainties in wind resources and water demand forecasts. The proposed integrated stochastic model is presented as a non-linear non-convex optimization problem, where the non-linear hydraulic constraints in the water network are linearized using piece-wise linearization technique, and the non-convexity is efficiently tackled with a Boolean solution methodology to convert the proposed model with JPCs to a tractable mixed-integer linear programming (MILP) formulation that can be quickly solved to optimality. The suggested framework is applied to a 15-node commercial-scale water network jointly operated with a power transmission system using a modified IEEE 57-bus test system. The numerical results demonstrate the of the proposed stochastic framework, resulting in cost reduction (13% on average when compared to the traditional setting) and energy saving of the integrated model under different realizations of uncertain renewable energy sources (RESs) and water demand scenarios. Additionally, the scalability of the proposed model is tested on a modified IEEE 118-bus test system connected to five water networks.

LQR based optimal co-design for linear control systems with input and state constraints

This paper considers a constrained co-design problem for linear time-invariant (LTI) systems. With a practical vision of real-world problems, constraints on the control signal and system states are involved in the proposed algorithm. The main goal is to design a sub-optimal controller by simultaneously obtaining the control policy and the model parameters. To this end, the conventional problem of solving the Hamiltonian-Jacobi-Bellman (HJB) equation is transformed into a nonlinear non-convex optimisation problem. Then, by reformulating the constraints of the optimisation problem, it is relaxed into a convex Semi-Definite Programming (SDP). By presenting an iterative method, the control cost, the performance, and the number of iterations are improved compared to conventional methods, and a closer result to the optimal solution is obtained. The performance and efficacy of the proposed algorithm are investigated through a case study on the physical load positioning system.

A robust BFGS algorithm for unconstrained nonlinear optimization problems

The traditional BFGS algorithm has been proved very efficient. It is convergent for convex nonlinear optimization problems. However, for non-convex nonlinear optimization problems, it is known that the BFGS algorithm may not be convergent. This paper proposes a robust BFGS algorithm in the sense that the algorithm superlinearly converges to a local minimum under some mild assumptions for both convex and non-convex nonlinear optimization problems. Numerical test on the CUTEst test set is reported to demonstrate the merit of the proposed robust BFGS algorithm. This result shows that the robust BFGS algorithm is very efficient and effective.

Chance constrained dynamic optimization approach for single machine scheduling involving flexible maintenance, production, and uncertainty

Chance constraints are suitable for industrial process modeling under uncertain conditions, where constraints cannot be strictly satisfied or do not need to be fully satisfied. In this paper, a single machine scheduling problem involving flexible maintenance, production, and uncertainty is modeled as a chance constrained dynamic optimization problem (CCDOP). A novel method is proposed for transforming the CCDOP into an equivalent deterministic dynamic optimization problem (DOP) with fixed state jump times. Furthermore, by using the idea of l1 penalty function and a smooth approximation technique, the resulting deterministic DOP becomes a smoothing penalty problem, which is a non-convex nonlinear parameter optimization problem (NNPOP) with simple bounds on the variables. To solve the NNPOP, a gradient-based stochastic search algorithm (GSSA) is developed based on a gradient-based adaptive search algorithm (GASA) and a novel stochastic search algorithm (NSSA). The convergence analysis result shows that the GSSA is a globally convergent algorithm. Finally, two numerical examples are used to illustrate the effectiveness of the proposed method. Numerical results show that the GSSA has excellent convergence behavior with robust computation feature, providing better results compared with the other typical methods.

Optimal economic-environmental dispatch in MT-HVDC systems via sine-cosine algorithm

This paper addresses the problem of optimal economic-environmental dispatch in Multi-Terminal High-Voltage Direct Current (MT-HVDC) networks using the Sine-Cosine Algorithm (SCA). This optimization methodology allows working with nonlinear non-convex large-scale optimization problems via sequential programming. The SCA works with an initial population and rules of advance based on the best current solution and sine and cosine functions that define the direction of the next solution. Three variants of the SCA are evaluated in a standard six-node MT-HVDC system considering a linear combination of the objective functions (i.e., greenhouse emissions and energy production costs). The main advantage of the proposed evolutionary approach lies in its pure algorithmic structure. Thus, it can be easily adapted to any continuous optimization problem. All numerical calculations are performed using MATLAB software.

Parametric estimation in photovoltaic modules using the crow search algorithm

<p>The problem of parametric estimation in photovoltaic (PV) modules considering manufacturer information is addressed in this research from the perspective of combinatorial optimization. With the data sheet provided by the PV manufacturer, a non-linear non-convex optimization problem is formulated that contains information regarding maximum power, open-circuit, and short-circuit points. To estimate the three parameters of the PV model (i.e., the ideality diode factor (a) and the parallel and series resistances (R<sub>p</sub> and R<sub>s</sub>)), the crow search algorithm (CSA) is employed, which is a metaheuristic optimization technique inspired by the behavior of the crows searching food deposits. The CSA allows the exploration and exploitation of the solution space through a simple evolution rule derived from the classical PSO method. Numerical simulations reveal the effectiveness and robustness of the CSA to estimate these parameters with objective function values lower than 1 × 10<sup>−28</sup> and processing times less than 2 s. All the numerical simulations were developed in MATLAB 2020a and compared with the sine-cosine and vortex search algorithms recently reported in the literature.</p>

Lifting convex inequalities for bipartite bilinear programs

The goal of this paper is to derive new classes of valid convex inequalities for quadratically constrained quadratic programs (QCQPs) through the technique of lifting. Our first main result shows that, for sets described by one bipartite bilinear constraint together with bounds, it is always possible to sequentially lift a seed inequality that is valid for a restriction obtained by fixing variables to their bounds, when the lifting is accomplished using affine functions of the fixed variables. In this setting, sequential lifting involves solving a non-convex nonlinear optimization problem each time a variable is lifted, just as in Mixed Integer Linear Programming. To reduce the computational burden associated with this procedure, we develop a framework based on subadditive approximations of lifting functions that permits sequence-independent lifting of seed inequalities for separable bipartite bilinear sets. In particular, this framework permits the derivation of closed-form valid inequalities. We then study a separable bipartite bilinear set where the coefficients form a minimal cover with respect to the right-hand-side. For this set, we introduce a bilinear cover inequality, which is second-order cone representable. We argue that this bilinear cover inequality is strong by showing that it yields a constant-factor approximation of the convex hull of the original set. We study its lifting function and construct a two-slope subadditive upper bound. Using this subadditive approximation, we lift fixed variable pairs in closed-form, thus deriving a lifted bilinear cover inequality that is valid for general separable bipartite bilinear sets with box constraints.

MAD Dispersion Measure Makes Extremal Queue Analysis Simple

A notorious problem in queueing theory is to compute the worst possible performance of the GI/G/1 queue under mean-dispersion constraints for the interarrival- and service-time distributions. We address this extremal queue problem by measuring dispersion in terms of mean absolute deviation (MAD) instead of the more conventional variance, making available methods for distribution-free analysis. Combined with random walk theory, we obtain explicit expressions for the extremal interarrival- and service-time distributions and, hence, the best possible upper bounds for all moments of the waiting time. We also obtain tight lower bounds that, together with the upper bounds, provide robust performance intervals. We show that all bounds are computationally tractable and remain sharp also when the mean and MAD are not known precisely but are estimated based on available data instead. Summary of Contribution: Queueing theory is a classic OR topic with a central role for the GI/G/1 queue. Although this queueing system is conceptually simple, it is notoriously hard to determine the worst-case expected waiting time when only knowing the first two moments of the interarrival- and service-time distributions. In this setting, the exact form of the extremal distribution can only be determined numerically as the solution to a nonconvex nonlinear optimization problem. Our paper demonstrates that using mean absolute deviation (MAD) instead of variance alleviates the computational intractability of the extremal GI/G/1 queue problem, enabling us to state the worst-case distributions explicitly.

Dynamic Power Allocation for Uplink NOMA With Statistical Delay QoS Guarantee

Most existing optimization objectives considered in non-orthogonal multiple access (NOMA) power allocation schemes are non-delay-sensitive metrics. In order to apply NOMA to various Internet of Things scenarios, the delay must be considered. The effective capacity of users, which characterizes the capacity under specific expiration probabilities, can potentially be a performance metric of statistical delay quality of service (QoS). In this paper, we propose two novel dynamic power allocation schemes with statistical delay QoS guarantee in the uplink NOMA system with paired users. One of the schemes maximizes the sum effective capacity (SEC) of the strong and weak users, which is a non-convex nonlinear optimization problem and is solved by Lagrangian dual decomposition and successive convex approximation (SCA). The other one maximizes the effective energy efficiency (EEE) of uplink NOMA, which is a fractional optimization problem and is solved by integrating the Dinkelbach method, SCA, and Lagrangian dual decomposition. Numerical results show that the SEC and EEE can be significantly improved by the proposed schemes, compared to the existing NOMA and orthogonal multiple access power allocation schemes.

Numerical algorithm for optimal control of switched systems and its application in cancer chemotherapy

This paper considers an optimal control problem of switched dynamical systems with control input and system state constraints. Unlike in traditional switched dynamical systems, the switching times cannot be specified directly and they are governed by a state-dependent switching condition. Thus, the existing methods cannot be directly used to solve this problem. To overcome this difficulty, the switching conditions are transformed into a continuous-time inequality constraint by introducing an integer constraint. Further, the original optimal control problem is approximated by using a sequence of constrained non-convex nonlinear parameter optimization problems by using a relaxation method, a control vector parameterization technique, and a time-scaling transformation. Following that, a penalty function-based intelligent optimization algorithm is proposed for obtaining a global optimal solution based on a more effective penalty function method and a more effective intelligent optimization algorithm. The convergence results show that the proposed method is globally convergent. Numerical simulation results show that the proposed method is lower time-consuming, has faster convergence speed, can obtain a better objective function value than the existing typical algorithms, and can achieve a stable and robust performance when considering the small perturbations in constraint conditions or the small perturbations of the model parameters.

Energy Efficiency Maximization for UAV-Assisted Emergency Communication Networks

Using unmanned aerial vehicles (UAVs) in emergency communications is a promising technology because of their flexible deployment, low cost, and high mobility. However, due to the limited energy of the onboard battery, the service duration of the UAV is greatly limited. In this paper, we study an emerging energy-efficient UAV emergency network, where a UAV works as an aerial base station to serve a group of users with different statistical quality-of-service (QoS) constraints in the downlink. In particular, the energy efficiency of the UAV is defined as the sum effective capacity of the downlink users divided by the energy consumption of the UAV, which includes the energy consumed by communication and the energy consumed by hovering. Then, we formulate an optimization problem to maximize the energy efficiency of the UAV by jointly optimizing the UAV’s altitude, downlink transmit power, and bandwidth allocation while meeting a statistical delay QoS requirement for each user. The formulated optimization problem is a nonlinear nonconvex optimization problem of fractional programming, which is difficult to solve. In order to deal with the nonconvex optimization problem, the following two steps are used. First, we transform the fractional objective function into a tractable subtractive function. Second, we decompose the original optimization problem into three subproblems, and then, we propose an efficient iterative algorithm to obtain the energy efficiency maximization value by using the Dinkelbach method, the block coordinate descent, and the successive convex optimization technique. Extensive simulation results show that our proposed algorithm has significant energy savings compared with a benchmark scheme.

A novel finite-time q-power recurrent neural network and its application to uncertain portfolio model

This paper presents a novel finite-time q-power recurrent neural network (FT-QPNN) for uncertain portfolio model. An uncertain mean–variance-skewness model under concave transaction costs is discussed. This portfolio model is essentially a nonconvex nonlinear optimization problem with a non-positive definite Hessian matrix of the Lagrange function. The non-positive definite Hessian matrix leads to the failure of many recurrent neural network methods in solving the problem, and many recurrent neural networks cannot converge to the equilibrium point in finite time. To overcome these difficulties, the FT-QPNN is proposed. Combined with finite-time activation function and local convexification method, the FT-QPNN can solve the optimization problem with non-positive definite Hessian matrix and converge to the equilibrium point in finite time. The global finite-time stability and robustness properties of the FT-QPNN are proved theoretically and verified by numerical experiments. Furthermore, the proposed FT-QPNN is applied to solve the uncertain portfolio model. The application simulation results and comparative experiments with other methods respectively illustrate the feasibility and superiority of the FT-QPNN.

Optimal placement ofFACTSdevices and power‐flow solutions for a power network system integrated with stochastic renewable energy resources using new metaheuristic optimization techniques

Flexible AC transmission systems (FACTS) and optimal power-flow (OPF) solutions play an important role in solving power operation problems. The volatile nature of the power generation profiles from renewable energy sources, solar and wind systems, and determining the optimal locations and sizes of FACTS devices increase the complexity of the OPF problems in modern power network models, such as transmission power loss, power generation operation cost and voltage deviation, as a highly nonlinear-nonconvex optimization problem. Therefore, this article introduces and employs four new independent, reliable and efficient optimization algorithms inspired by nature and biological nature, namely: Slime Mould Algorithm (SMA), Artificial Ecosystem-based Optimization (AEO), Marine Predators Algorithm (MPA) and Jellyfish Search (JS), for solving both multi- and single-OPF objective problems for a power network incorporating FACTS and stochastic renewable energy sources. The proposed new metaheuristic optimization techniques are compared to the common and available alternatives in the literature, Particle Swarm Optimization (PSO), Moth Flame Optimization (MFO) and Grey Wolf Optimizer (GWO), using IEEE 30-bus test system. To consider and address the challenges of the OPF in modern power network models, the proposed optimization techniques tested under different operation cases such as an increasing in the load, with and without FCTAS and renewable energy sources, different renewable energy sources locations on the network. The result showed that the MPA, SMA, JS and AEO algorithms are more effective solvers for the OPF problems cases compared to the PSO, GWO and MFO algorithms. For example, the AEO obtained 0.0844 p.u. in case of minimizing the voltage deviation compared to 0.1155 p.u. for PSO, which means that the AEO algorithm improved the voltage deviation term by 27% compared to the PSO algorithm.

Sequence of polyhedral relaxations for nonlinear univariate functions

Given a nonlinear, univariate, bounded, and differentiable function $f(x)$, this article develops a sequence of Mixed Integer Linear Programming (MILP) and Linear Programming (LP) relaxations that converge to the graph of $f(x)$ and its convex hull, respectively. Theoretical convergence of the sequence of relaxations to the graph of the function and its convex hull is established. For nonlinear non-convex optimization problems, the relaxations presented in this article can be used to construct tight MILP and LP relaxations. These MILP and the LP relaxations can also be used with MILP-based and spatial branch-and-bound based global optimization algorithms, respectively.

Microgrid dynamic combined power–heat economic-emission dispatch with deferrable loads and price-based energy storage elements and power exchange

This research effort formulates and solves a biobjective dynamic combined heat and power economic-emission dispatch (DCHPED) of generation in a microgrid (MG). The MG encompasses fossil-fuel power and heat units, renewable resources, price-operated electrical and thermal energy storage elements, bidirectional-exchange of electrical power with the external grid, and fixed nondefferrable and deferrable loads. The two objective functions involve the total operation cost and the total gas emission produced by the fossil fuel units participating in the MG dynamic dispatch. Costs of emission tax and price-based energy storage and exchanges with the external grid are ingredients of the total operating costs. The Bernstein-search differential evolution (BSDE) algorithm is utilized to deal with the problem to obtain the day-ahead optimal schedules of available resources. The nonlinear nonconvex MG DCHPED optimization problem that has a large number of decision variables involves equality and inequality constraints, production costs, emission levels, taxes of released gases, ramp-rate limits of power-producing units, effects of valve-points in generating units, active power transmission losses, and feasible operation regions of cogenerating (CHP) units. The paper investigates the positive impacts of utilizing price-based electrical and thermal storage elements, deferrability of loads, and price-based power exchanges on the day-ahead MG operation cost. The paper also describes an approach to achieve the best compromise solution (BCS) of the MG DCHPED employing the fuzzy satisfying method (FSM). In addition, it shows that gas emissions and emission taxes in the MG can be both controlled by exchanging power with the external grid.