Convex Method Research Articles

A subspace derivative-free projection method for convex constrained nonlinear equations

A subspace derivative-free projection method for convex constrained nonlinear equations

A stochastic moving ball approximation method for smooth convex constrained minimization

AbstractIn this paper, we consider constrained optimization problems with convex objective and smooth convex functional constraints. We propose a new stochastic gradient algorithm, called the Stochastic Moving Ball Approximation (SMBA) method, to solve this class of problems, where at each iteration we first take a (sub)gradient step for the objective function and then perform a projection step onto one ball approximation of a randomly chosen constraint. The computational simplicity of SMBA, which uses first-order information and considers only one constraint at a time, makes it suitable for large-scale problems with many functional constraints. We provide a convergence analysis for the SMBA algorithm using basic assumptions on the problem, that yields new convergence rates in both optimality and feasibility criteria evaluated at some average point. Our convergence proofs are novel since we need to deal properly with infeasible iterates and with quadratic upper approximations of constraints that may yield empty balls. We derive convergence rates of order $${\mathcal {O}} (k^{-1/2})$$ O ( k - 1 / 2 ) when the objective function is convex, and $${\mathcal {O}} (k^{-1})$$ O ( k - 1 ) when the objective function is strongly convex. Preliminary numerical experiments on quadratically constrained quadratic problems demonstrate the viability and performance of our method when compared to some existing state-of-the-art optimization methods and software.

A new distributed robust H∞ control strategy for a class of uncertain interconnected large-scale time-delay systems subject to actuator saturation and disturbance

In the realm of large-scale systems, the complexity of controller design has long been exacerbated by the proliferation of decision variables and inherent conservatism. This study introduces a novel approach to address these challenges, presenting a new distributed robust controller design methodology tailored for large-scale systems grappling with disturbances, uncertainties, and actuator saturations. The primary objectives include reducing conservatism, minimizing decision variables, and significantly curtailing computation time. To surmount these hurdles, the research leverages descriptive and reciprocally convex methods, formulating the design procedure using linear matrix inequalities. This enables the adjustment of uncertain parameters and robust disturbance rejection, thereby ensuring stability in large-scale systems. Additionally, a feedback control law is proposed to accommodate saturation constraints and ensure the closed-loop system’s stability. Notably, the effectiveness of the proposed control scheme is demonstrated through the evaluation of a full-car active suspension system, which is partitioned into interconnected subsystems to a large-scale system. Comparative analyses underscore the superior performance and technical advancements offered by the proposed methodology over existing approaches.

Collision-free trajectory planning for UAVs based on sequential convex programming

This paper proposes an optimization-based approach to solve the collision-free trajectory planning problem for unmanned aerial vehicles (UAVs) with nonlinear dynamical systems. The problem is modeled as a non-convex optimization problem, and then iteratively solved by sequential convex programming algorithm which approximates non-convex constraints by convexification methods. We linearize the dynamical equations of the UAV and penalize collision-free constraints with a hinge loss. Furthermore, a variable trust region method is employed to ensure the convergence and stability of the algorithm. We apply the algorithm to three-degree-of-freedom UAV case studies. Numerical results validate that our method can provide collision-free trajectories with a 27.19% reduction in total flight time and a 93.61% reduction in computation time on average compared with the genetic algorithm method.

Optimal methods for convex nested stochastic composite optimization

Optimal methods for convex nested stochastic composite optimization

Convex Approach to Real-Time Multiphase Trajectory Optimization for Urban Air Mobility

Electric vertical takeoff and landing technology has emerged as a promising solution to alleviate ground transportation congestion. However, the limited capacity of onboard batteries enforces hard time constraints for vehicle operations in a complex urban environment. Therefore, a constrained, real-time trajectory optimization method is necessary to enable safe and energy-efficient vehicle flight. Unfortunately, most existing trajectory optimization methods suffer from low computational efficiency, unpredictable convergence processes, and strong dependence on a good initial trajectory. In this paper, we employ a successive convex programming approach to solve the multiphase minimum-energy-cost electric vertical takeoff and landing vehicle trajectory optimization problem for urban air mobility applications. The main contribution is the development and validation of two novel convexification methods that find approximate optimal solutions to the multiphase nonlinear trajectory optimization problem in real time by solving a sequence of convex subproblems. Specifically, the first method transforms the original nonlinear problem into a sequence of second-order cone programming problems through a convenient change of variables and lossless convexification, while the second approach achieves a similar goal without the need for control convexification. The resulting convex subproblems can be solved reliably in real time by advanced interior point methods. The proposed methods are demonstrated through numerical simulations of two different urban air mobility scenarios and compared with the solution obtained from GPOPS-II, a state-of-the-art general-purpose optimal control solver. Our proposed sequential convex programming methods can obtain near-optimal solutions with faster computational speed than GPOPS-II.

Spacecraft Close Proximity to Noncooperative Target Based on Pseudospectral Convex Method

Spacecraft Close Proximity to Noncooperative Target Based on Pseudospectral Convex Method

Mosquito suppression via Filippov incompatible insect technique

Mosquito-borne diseases persist as a global health challenge despite ongoing control efforts, necessitating the exploration of alternative control approaches. This research proposes a Filippov incompatible insect technique (IIT) model with a threshold policy control for suppressing mosquito population. The model implements biological and chemical control strategies only when wild mosquito density surpasses an economic threshold. The biological strategy involves releasing Wolbachia-infected male mosquitoes, offering advantages such as a shortened lifespan and disease blocking. Simultaneously, moderate insecticide and larvicide spraying serve as a chemical approach to expedite the control process. The model provides economic advantages by minimizing control costs and environmental benefits through the integration of biological strategies, thus reducing reliance on chemicals. By employing Filippov's convex method, we identify necessary conditions for the existence of sliding segments and determine the dynamics of the switching line. Theoretical analysis reveals the model's long-term dynamics, including sliding bifurcations, pseudo-equilibria, and their stability. The model exhibits boundary equilibrium bifurcations and transcritical bifurcations in both free and controlled systems. Varying threshold values significantly impact the control outcomes effectiveness. The main findings demonstrate that integrated strategies efficiently decrease the population density of wild mosquitoes.

Robust economic–environmental dispatch problem for BESS and PV generators in MT-HVDC systems: A MI-MPC approach

The research addresses the energy management problem (EMP) in multi-terminal high-voltage direct current (MT-HVDC) systems using a mixed-integer model predictive control (MI-MPC). The MT-HVDC system consists of conventional thermal generation plants (fossil sources), renewable generation sources (solar photovoltaic plants), and battery energy storage systems (BESS). This combination creates a complex nonlinear EMP, as it introduces the power losses of the transmission networks through B-coefficients. The main contributions of this research can be summarized in three elements: (i) the formulation of a measurement- or simulation-based approach to determine B−coefficients in MT-HVDC systems for calculating power losses; (ii) the convexification of the EMP model through MI-MPC approaches, allowing to minimize economic and environmental objective functions; and (iii) the inclusion of uncertainties through the robust convex method. Numerical validations in the 11-bus grid, including the linear transportation model for the MT-HVDC system, confirm the multi-objective nature of the EMP when considering economic and environmental indices. In addition, the effect of uncertainties on generation and demand causes significant variations in the optimal Pareto front when compared to the deterministic scenario.

Convergence rate analysis of the gradient descent–ascent method for convex–concave saddle-point problems

In this paper, we study the gradient descent–ascent method for convex–concave saddle-point problems. We derive a new non-asymptotic global convergence rate in terms of distance to the solution set by using the semidefinite programming performance estimation method. The given convergence rate incorporates most parameters of the problem and it is exact for a large class of strongly convex-strongly concave saddle-point problems for one iteration. We also investigate the algorithm without strong convexity and we provide some necessary and sufficient conditions under which the gradient descent–ascent enjoys linear convergence.

An inertial Dai-Liao conjugate method for convex constrained monotone equations that avoids the direction of maximum magnification

An inertial Dai-Liao conjugate method for convex constrained monotone equations that avoids the direction of maximum magnification

Discrete-time Euler-smoothing methods for time-varying convex constrained optimization

In this paper, we propose two kinds of prediction–correction methods, discrete-time Euler-smoothing methods, for solving the general time-varying convex constrained optimization problem. By using the complementarity function and smoothing technique, the Karush–Kuhn–Tucker (KKT) trajectory of the problem can be reformulated as a system of smooth equations and considering the dynamics of the system of smooth equations in a discrete constant time-sampling periods scheme. The proposed methods mainly consist of the prediction step derived by Euler approximation and the correction step generated by one or multiple Newton iterations. Under some reasonable conditions, the approximate solution trajectories produced by the methods achieve the asymptotic error bound behaving as O(h4) with respect to optimal trajectory both in the primal and dual space as the smoothing parameter approaches to zero, where h is the sampling period. Finally, the experimental results show that our methods are efficient, and the steady-state tracking errors are much smaller than the other prediction–correction methods under the same conditions.

AN-SPS: adaptive sample size nonmonotone line search spectral projected subgradient method for convex constrained optimization problems

We consider convex optimization problems with a possibly nonsmooth objective function in the form of a mathematical expectation. The proposed framework (AN-SPS) employs Sample Average Approximations (SAA) to approximate the objective function, which is either unavailable or too costly to compute. The sample size is chosen in an adaptive manner, which eventually pushes the SAA error to zero almost surely (a.s.). The search direction is based on a scaled subgradient and a spectral coefficient, both related to the SAA function. The step size is obtained via a nonmonotone line search over a predefined interval, which yields a theoretically sound and practically efficient algorithm. The method retains feasibility by projecting the resulting points onto a feasible set. The a.s. convergence of AN-SPS method is proved without the assumption of a bounded feasible set or bounded iterates. Preliminary numerical results on Hinge loss problems reveal the advantages of the proposed adaptive scheme. In addition, a study of different nonmonotone line search strategies in combination with different spectral coefficients within AN-SPS framework is also conducted, yielding some hints for future work.

Rich dynamics of a delayed Filippov avian-only influenza model with two-thresholds policy

A nonsmooth Filippov avian-only influenza model with threshold strategies of culling susceptible and/or infected birds under different conditions is proposed to control the spread of avian influenza. The time delay representing the incubation period of avian influenza is incorporated into our model to make it more realistic, which is different from the traditional Filippov model. The stability of various types of equilibria and the existence of Hopf bifurcation are researched. Moreover, the existence of the sliding mode and its dynamics are investigated by Filippov convexity method. The theoretical analysis and numerical simulations indicate that, according to the value of thresholds and time delay, all solutions eventually converge to the regular equilibrium, pseudoequilibrium or stable periodic solution. We also indicate that time delay has a great influence on the sliding mode. Due to the many factors involved, and our main purpose is to study the impact of time delay on system stability, we have only conducted a simple analysis of the impact of time delay on the sliding mode. Furthermore, the boundary bifurcation switching stable regular equilibrium or stable limit cycle to a stable pseudoequilibrium can be exhibited by numerical simulations. Finally, with the increase of time delay, the global bifurcations from grazing bifurcation to buckling bifurcation and then to cross bifurcation are obtained. Our results indicate that although Filippov control strategies can effectively control the number of infected birds in many cases, the existence of time delay may challenge influenza control by the emergence of buckling bifurcation and cross bifurcation.

Employing advanced control, energy storage, and renewable technologies to enhance power system stability

As the world witnesses a surge in the adoption of renewable energy sources to meet the surging global power demands, the dynamic and intermittent nature of these sources emerges as a significant hurdle. This article extensively explores the potential of advanced control systems, energy storage technologies, and renewable resources to fortify stability within power systems. Advanced control methodologies are strategically amalgamated with energy storage deployment and the utilization of renewable energy, to advance the reliability, predictability, and sustainability of power systems. The stability analysis, with a dedicated focus on Input-to-input-to-state stability (ISS), is conducted meticulously by applying the Lyapunov function and the Reciprocally Convex Approach, resulting in an impressive stability rate of 23.6%. Additionally, Positive Realness is substantiated by extracting Linear Matrix Inequalities (LMI) in the context of Enhancing Grid Stability with Wind Power. The study places particular emphasis on evaluating ISS and Passivity in both delayed and non-delayed systems, with a specific focus on neutral time-delay systems. This evaluation involves the selection of an appropriate Lyapunov-Krasovskii Functional (LKF) and its derivation, coupled with the integration of reciprocally convex methods, descriptive approach, and Jensen inequality. The outcomes of these analyses shed light on the causes of excess energy and its effective storage, along with highlighting the synergistic impact of integrating renewable sources and controlling grid frequency, voltage, and power in real time. The study also accentuates the robustness achieved through Enhanced Stability and the mitigation of Reduced Fluctuations, especially in the context of renewable energy sources.

FILIPPOV TYPE PREY–PREDATOR SYSTEM FOR SELECTIVE HARVESTING OF PREY

In this paper, the dynamics of a Filippov system with logistically growing prey and Holling Type II predation is explored. The selective nonlinear harvesting of prey is carried out when the ratio of prey to predator is above a specified threshold value. In order to prevent the extinction of both the species and sustainable economic gains, no harvesting is allowed when the ratio is below a threshold value. The equilibrium states and their stability for the individual smooth are analyzed. Filippov convex method is applied to determine the dynamics on the boundary. The sliding mode dynamics is scrutinized, and it is observed that there is no escaping region. The existence of boundary points, tangent points and pseudo-equilibrium points are established. The conditions are obtained for the visibility of the tangent points. The complex behavior within the sliding region is studied by experimenting with different initial conditions. The increase in harvesting effort may lead to extinction of both the species through the sliding region of the boundary. The change in the stability behavior of both the interior equilibrium states is investigated with respect to model parameters. Border cross bifurcates with threshold value as the bifurcation parameter is shown for both the interior equilibrium states.

Dynamic analysis of a Filippov blood glucose insulin model

<abstract><p>This paper proposed a Filippov blood glucose insulin model with threshold control strategy and studied its dynamic properties. Using Filippov's convex method, we proved the global stability of its two subsystems, the existence and conditions of the sliding region of the system were also given, and different types of equilibrium states of the system were also addressed. The existence and stability of pseudo equilibrium points were thoroughly discussed. Through numerical simulations, we have demonstrated that it is possible to effectively control blood sugar concentrations to achieve more cost-effective treatment levels by selecting an appropriate threshold range for insulin injection.</p></abstract>

An improved sampled-data control for a nonlinear dynamic positioning ship with Takagi-Sugeno fuzzy model.

This article considered the sampled-data control issue for a dynamic positioning ship (DPS) with the Takagi-Sugeno (T-S) fuzzy model. By introducing new useful terms such as second-order term of time, an improved Lyapunov-Krasovskii function (LKF) was constructed. Additionally, the reciprocally convex method is introduced to bound the derivative of LKF. According to the constructed LKF, the sampling information during the whole sampling period was fully utilized, and less conservatism was obtained. Then, the stability condition, robust performance, mode uncertainty and sampled-data controller design were analyzed by means of the linear matrix inequality (LMI). Finally, an example was given to demonstrate the effectiveness of the proposed method.

An optimization-based discrete element model for dry granular flows: Application to granular collapse on erodible beds

Erosion processes and the associated static/flowing transition in granular flows are still poorly understood despite their crucial role in natural hazards such as landslides and debris flows. Continuum models do not yet adequately reproduce the observed increase of runout distance of granular flows on erodible beds or the development of waves at the bed/flow interface. Discrete Element Methods, which simulate each grain's motion and their complex interactions, provide a unique tool to investigate these processes numerically. Among them, Convex Methods (CM), resulting from the convexification of Contact Dynamics methods, benefit from a robust theoretical framework, ensuring the convergence of the numerical solution at every time iteration. They are also intrinsically more stable than classical Molecular Dynamics methods. However, although already implemented in engineering fields, CMs have not yet been tested in the framework of flows on erodible beds. We present here a Convex Optimization Contact Dynamics (COCD) method and prove that it generates a numerical solution verifying Coulomb's law at each contact and iteration. After its calibration and validation with experiments and another widely used Contact Dynamics method, we show that our simulations accurately reproduce qualitative and even many quantitative characteristics of experimental granular flows on erodible beds, including the increase of runout distance with the thickness of the erodible bed, the change of the static/flowing interface and the presence of erosion waves behind the flow front. Beyond erosion processes, our article endorses CMs as potential accurate tools for exploring complex granular mechanisms.

A hybrid direct search and projected simplex gradient method for convex constrained minimization

We propose a new Derivative-free Optimization (DFO) approach for solving convex constrained minimization problems. The feasible set is assumed to be the non-empty intersection of a finite collection of closed convex sets, such that the projection onto each of these individual convex sets is simple and inexpensive to compute. Our iterative approach alternates between steps that use Directional Direct Search (DDS), considering adequate poll directions, and a Spectral Projected Gradient (SPG) method, replacing the real gradient by a simplex gradient, under a DFO approach. In the SPG steps, if the convex feasible set is simple, then a direct projection is computed. If the feasible set is the intersection of finitely many convex simple sets, then Dykstra's alternating projection method is applied. Convergence properties are established under standard assumptions usually associated to DFO methods. Some preliminary numerical experiments are reported to illustrate the performance of the proposed algorithm, in particular by comparing it with a classical DDS method. Our results indicate that the hybrid algorithm is a robust and effective approach for derivative-free convex constrained optimization.