General Nonlinear Optimization Problems Research Articles

Error analysis of Haar wavelet-based Galerkin numerical method with application to various nonlinear optimal control problems

ABSTRACT First, this paper defines a general nonlinear optimal control problem with state/control constraints and its approximation problem as the Haar wavelet Galerkin optimal control problem (HWGOCP). Then, a Haar wavelet-based Galerkin numerical method has been developed, which converts it to a nonlinear optimization problem. We theoretically prove that a Haar wavelet feasible solution of HWGOCP will exist. We also show that the approximate solutions of HWGOCP are consistent and converge to the optimal solution of the problem. A variety of application problems have been considered, which include optimal control of tumour growth using Chemotherapy drugs, optimal control of infection via the SIS model using treatment, the Brachistochrone problem in mechanics, optimal control of mold using a fungicide, optimal control of pH value of a chemical reaction to determine the quality of a product, etc.

A cooperative neural dynamic model for solving general convex nonlinear optimization problems with fuzzy parameters and an application in manufacturing systems

SummaryIn the presented study, the solution of the fuzzy nonlinear optimization problems (FNLOPs) is calculated using a recurrent neural network (RNN) model. Since there is a few research for solving FNLOP by RNN's, we give a new approach to solve the problem. By reducing the original program to an interval problem and then weighting problem, the Karush–Kuhn–Tucker (KKT) conditions are given. Moreover, we use the KKT conditions into a RNN as an important tool to solve the problem. Besides, the global convergence properties and the Lyapunov stability of the dynamic model are studied in this study. In the final step, some illustrative examples are considered to establish the obtained results. Reported results are compared with some others network models.

Receding Horizon Control Using Graph Search for Multi-Agent Trajectory Planning

It is hard to find the global optimum of general nonlinear and nonconvex optimization problems in a reasonable time. This article presents a method to transfer the receding horizon control approach, where nonlinear, nonconvex optimization problems are considered, into graph-search problems. Specifically, systems with symmetries are considered to transfer system dynamics into a finite-state automaton. In contrast to traditional graph-search approaches where the search continues until the goal vertex is found, the transfer of a receding horizon control approach to graph-search problems presented in this article allows to solve them in real time. We prove that the solutions are recursively feasible by restricting the graph search to end in accepting states of the underlying finite-state automaton. The approach is applied to trajectory planning for multiple networked and autonomous vehicles. We evaluate its effectiveness in simulation and experiments in the Cyber-Physical Mobility Lab, an open-source platform for networked and autonomous vehicles. We show real-time capable trajectory planning with collision avoidance in experiments on off-the-shelf hardware and code in MATLAB for two vehicles.

Online Smooth Backfitting for Generalized Additive Models

We propose an online smoothing backfitting method for generalized additive models coupled with local linear estimation. The idea can be extended to general nonlinear optimization problems. The strategy is to use an appropriate-order expansion to approximate the nonlinear equations and store the coefficients as sufficient statistics which can be updated in an online manner by the dynamic candidate bandwidth method. We investigate the statistical and algorithmic convergences of the proposed method. By defining the relative statistical efficiency and computational cost, we further establish a framework to characterize the tradeoff between estimation performance and computation performance. Simulations and real data examples are provided to illustrate the proposed method and algorithm. Supplementary materials for this article are available online.

A Noise-Enduring and Finite-Time Zeroing Neural Network for Equality-Constrained Time-Varying Nonlinear Optimization

This article focuses on the research of a general time-varying nonlinear optimization (TVNO) problem solving especially in a noise-disturbance environment. For addressing this problem more efficiently, a new noise-enduring and finite-time convergent design formula is suggested to establish a novel zeroing neural network (NZNN). In contrast to the initial zeroing neural network or the noising-enduring zeroing neural network, which either only achieves finite-time convergence or only suppresses external disturbances, the merit of the proposed NZNN model is able to find an error-free optimal solution in a finite time under various different types of external noises. In addition, the detailed mathematical analyses about finite-time convergence and noise endurance are given to prove the excellent characteristics of the NZNN model. Numerical comparative results are provided to demonstrate the accuracy, efficiency, and advantages of the NZNN model for TVNO under various types of external disturbances. Robotic tracking example further validates the applicability of the NZNN model especially in a noise-disturbance environment.

Development of iterative algorithms for solving the inverse problem using inverse calculations

Iterative algorithms for solving the inverse problem, presented as a quadratic programming problem, developed by modifying algorithms based on the inverse calculation mechanism are proposed. Iterative algorithms consist in a sequential change of the argument values using iterative formulas until the function reaches the value that most corresponds to the constraint. Two solutions are considered: by determining the shortest distance to the line of the given level determined by the constraint, and by moving along the gradient. This approach was also adapted to solve more general nonlinear programming optimization problems. The solution of four problems is considered: formation of production output and storage costs, optimization of the securities portfolio and storage costs for the given volume of purchases. It is shown that the solutions obtained using iterative algorithms are consistent with the result of using classical methods (Lagrange multiplier, penalty), standard function of the MathCad package. In this case, the greatest degree of compliance was obtained using the method based on constructing the level line; the method based on moving along the gradient is more universal. The advantage of the algorithms is a simpler computer implementation of iterative formulas, the ability to get a solution in less time than known methods (for example, the penalty method, which requires multiple optimizations of a modified function with a change in the penalty parameter). The algorithms can also be used to solve other nonlinear programming problems of the presented kind. The paper can be useful for specialists when solving problems in the field of economics, as well as developing decision support systems.

ROI: An Extensible R Optimization Infrastructure

Optimization plays an important role in many methods routinely used in statistics, machine learning and data science. Often, implementations of these methods rely on highly specialized optimization algorithms, designed to be only applicable within a specific application. However, in many instances recent advances, in particular in the field of convex optimization, make it possible to conveniently and straightforwardly use modern solvers instead with the advantage of enabling broader usage scenarios and thus promoting reusability. This paper introduces the R optimization infrastructure ROI which provides an extensible infrastructure to model linear, quadratic, conic and general nonlinear optimization problems in a consistent way. Furthermore, the infrastructure administers many different solvers, reformulations, problem collections and functions to read and write optimization problems in various formats.

A bilevel optimal motion planning (BOMP) model with application to autonomous parking

In this paper, we present a bilevel optimal motion planning (BOMP) model for autonomous parking. The BOMP model treats motion planning as an optimal control problem, in which the upper level is designed for vehicle nonlinear dynamics, and the lower level is for geometry collision-free constraints. The significant feature of the BOMP model is that the lower level is a linear programming problem that serves as a constraint for the upper-level problem. That is, an optimal control problem contains an embedded optimization problem as constraints. Traditional optimal control methods cannot solve the BOMP problem directly. Therefore, the modified approximate Karush–Kuhn–Tucker theory is applied to generate a general nonlinear optimal control problem. Then the pseudospectral optimal control method solves the converted problem. Particularly, the lower level is the J_2-function that acts as a distance function between convex polyhedron objects. Polyhedrons can approximate objects in higher precision than spheres or ellipsoids. As a result, a fast high-precision BOMP algorithm for autonomous parking concerning dynamical feasibility and collision-free property is proposed. Simulation results and experiment on Turtlebot3 validate the BOMP model, and demonstrate that the computation speed increases almost two orders of magnitude compared with the area criterion based collision avoidance method.

A hybrid evolutionary-simplex search method to solve nonlinear constrained optimization problems

This research article presents a novel design of a hybrid evolutionary-simplex search method to solve the class of general nonlinear constrained optimization problems. In this article, the particle swarm optimization (PSO) method and the Nelder–Mead (NM) simplex search algorithm are utilized in a unified way to enhance the overall performance of the proposed solution method. The NM algorithm is used as an integrative step in the PSO method to reinforce the convergence of the PSO method and overcome the global search weakness in the NM algorithm. On the other hand, a penalty function technique is embedded in the proposed method to solve constrained optimization problems. Two levels of numerical experiments were conducted to evaluate the proposed method. First, a comparison is conducted with well-known benchmark problems. Second, the proposed method is tested in solving three engineering design optimization problems. In addition, the results of the proposed method were compared to optimization methods published in the literature in three main criteria: effectiveness, efficiency and robustness. The results show the competitive performance of the proposed method in this article.

Continuity of Pontryagin Extremals with Respect to Delays in Nonlinear Optimal Control

Consider a general nonlinear optimal control problem in finite dimension, with constant state and/or control delays. By the Pontryagin Maximum Principle, any optimal trajectory is the projection of a Pontryagin extremal. We establish that, under appropriate assumptions, Pontryagin extremals depend continuously on the parameter delays, for adequate topologies. The proof of the continuity of the trajectory and of the control is quite easy, however, for the adjoint vector, the proof requires a much finer analysis. The continuity property of the adjoint with respect to the parameter delay opens a new perspective for the numerical implementation of indirect methods, such as the shooting method. We also discuss the sharpness of our assumptions.

Accurate Computation of Vocal Tract Filter Parameters Using a Hybrid Genetic Algorithm

ABSTRACTThis paper reports an original technique for accurately estimating the parameters of human vocal tract filters for vowels in English for speech processing applications such as voice recognition. In this paper, the vocal tract filter design problem is reformulated as a general nonlinear optimization problem and solved using a hybrid Genetic Algorithm (GA). The hybrid GA computes a rough estimate of the global minimum using GA and refines using computationally cheap local search. Issues that are of concern in digital filtering such as achieving stability and overcoming finite precision effects are addressed. The objective function for optimization used in this paper is formulated in terms of poles and zeros of the filters to avoid ill-conditioning and to take advantage of symmetries in the location of poles and zeros. Simulation results indicate that the approach presented in this paper provides a close fit in terms of mean square error between the experimental and designed filters.

Optimal Operation of a Plug-In Hybrid Vehicle

We present a convex optimization control method that has been shown in simulations to increase the fuel efficiency of a plug-in hybrid electric vehicle by over 10%. Using information on energy demand and energy use profiles, the problem is defined to preferentially use battery resources sourced from the grid over petroleum resources. We pose the general nonlinear optimal resource management problem over a predetermined route as a convex optimization problem using a reduced model of the vehicle. This problem is computationally efficient enough to be optimized “on the fly” on the on-board vehicle computer and is thus able to adapt to changing vehicle conditions in real time. Using this reduced model to generate control inputs for the detailed vehicle simulator autonomie, we record efficiency gains of over 10% as compared to the industry standard charge depleting charge sustaining controller over synthetic mixed urban-suburban routes.

Duality in nonlinear programming problems under fuzzy environment with exponential membership functions

In this paper, we have established appropriate duality relations for a general nonlinear optimization problem under fuzzy environment, taking exponential membership functions and using the aspiration level approach. A numerical example has also been shown to justify the results presented in the paper.

Linear optimization of antenna array for radiation pattern defined on a planar surface

ABSTRACTIn this contribution, the antenna array synthesis for radiation pattern defined on a planar surface is described. This can be required when the electric field r-decay factor effect cannot be neglected, e.g. an antenna array mechanically tilted and a specific pattern defined in terms of cartesian coordinates. Differently from the classical sidelobe level reduction optimization problem, the case in which both a constant power bounded area and a sidelobe suppression region are defined and required to be synthesized is herein addressed. The general nonlinear optimization problem is first presented; then, the derivation of some sub-optimal linear problems based on practical implementation hypothesis is described, with the advantage of a reduced computational complexity. Numerical results validate the effectiveness of the proposed method.

Constrained Multiobjective Nonlinear Optimization: A User Preference Enabling Method.

A novel user preference enabling (UPE) method is developed to solve general constrained nonlinear multiple objective optimization (MOO) problems. User wish lists on the preferred range of each objective function are introduced and incorporated into the MOO formulation to form a user-preferred (UP) MOO problem. A theoretical foundation of the UP feasible region of MOO problems is developed. The developed theoretical work leads to the development of a UPE method for effectively solving the UP-MOO problems. Distinguishing features of the proposed method include its ability to compute a targeted Pareto solution and to serve as a complement to existing MOO methods, in the sense that the proposed method assists the existing MOO methods in computing the Pareto front by providing feasible solutions and UP feasible solutions. Both proposed UPE method and derived theoretical developments are evaluated on several test systems with promising results.

On the steplength selection in gradient methods for unconstrained optimization

The seminal paper by Barzilai and Borwein (1988) has given rise to an extensive investigation, leading to the development of effective gradient methods. Several steplength rules have been first designed for unconstrained quadratic problems and then extended to general nonlinear optimization problems. These rules share the common idea of attempting to capture, in an inexpensive way, some second-order information. However, the convergence theory of the gradient methods using the previous rules does not explain their effectiveness, and a full understanding of their practical behaviour is still missing. In this work we investigate the relationships between the steplengths of a variety of gradient methods and the spectrum of the Hessian of the objective function, providing insight into the computational effectiveness of the methods, for both quadratic and general unconstrained optimization problems. Our study also identifies basic principles for designing effective gradient methods.

Guided particle swarm optimization method to solve general nonlinear optimization problems

ABSTRACTThe development of hybrid algorithms is becoming an important topic in the global optimization research area. This article proposes a new technique in hybridizing the particle swarm optimization (PSO) algorithm and the Nelder–Mead (NM) simplex search algorithm to solve general nonlinear unconstrained optimization problems. Unlike traditional hybrid methods, the proposed method hybridizes the NM algorithm inside the PSO to improve the velocities and positions of the particles iteratively. The new hybridization considers the PSO algorithm and NM algorithm as one heuristic, not in a sequential or hierarchical manner. The NM algorithm is applied to improve the initial random solution of the PSO algorithm and iteratively in every step to improve the overall performance of the method. The performance of the proposed method was tested over 20 optimization test functions with varying dimensions. Comprehensive comparisons with other methods in the literature indicate that the proposed solution method is promising and competitive.

A stabilized filter SQP algorithm for nonlinear programming

This paper presents a stabilized filter sequential quadratic programming (SQP) method for the general nonlinear optimization problems. The technique of stabilizing the inner quadratic programmings is an efficient strategy for the degenerate problem and brings the local superlinear convergence, while the integrated filter technique works effectively and guarantees the global convergence. The new algorithm works on both the primal and dual variables and solves the problem within the computational complexity comparable to the classical SQP algorithm. For the convergence, we show that (1) it converges either to a Karush---Kuhn---Tucker point at which the cone-continuity property holds, or to a stationary point in the sense of minimizing the constraint violation, and (2) under some second-order sufficient conditions, it converges locally superlinearly without any constraint qualifications. Our preliminary numerical results on a set of CUTEr test problems as well as on degenerate problems demonstrate the efficiency of the new algorithm.

Optimal shakedown analysis of plane reinforced concrete frames according to Eurocodes

We present an updated mathematical model of shakedown optimization for reinforced concrete plane frames subjected to variable and repeated uncertain loading within a known domain. In such structures, plastic redistribution of forces is known to occur, and various mechanisms of system collapse at shakedown have been identified, such as plastic yielding and sign-changing. We develop a general nonlinear mixed-integer optimization problem that reduces to a linear programming problem, and we demonstrate the duality of the linear programming problem for the static and kinematic formulations. We derive strength conditions according to Eurocode 2 and an iterative process of optimization, where stiffness properties of frame elements are allowed to vary. The frame cross-sections are rectangular and made from doubly reinforced concrete; the material is considered composite. We successfully demonstrate the numerical optimization procedure on a two-storey reinforced concrete plane frame. We present variations of interaction loci of each optimized section in graphical form.

Bat Algorithm: A Survey of the State-of-the-Art

The area of metaheuristic optimization algorithms has been attracting researchers for many years. These algorithms have built in capability to explore a large region of the solution space, are computationally robust, efficient and can avoid premature convergence. They have been extensively tested and applied on many hard optimization problems where conventional computing techniques perform unsatisfactorily. They are capable of solving general N-dimensional, linear, nonlinear and complex global optimization problems. One of the latest entrants in this field is the Bat algorithm which is based on the echolocation behaviour of bats. It has been proven to have good convergence properties on different benchmark functions and seems promising for dealing with optimization problems. The aim of this paper is to provide a survey of the state of the art on Bat algorithm. A concise effort has been made so that the readers get a rapid insight into some of the applications upon which bat algorithm has been applied till date in specialized fields of science and engineering. Some of the variants of the bat algorithm as reported in the literature have also been discussed.