Optimization Problem In Continuous Variables Research Articles

Probability and Certainty in the Performance of Evolutionary and Swarm Optimization Algorithms

Reporting the empirical results of swarm and evolutionary computation algorithms is a challenging task with many possible difficulties. These difficulties stem from the stochastic nature of such algorithms, as well as their inability to guarantee an optimal solution in polynomial time. This research deals with measuring the performance of stochastic optimization algorithms, as well as the confidence intervals of the empirically obtained statistics. Traditionally, the arithmetic mean is used for measuring average performance, but we propose quantiles for measuring average, peak and bad-case performance, and give their interpretations in a relevant context for measuring the performance of the metaheuristics. In order to investigate the differences between arithmetic mean and quantiles, and to confirm possible benefits, we conducted experiments with 7 stochastic algorithms and 20 unconstrained continuous variable optimization problems. The experiments showed that median was a better measure of average performance than arithmetic mean, based on the observed solution quality. Out of 20 problem instances, a discrepancy between the arithmetic mean and median happened in 6 instances, out of which 5 were resolved in favor of median and 1 instance remained unresolved as a near tie. The arithmetic mean was completely inadequate for measuring average performance based on the observed number of function evaluations, while the 0.5 quantile (median) was suitable for that task. The quantiles also showed to be adequate for assessing peak performance and bad-case performance. In this paper, we also proposed a bootstrap method to calculate the confidence intervals of the probability of the empirically obtained quantiles. Considering the many advantages of using quantiles, including the ability to calculate probabilities of success in the case of multiple executions of the algorithm and the practically useful method of calculating confidence intervals, we recommend quantiles as the standard measure of peak, average and bad-case performance of stochastic optimization algorithms.

Joint Resource Allocation on Slot, Space and Power Towards Concurrent Transmissions in UAV Ad Hoc Networks

With innovative applications of unmanned aerial vehicle (UAV) ad hoc networks in various areas, their demands on broad bandwidth, large capacity and low latency become prominent. The combination of millimeter wave, directional antenna and time division multiple access techniques, which enables concurrent transmissions, is promising to deal with it. In this paper, we study the resource allocation problem in UAV ad hoc networks. Specifically, the slot assignment, antenna boresight and transmit power are jointly optimized to promote the network capacity. First, we formulate the optimization problem as the maximization of the fairness-weighted network capacity, subject to the constraint on priority guarantee. Then, because the formulated problem is a mixed integer non-linear programming problem (MINLP), which is NP-hard, two algorithms called dual-based iterative search algorithm (DISA) and sequential exhausted allocation algorithm (SEAA) are respectively proposed to efficiently solve it with acceptable complexity. DISA slacks the MINLP into a continuous-variable optimization problem and solves it with the Lagrangian dual method in an iterative manner. As a heuristic method, SEAA schedules links sequentially, i.e., from high-priority to low-priority ones. Numerical results demonstrate that both DISA and SEAA can efficiently allocate resources for UAVs, while guaranteeing the fairness and priority of links.

A Modified Jaya Algorithm for Mixed-Variable Optimization Problems

Abstract Mixed-variable optimization problems consist of the continuous, integer, and discrete variables generally used in various engineering optimization problems. These variables increase the computational cost and complexity of optimization problems due to the handling of variables. Moreover, there are few optimization algorithms that give a globally optimal solution for non-differential and non-convex objective functions. Initially, the Jaya algorithm has been developed for continuous variable optimization problems. In this paper, the Jaya algorithm is further extended for solving mixed-variable optimization problems. In the proposed algorithm, continuous variables remain in the continuous domain while continuous domains of discrete and integer variables are converted into discrete and integer domains applying bound constraint of the middle point of corresponding two consecutive values of discrete and integer variables. The effectiveness of the proposed algorithm is evaluated through examples of mixed-variable optimization problems taken from previous research works, and optimum solutions are validated with other mixed-variable optimization algorithms. The proposed algorithm is also applied to two-plane balancing of the unbalanced rigid threshing rotor, using the number of balance masses on plane 1 and plane 2. It is found that the proposed algorithm is computationally more efficient and easier to use than other mixed optimization techniques.

Finance-based scheduling using meta-heuristics: discrete versus continuous optimization problems

Purpose – The purpose of this paper is to compare the performance of the genetic algorithm (GA), simulate annealing (SA) and shuffled frog-leaping algorithm (SFLA) in solving discrete versus continuous-variable optimization problems of the finance-based scheduling. This involves the minimization of the project duration and consequently the time-related cost components of construction contractors including overheads, finance costs and delay penalties. Design/methodology/approach – The meta-heuristics of the GA, SA and SFLA have been implemented to solve non-deterministic polynomial-time hard (NP-hard) finance-based scheduling problem employing the objective of minimizing the project duration. The traditional problem of generating unfeasible solutions in scheduling problems is adequately tackled in the implementations of the meta-heuristics in this paper. Findings – The obtained results indicated that the SA outperformed the SFLA and GA in terms of the quality of solutions as well as the computational cost based on the small-size networks of 30 activities, whereas it exhibited the least total duration based on the large-size networks of 120 and 210 activities after prolonged processing time. Research limitations/implications – From researchers’ perspective, finance-based scheduling is one of the few domain problems which can be formulated as discrete and continuous-variable optimization problems and, thus, can be used by researchers as a test bed to give more insight into the performance of new developments of meta-heuristics in solving discrete and continuous-variable optimization problems. Practical implications – Finance-based scheduling discrete-variable optimization problem is of high relevance to the practitioners, as it allows schedulers to devise finance-feasible schedules of minimum duration. The minimization of project duration is focal for the minimization of time-related cost components of construction contractors including overheads, finance costs and delay penalties. Moreover, planning for the expedient project completion is a major time-management aspect of construction contractors towards the achievement of the objective of client satisfaction through the expedient delivery of the completed project for clients to start reaping the anticipated benefits. Social implications – Planning for the expedient project completion is a major time-management aspect of construction contractors towards the achievement of the objective of client satisfaction. Originality/value – SFLA represents a relatively recent meta-heuristic that proved to be promising, based on its limited number of applications in the literature. This paper is to implement SFLA to solve the discrete-variable optimization problem of the finance-based scheduling and assess its performance by comparing its results against those of the GA and SA.

Predatory Search Strategy Based on Swarm Intelligence for Continuous Optimization Problems

We propose an approach to solve continuous variable optimization problems. The approach is based on the integration of predatory search strategy (PSS) and swarm intelligence technique. The integration is further based on two newly defined concepts proposed for the PSS, namely, “restriction” and “neighborhood,” and takes the particle swarm optimization (PSO) algorithm as the local optimizer. The PSS is for the switch of exploitation and exploration (in particular by the adjustment of neighborhood), while the swarm intelligence technique is for searching the neighborhood. The proposed approach is thus named PSS-PSO. Five benchmarks are taken as test functions (including both unimodal and multimodal ones) to examine the effectiveness of the PSS-PSO with the seven well-known algorithms. The result of the test shows that the proposed approach PSS-PSO is superior to all the seven algorithms.

Improved Differential Evolution Algorithm for Structural Optimization Design of Flumes with Hybrid Discrete Variables

The traditional method applying to solve continuous variable optimization problems is not suit for flume structural optimization design with hybrid discrete variable. According to the mathematical model of structural optimum design of the prestressed U-shell flumes, differential evolution (DE) algorithm was introduced to flume structural optimization design. In order to improve the population’s diversity and the ability of escaping from the local optimum, a self-adapting crossover probability factor was presented. Furthermore, a chaotic sequence based on logistic map was employed to self-adaptively adjust mutation factor based on linear crossover, which can improve the convergence of DE algorithm. Dynamic penalty function, to transform the constrained problem to unconstrained one, was employed. The result shows that, compared with the original design scheme, the optimization design scheme can greatly reduce the amount of prestressed reinforcement. The construction cost of both the flume and the whole project can be reduced accordingly.

An Algorithm for Gradient-Free Simulation Optimization using Sampling Control

This article considers a class of continuous-variable optimization problems in which the objective function cannot be computed exactly but must instead be estimated by using, typically, a simulation. As the simulation output is stochastic, iterative optimization algorithms these problems are often augmented with noise-removal features to ensure convergence to an optimal solution. Two broad noise-removal approaches are considered: stepsize-control, which involves a decreasing stepsize, or sampling-control, which involves an increasing sample size. Of these two, stepsize-control predates sampling-control by several decades and is the most commonly used approach, although, as we show, sampling-control has some advantages over stepsize-control. Our particular focus is on optimization problems for which direct gradient estimates are not available, and that instead must be approximated using estimates of the objective function. The classic Kiefer-Wolfowitz algorithm, using stepsize-control, is one such algorithm that estimates a divided difference approximation of the gradient. This article presents a sampling-controlled version of this algorithm that also uses divided difference estimates and has the benefit of being easily parallelizable. A convergence proof and some simulation results are also included. Note that one of the advantages of the sampling-controlled method is that it is better suited to estimators that have small-sample bias but are asymptotically unbiased: because the sampling rate increases, estimator bias is gradually decreased as the number of samples increases.

The long-term hydro-scheduling problem—a new algorithm

Abstract A new efficient algorithm to solve the long-term hydro-scheduling problem (LTHSP) is presented in this paper. The algorithm is based on using the short-term memory of the Tabu search (TS) approach to solve the nonlinear optimization problem in continuous variables of the LTHSP. The paper introduces new rules for generating feasible solutions with an adaptive step vector adjustment. Moreover, a Tabu list for the continuous variables has been designed. The proposed implementation contributes to the enhancement of speed and convergence of the original Tabu search algorithm (TSA). A significant reduction in the objective function over previous classical optimization methods and a simulated annealing algorithm has been achieved. Moreover, the proposed TS requires less iterations to converge than simulated annealing. The proposed algorithm has been applied successfully to solve a system with four series cascaded reservoirs. Numerical results show an improvement in the solution compared with previously obtained results.

An innovative simulated annealing approach to the long-term hydroscheduling problem

This paper presents a new simulated annealing algorithm (SAA) to solve the long-term hydroscheduling problem. A new algorithm for randomly generating feasible trial solutions is introduced. The problem is a hard nonlinear optimization problem in continuous variables. An adaptive cooling schedule and a new method for variables discretization are implemented to enhance the speed and convergence of the original SAA. A significant reduction in the number of the objective function evaluations, and consequently less iteration are required to reach the optimal solution. The proposed algorithm has been applied successfully to solve a system with four series cascaded reservoirs. Numerical results show an improvement in the solutions compared to previously obtained results.

OPTIMAL DESIGN WITH DISCRETE VARIABLES: SOME NUMERICAL EXPERIMENTS

Continuous–discrete variable non-linear optimization problems are defined and categorized into six different types. These include a full range of problems from continuous to purely discrete and non-differentiable. Methods for solution of these problems are studied and their characteristics are catalogued. The branch and bound, simulated annealing and genetic algorithms are found to be the most general methods for solving discrete problems. After some enhancements, these and two other methods are implemented into a program for certain applications. Several example problems are solved to study performance of the methods. It is concluded that solution of the mixed variable non-linear optimization problems usually requires considerable more computational effort compared to the continuous variable optimization problems. In addition, there is no guarantee that the best solution has been obtained; however, good practical solutions are usually obtained. © 1997 by John Wiley & Sons, Ltd.

NONLINEAR DISCRETE STRUCTURAL OPTIMIZATION

Structural design optimization will be more convenient to formulate the design problem with discrete variable than it would be if the variables were assumed to be continuous. In order to solve a structural design problem with discrete variable only, two completely different techniques, 1) Integer gradient direction, which is later supported by subsequential search interval technique and 2) Modified Rosenbrocks orthognalization techniques have hybridized. Rosenbrocks original procedure is a well established method to solve continuous variable optimization problem; but to suit to discrete variable problem solution some modifications are needed and reported here. By this hybridizing most of the practical difficulties, usually, encountered in the discrete optimization can be overcome. Details of the techniques are discussed and by their combination a solution code has been generated. A constrained problem is first converted into a sequence of unconstrained problem by use of interior penalty function and then solved by the generated code. The efficiency of the generated code is revealed by solving several test problems.