Continuous Constrained Optimization Problems Research Articles

A novel state transition algorithm with adaptive fuzzy penalty for multi-constraint UAV path planning

Unmanned aerial vehicles (UAVs) require pre-planned flight paths that are energy-efficient, safe and smooth across their wide range of application scenarios. In this study, a novel UAV path planning method is proposed. Firstly, the UAV path planning under numerous obstacles is modeled as a continuous constrained optimization problem. The cost function is formulated as a linear combination of length, height variation, and smoothness, while the constraints include obstacle avoidance, height limitation, and the maneuverability of UAV. Subsequently, a novel constrained state transition algorithm with adaptive fuzzy penalty (AFSTA) is proposed to solve the optimization problem. In AFSTA, a novel adaptive fuzzy penalty function is designed to leverage expert knowledge to establish a reasonable mapping relationship from the fitness value and the degree of constraint violation to the penalty factor for a candidate solution. Meanwhile, the state transition algorithm (STA) is used as the search engine for both global and local search. Experimental results illustrate that the proposed method can find energy-efficient, safe, and maneuverable flight paths successfully with the superiority over other state-of-the-art metaheuristic algorithms.

Inexact restoration for minimization with inexact evaluation both of the objective function and the constraints

In a recent paper an Inexact Restoration method for solving continuous constrained optimization problems was analyzed from the point of view of worst-case functional complexity and convergence. On the other hand, the Inexact Restoration methodology was employed, in a different research, to handle minimization problems with inexact evaluation and simple constraints. These two methodologies are combined in the present report, for constrained minimization problems in which both the objective function and the constraints, as well as their derivatives, are subject to evaluation errors. Together with a complete description of the method, complexity and convergence results will be proved.

An optimization numerical spiking neural P system for solving constrained optimization problems

An optimization spiking neural P (OSN P) system is a discrete optimization model without the aid of evolutionary operators of evolutionary algorithms or swarm intelligence algorithms. However, since the processing object of OSN P systems is a spike, where information is encoded by the timing of spikes or the number of spikes in neurons, OSN P systems are limited for solving continuous optimization problems. To break this limitation, an extended numerical spiking neural (ENSN P) system is proposed based on numerical spiking neural P (NSN P) systems and multiple (ENSN P) systems, called optimization numerical spiking neural P systems (ONSN P systems or ONSNPS), are designed to solve continuous constrained optimization problems. More specifically, in ENSN P systems, the production functions are selected by probability to achieve updated parameters. In OSN P systems, a guider algorithm is introduced to finish individuals’ crossover and selection. The extensively experimental results in five benchmarks, thirty-two optimization problems including five benchmark problems, seventeen manufacturing design optimization problems and ten benchmarks from CEC show that ONSN P systems in this paper outperform or are competitive to twenty-eight optimization algorithms. Finally, algorithm complexity and Holm-Bonferroni procedure based on statistical results is used to test the complexity changing when we use different dimensionality of the search space and the difference in terms of statistical performance. The testing results indicate that the time complexity of ONSN P systems grows linearly with problem dimensions and ONSN P systems are better performance than the most algorithms.

Band Selection for HSI Classification Using Binary Constrained Optimization

Hyperspectral images (HSIs) containing tens to hundreds of bands can be used in various image classification tasks. However, due to the high data redundancy of the spectral information, the acquiring and analysis of HSIs are usually relatively time-consuming and wasteful of storage space, and therefore limit the practical application of HSIs. Selecting a subset of bands without sacrificing classification accuracy is a strategy to relieve such problems. In this letter, we present an optimization-based method, which can jointly optimize the band selection (BS) and the classification network parameters for HSIs. The proposed method regards the discrete selection problem as a continuous constrained optimization problem and adaptively selects the informative band subsets for classification. Besides, the experimental results on three public datasets show that our BS method outperforms the state-of-the-art methods in terms of classification accuracy.

Constraint-Objective Cooperative Coevolution for Large-scale Constrained Optimization

Large-scale optimization problems and constrained optimization problems have attracted considerable attention in the swarm and evolutionary intelligence communities and exemplify two common features of real problems, i.e., a large scale and constraint limitations. However, only a little work on solving large-scale continuous constrained optimization problems exists. Moreover, the types of benchmarks proposed for large-scale continuous constrained optimization algorithms are not comprehensive at present. In this article, first, a constraint-objective cooperative coevolution (COCC) framework is proposed for large-scale continuous constrained optimization problems, which is based on the dual nature of the objective and constraint functions: modular and imbalanced components. The COCC framework allocates the computing resources to different components according to the impact of objective values and constraint violations. Second, a benchmark for large-scale continuous constrained optimization is presented, which takes into account the modular nature, as well as both imbalanced and overlapping characteristics of components. Finally, three different evolutionary algorithms are embedded into the COCC framework for experiments, and the experimental results show that COCC performs competitively.

A Two-stage Simulation Assisted Differential Evolution Algorithm for Reliable Chance Constrained Programming with Minimum Risk Level

Recently the literature of Simulation Optimization for solving stochastic constrained optimization problems has been substantially increasing. From the optimization perspective, the population based meta-heuristics algorithms, such as Differential Evolutionary (DE), has shown tremendous success in solving continuous constrained optimization problems. While from the simulation and stochastic perspectives, Monte-Carlo Simulation alongside Chance Constrained Programming (CCP), is one of the most successful combined approaches to handle constraint uncertainty, also called chance constraints. Risk levels are used in CCP to bound the stochastic constraints infeasibility. However, the determination of the risk levels is not precise and is based on subjective personal judgments; due to the lack of a scientific approach to determine their values especially in critical problems that require the minimum possible risk levels. Furthermore, reliability in such problems is a vital requirement, especially for studies under worst-case scenarios. Therefore, we propose a novel approach, called Two-stage simulation assisted Differential Evolutionary for Chance constrained programming algorithm (Two-stage DE-CCP), with two major contributions. The first contribution aims to obtain minimum boundaries for the chance constrains’ risk levels, by utilizing the meta-heuristics DE implicit capabilities with a two stages modification. While the second contribution proposed as a Reliable Chance Constrained Programming (RCCP) with a novel scheme for chance constraints’ feasibility checking, that is called Extreme Scenario Feasibility Checking (ESFC). This name reveals that ESFC aims to obtain more reliable solutions, through comparing the extreme/worst-case scenarios of the chance constraints. The proposed algorithm is applied under normal CCP feasibility checking and ESFC for RCCP, while a comparison between them is performed and an empirical Reliability ratio (Rr) is proposed. The experimental results on benchmark problems and real application show that the proposed algorithm is able to provide decision makers with the minimum possible risk levels. While a substantially high Rr is obtained from the proposed RCCP, as compared to traditional CCP.

Inverse design in photonics by topology optimization: tutorial

Topology optimization (TopOpt) methods for inverse design of nano-photonic systems have recently become extremely popular and are presented in various forms and under various names. Approaches comprise gradient- and non-gradient-based algorithms combined with more or less systematic ways to improve convergence, discreteness of solutions, and satisfaction of manufacturing constraints. We here provide a tutorial for the systematic and efficient design of nano-photonic structures by TopOpt. The implementation is based on the advanced and systematic approaches developed in TopOpt for structural optimization during the last three decades. The tutorial presents a step-by-step guide for deriving the continuous constrained optimization problem forming the foundation of the TopOpt method, using a cylindrical metalens design problem as an example. It demonstrates the effect and necessity of applying a number of auxiliary tools in the design process to ensure good numerical modeling practice and to achieve physically realizable designs. Application examples also include an optical demultiplexer.

Bat algorithm applied to continuous constrained optimization problems

Many optimization techniques have been proposed for solving economic load and or emission dispatch problems in power generation plants. Among them, bio-inspired techniques have become popular due to their performance and simplicity. Bat Algorithm (BA) is one of the newer additions in the field although it has been applied in a series of evaluation tests. The method is trying to replicate the echolocation behavior of bats with artificial ones. In this paper, we apply BA to the well-known problem of economic load dispatch, in different power demand scenarios. Its performance in terms of solution fitness is compared with other optimizations techniques proposed in the literature for the same scenarios.

Event based guaranteed cost consensus for distributed multi-agent systems

To investigate the energy consumption involved in an event based control scheme, the problem of event based guaranteed cost consensus for distributed multi-agent systems with general linear time invariant dynamics is considered in this paper. A delay system method is used to transform the multi-agent systems into a special delay system based on a sampled-data event triggering mechanism, which only requires supervision of system states at discrete instants. Sufficient conditions to achieve the consensus with guaranteed cost are presented and expressed as a continuous constrained optimization problem with a linear objective function, linear and bilinear matrix inequalities constraints, involving the co-design of the controller gain matrix and event triggering parameters. An illustrative example is given to show the effectiveness of the proposed approach.

Multi-objective trajectory planning over terrain using label-setting greedy-based algorithm

In this paper, the multi-objective trajectory planning problem for an unmanned aerial vehicle on a low altitude terrain following/threat avoidance mission is studied. Using a grid-based discrete scheme, the continuous constrained optimization problem into a combination of search and decision problem is transformed over a finite network. A variant of the minimum cost network flow approach, which is referred to as label-setting greedy-based algorithm, is then applied to this problem. Using the digital terrain elevation data and discrete dynamic equations of motion, an optimal four-dimensional trajectory (three spatial and one time dimensions) from a source to a destination is obtained deterministically by minimizing a nonlinear cost functional subject to dynamic and mission constraints of the unmanned aerial vehicle. For each arc in the grid, a cost functional is considered as a combination of the path length, fuel consumption, flight time, and risk-of-threat region. Moreover, due to the increasing deviation of inertial navigation system in terms of time, having a safe flight and collision avoidance with terrain at low altitude is a significant problem in the trajectory design of this type of the vehicles. In this work, it’s tried to meet this constraint in the trajectory planning. The simulation studies on the unmanned aerial vehicle flight planning problem are presented to verify the capability of the proposed algorithm to generate admissible trajectory in minimum possible time comparison to previous works.

Hybrid ant colony optimization algorithms for mixed discrete–continuous optimization problems

This paper presents three new hybrid ant colony optimization algorithms that are extended from the ACOR developed by Socha and Dorigo for solving mixed discrete–continuous constrained optimization problems. The first two hybrids, labeled ACOR-HJ and ACOR-DE, differs in philosophy with the former integrating ACOR with the effective Hooke and Jeeves local search method and the latter a cooperative hybrid between ACOR and differentia evolution. The third hybrid, labeled ACOR-DE-HJ, is the second cooperative hybrid enhanced with the Hooke and Jeeves local search. All three algorithms incorporate a method to handle mixed discrete–continuous variables and the Deb’s parameterless penalty method for handling constraints. Fourteen problems selected from various domains were used for testing the performance of both algorithms. It was showed that all three algorithms greatly outperform the original ACOR in finding the exact or near global optima. An investigation was also carried out to determine the relative performance of applying local search with a fixed probability or varying probability.

Constrained simulated annealing for stability margin computation in a time‐delay system

AbstractThe stability margin of a time‐delay system is formulated via factorization. This paper provides a numerical method for computing the stability margin of time‐delay linear time‐invariant systems with delay dependence by using a constrained simulated annealing algorithm. The constrained simulated annealing is used to solve a nonlinear continuous constrained optimization problem, which is derived from computing the stability margin of a delay system. Illustrative examples show that the established method can verify the stability for a class of time‐delay systems. Copyright © 2006 John Wiley & Sons, Ltd.

Efficient and Adaptive Lagrange-Multiplier Methods for Nonlinear Continuous Global Optimization

Lagrangian methods are popular in solving continuous constrained optimization problems. In this paper, we address three important issues in applying Lagrangian methods to solve optimization problems with inequality constraints. First, we study methods to transform inequality constraints into equality constraints. An existing method, called the slack-variable method, adds a slack variable to each inequality constraint in order to transform it into an equality constraint. Its disadvantage is that when the search trajectory is inside a feasible region, some satisfied constraints may still pose some effect on the Lagrangian function, leading to possible oscillations and divergence when a local minimum lies on the boundary of the feasible region. To overcome this problem, we propose the MaxQ method that carries no effect on satisfied constraints. Hence, minimizing the Lagrangian function in a feasible region always leads to a local minimum of the objective function. We also study some strategies to speed up its convergence. Second, we study methods to improve the convergence speed of Lagrangian methods without affecting the solution quality. This is done by an adaptive-control strategy that dynamically adjusts the relative weights between the objective and the Lagrangian part, leading to better balance between the two and faster convergence. Third, we study a trace-based method to pull the search trajectory from one saddle point to another in a continuous fashion without restarts. This overcomes one of the problems in existing Lagrangian methods that converges only to one saddle point and requires random restarts to look for new saddle points, often missing good saddle points in the vicinity of saddle points already found. Finally, we describe a prototype Novel (Nonlinear Optimization via External Lead) that implements our proposed strategies and present improved solutions in solving a collection of benchmarks.