Large Optimization Problems Research Articles

Constrained improved particle swarm optimization algorithm for optimal operation of large scale reservoir: proposing three approaches

In this paper, the improved particle swarm optimization (IPSO) algorithm is used to solve large scale reservoir operation optimization problem proposing unconstrained and two constrained versions of this algorithm. In the two constrained versions proposed for the IPSO algorithm, named PCIPSO and FCIPSO, each particle may be forced to satisfy problem constraints during solution building. By considering water releases or storage volumes at each operation time period as decision variable of the problem, here, two formulations are proposed for each version. In the second proposed constrained version algorithm (FCIPSO), at first, the water storage volume bounds are modified in order to recognize the infeasible components of the search space and exclude from the search process before the main search starts. This mechanism leads to smaller search space size for the problem and finally better results. The simple and hydropower operation problems of “Dez” reservoir in the southern Iran over 60, 240 and 480 monthly operations time periods are solved here using both proposed formulations of theses algorithms and the results are presented and compared with other available results. The results show the capability of the proposed algorithms and especially the second constrained version of the IPSO algorithm, FCIPSO, to optimally solve the reservoir operation optimization problem. In other words, the results of both formulations of constrained IPSO and especially FCIPSO algorithm are improved significantly in comparison with unconstrained IPSO algorithm over all operations time periods of simple and hydropower operation of the reservoir.

A modified competitive swarm optimizer for large scale optimization problems

In the recent literature a popular algorithm namely ‘Competitive Swarm Optimizer (CSO)’ has been proposed for solving unconstrained optimization problems that updates only half of the population in each iteration. A modified CSO (MCSO) is being proposed in this paper where two thirds of the population swarms are being updated by a tri-competitive criterion unlike CSO. A small change in CSO makes a huge difference in the solution quality. The basic idea behind the proposition is to maintain a higher rate of exploration to the search space with a faster rate of convergence. The proposed MCSO is applied to solve the standard CEC2008 and CEC2013 large scale unconstrained benchmark optimization problems. The empirical results and statistical analysis confirm the better overall performance of MCSO over many other state-of-the-art meta-heuristics, including CSO. In order to confirm the superiority further, a real life problem namely ‘sampling-based image matting problem’ is solved. Considering the winners of CEC 2008 and 2013, MCSO attains the second best position in the competition.

Regularized decomposition of large scale block-structured robust optimization problems

We consider a general robust block-structured optimization problem, coming from applications in network and energy optimization. We propose and study an iterative cutting-plane algorithm, generic for a large class of uncertainty sets, able to tackle large-scale instances by leveraging on their specific structure. This algorithm combines known techniques (cutting-planes, proximal stabilizations, efficient heuristics, warm-started bundle methods) in an original way for better practical efficiency. We provide a theoretical analysis of the algorithm and connections to existing literature. We present numerical illustrations on real-life problems of electricity generation under uncertainty. These clearly show the advantage of the proposed regularized algorithm over classic cutting plane approaches. We therefore advocate that regularized cutting plane methods deserve more attention in robust optimization.

Constrained gravitational search algorithm for large scale reservoir operation optimization problem

The gravitational search algorithm (GSA) is used in this paper to solve large scale reservoir operation optimization problem. Here, two constrained versions of GSA are proposed to solve this problem in which masses may be forced to satisfy problem constraints during solution building. This approach is very useful when attempting to solve large scale optimization problem as it will lead to a considerable reduction of the search space size. Here, in the second version of GSA, the storage volume bounds of the reservoir are modified prior to the main search to recognize the infeasible components of the search space and exclude from the search process before the main search starts. Two formulations are also proposed here for each proposed algorithm considering water releases or storage volumes at each operation time period as decision variable of the problem. Proposed algorithms are used to solve the simple and hydropower operation problem of “Dez” reservoir in Iran and the results are presented and compared with using original form of the GSA and any available results. The results indicate the ability of the proposed algorithm and especially the second constrained version of GSA to optimally solve the reservoir operation optimization problem.

Dual mean field search for large scale linear and quadratic knapsack problems

An implementation of mean field annealing to deal with large scale linear and non linear binary optimization problems is given. Mean field annealing is based on the analogy between combinatorial optimization and interacting physical systems at thermal equilibrium. Specifically, a mean field approximation of the Boltzmann distribution given by a Lagrangian that encompass the objective function and the constraints is calculated. The original discrete task is in this way transformed into a continuous variational problem. In our version of mean field annealing, no temperature parameter is used, but a good starting point in the dual space is given by a “thermodynamic limit” argument. The method is tested in linear and quadratic knapsack problems with sizes that are considerably larger than those used in previous studies of mean field annealing. Dual mean field annealing is capable to find high quality solutions in running times that are orders of magnitude shorter than state of the art algorithms. Moreover, as may be expected for a mean field theory, the solutions tend to be more accurate as the number of variables grow.

A Heuristic Initialized Stochastic Memetic Algorithm for MDPVRP With Interdependent Depot Operations.

The vehicle routing problem (VRP) is a widely studied combinatorial optimization problem. We introduce a variant of the multidepot and periodic VRP (MDPVRP) and propose a heuristic initialized stochastic memetic algorithm to solve it. The main challenge in designing such an algorithm for a large combinatorial optimization problem is to avoid premature convergence by maintaining a balance between exploration and exploitation of the search space. We employ intelligent initialization and stochastic learning to address this challenge. The intelligent initialization technique constructs a population by a mix of random and heuristic generated solutions. The stochastic learning enhances the solutions' quality selectively using simulated annealing with a set of random and heuristic operators. The hybridization of randomness and greediness in the initialization and learning process helps to maintain the balance between exploration and exploitation. Our proposed algorithm has been tested extensively on the existing benchmark problems and outperformed the baseline algorithms by a large margin. We further compared our results with that of the state-of-the-art algorithms working under MDPVRP formulation and found a significant improvement over their results.

Differential Evolution with Novel Mutation and Adaptive Crossover Strategies for Solving Large Scale Global Optimization Problems

This paper presents Differential Evolution algorithm for solving high-dimensional optimization problems over continuous space. The proposed algorithm, namely, ANDE, introduces a new triangular mutation rule based on the convex combination vector of the triplet defined by the three randomly chosen vectors and the difference vectors between the best, better, and the worst individuals among the three randomly selected vectors. The mutation rule is combined with the basic mutation strategy DE/rand/1/bin, where the new triangular mutation rule is applied with the probability of 2/3 since it has both exploration ability and exploitation tendency. Furthermore, we propose a novel self-adaptive scheme for gradual change of the values of the crossover rate that can excellently benefit from the past experience of the individuals in the search space during evolution process which in turn can considerably balance the common trade-off between the population diversity and convergence speed. The proposed algorithm has been evaluated on the 20 standard high-dimensional benchmark numerical optimization problems for the IEEE CEC-2010 Special Session and Competition on Large Scale Global Optimization. The comparison results between ANDE and its versions and the other seven state-of-the-art evolutionary algorithms that were all tested on this test suite indicate that the proposed algorithm and its two versions are highly competitive algorithms for solving large scale global optimization problems.

混合整数2次錐計画法による3次元骨組構造物のトポロジー最適化

Recently, methods to obtain truss or frame with high stiffness by the topology optimization problem of minimizing compliance has been studied actively. In general, cross-sectional areas or shapes of each member are defined as design variables in such a topology optimization problem. Practically, since the member cross-sections are to be selected from predetermined candidates, the design variables are treated as discrete variables. Therefore, heuristics or mixed-integer programming (MIP) is selected as the algorithm for solving truss or frame topology optimization problems. In particular, MIP has an advantage that the global optimality of the solution is guaranteed mathematically. However, it is difficult for MIP to treat large scale problems because of the high computational cost. In previous studies, the topology optimization using MIP is limited to small-scale problems, such as trusses or 2D-frames. In recent years, second-order cone programming (SOCP) has attracted attention since it can be solved efficiently by using the interior-point method. It is reported in the latest study that mixed-integer second-order cone programming (MISOCP), i.e., the problem obtained by adding the integrity to some of the variables of SOCP, can deal with the large scale problem compared with MIP. In this paper, the topology optimization of 3D-frame, which is difficult to treat by using MIP, is solved by using MISOCP. First, the 3D-frame topology optimization problem of minimizing compliance is formulated as MISCOP by using the dual problem of total potential energy minimization problem. From this formula, various 3D-frame topology optimization problems of minimizing compliance in which various sets of cross-sectional shapes are defined as design variables are considered. In this paper, 3D-cantilever with tip load, latticed dome with central concentrated load, and simple building with vertical and horizontal loads are optimized by SOCP in various design value cases. In almost all of the optimization cases, the solver has terminated with finding the global optimal solution. In only a few optimization cases which could not find the global optimal solutions, the MISOCP solver was run with time limit and the best feasible solutions found by the solver are reported. These solutions are not necesarily global optimal solutions, but these are good solutions in the cases of this paper. As numerical experiments for comparison, topology optimization problems treated in this paper were tried to be solved by using MIP as well, but the optimal solutions could not be found at all within realistic computational time. The obtained main results are as follows: •The 3D-frame topology optimization problem of minimizing compliance can be formulated as MISOCP by using the dual problem of total potential energy minimization problem. •MISOCP can solve various large scale 3D-frame topology optimization problems which are difficult for MIP to treat because of the high computational cost. •For a larger problem which is beyond the capability of MISOCP, MISOCP cannot find the global optimal solution but can find solution good enough. The commercial solvers which treat MISOCP have been developed extensively, and calculation speed is improved year after year. Against this background, there exists a possibility that the MISOCP will become a powerful topology optimization tool for building structures.

Modification of Nonlinear Conjugate Gradient Method with Weak Wolfe-Powell Line Search

Conjugate gradient (CG) method is used to find the optimum solution for the large scale unconstrained optimization problems. Based on its simple algorithm, low memory requirement, and the speed of obtaining the solution, this method is widely used in many fields, such as engineering, computer science, and medical science. In this paper, we modified CG method to achieve the global convergence with various line searches. In addition, it passes the sufficient descent condition without any line search. The numerical computations under weak Wolfe-Powell line search shows that the efficiency of the new method is superior to other conventional methods.

Convex Relaxation of Optimal Power Flow in Distribution Feeders with Embedded Solar Power

There is an increasing interest in using Distributed Energy Resources (DER) directly coupled to end user distribution feeders. This poses an array of challenges because most of today's distribution feeders are designed for unidirectional power flow. Therefore when installing DERs such as solar panels with uncontrolled inverters, the upper limit of installable capacity is quickly reached in many of today's distribution feeders. This problem can often be mitigated by optimally controlling the voltage angles of inverters. However, the optimal power flow problem in its standard form is a large scale non-convex optimization problem, and thus can’t be solved precisely and also is computationally heavy and intractable for large systems. This paper examines the use of a convex relaxation using Semi-definite programming to optimally control solar power inverters in a distribution grid in order to minimize the global line losses of the feeder. The mathematical model is presented in details. Further, case studies are completed with simulations involving a 15-bus radial distribution system. These simulations are run for 24 hour periods, with actual solar data and demand data.

Multidisciplinary design optimization of direct-drive PMSG considering the site wind profile

The paper proposes a multidisciplinary design optimization (MDO) of permanent magnet synchronous generators (PMSG) for wind energy conversion systems (WECS). Such a MDO seeks the best compromise between the PMSG cost and the lifetime energy production of the WECS. So far, most of the papers in this field presented design methods using quite a few optimization variables and requiring a lot of assumptions. In the proposed MDO the specifications consist of geometrical, economical, magnetic, electrical and thermal constraints. Several analytic and semi-analytic sub-models where used to deal with these constraints, resulting in a large non-linear optimization problem that uses a sequential quadratic programming optimization algorithm. The paper considers the annual wind profile to estimate the lifetime wind turbine energy production. Moreover, the best possible solutions are presented in a Pareto front showing the trade-off between the generator cost and the wind turbine energy production. Among them, one optimal solution machine was chosen to be further investigated and its results were verified by finite element analysis.

Fine-grained parallelization of fitness functions in bioinformatics optimization problems: gene selection for cancer classification and biclustering of gene expression data.

BackgroundMetaheuristics are widely used to solve large combinatorial optimization problems in bioinformatics because of the huge set of possible solutions. Two representative problems are gene selection for cancer classification and biclustering of gene expression data. In most cases, these metaheuristics, as well as other non-linear techniques, apply a fitness function to each possible solution with a size-limited population, and that step involves higher latencies than other parts of the algorithms, which is the reason why the execution time of the applications will mainly depend on the execution time of the fitness function. In addition, it is usual to find floating-point arithmetic formulations for the fitness functions. This way, a careful parallelization of these functions using the reconfigurable hardware technology will accelerate the computation, specially if they are applied in parallel to several solutions of the population.ResultsA fine-grained parallelization of two floating-point fitness functions of different complexities and features involved in biclustering of gene expression data and gene selection for cancer classification allowed for obtaining higher speedups and power-reduced computation with regard to usual microprocessors.ConclusionsThe results show better performances using reconfigurable hardware technology instead of usual microprocessors, in computing time and power consumption terms, not only because of the parallelization of the arithmetic operations, but also thanks to the concurrent fitness evaluation for several individuals of the population in the metaheuristic. This is a good basis for building accelerated and low-energy solutions for intensive computing scenarios.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-016-1200-9) contains supplementary material, which is available to authorized users.

A hybrid particle swarm optimization and genetic algorithm with population partitioning for large scale optimization problems

Abstract In this paper, a new hybrid particle swarm optimization and genetic algorithm is proposed to minimize a simplified model of the energy function of the molecule. The proposed algorithm is called Hybrid Particle Swarm Optimization and Genetic Algorithm (HPSOGA). The HPSOGA is based on three mechanisms. The first mechanism is applying the particle swarm optimization to balance between the exploration and the exploitation process in the proposed algorithm. The second mechanism is the dimensionality reduction process and the population partitioning process by dividing the population into sub-populations and applying the arithmetical crossover operator in each sub-population in order to increase the diversity of the search in the algorithm. The last mechanism is applied in order to avoid the premature convergence and avoid trapping in local minima by using the genetic mutation operator in the whole population. Before applying the proposed HPSOGA to minimize the potential energy function of the molecule size, we test it on 13 unconstrained large scale global optimization problems with size up to 1000 dimensions in order to investigate the general performance of the proposed algorithm for solving large scale global optimization problems then we test the proposed algorithm with different molecule sizes with up to 200 dimensions. The proposed algorithm is compared against the standard particle swarm optimization to solve large scale global optimization problems and 9 benchmark algorithms, in order to verify the efficiency of the proposed algorithm for solving molecules potential energy function. The numerical experiment results show that the proposed algorithm is a promising and efficient algorithm and can obtain the global minimum or near global minimum of the molecular energy function faster than the other comparative algorithms.

A selective genetic algorithm for multiobjective optimization of cross sections in 3D trussed structures based on a spatial sensitivity analysis

Purpose – There are many problems in civil or mechanical engineering related to structural design. In such a case, the solution techniques which lead to deterministic results are no longer valid due to the heuristic nature of design problems. The purpose of this paper is to propose a computational tool based on genetic algorithms, applied to the optimal design of cross-sections (solid tubes) of 3D truss structures. Design/methodology/approach – The main feature of this genetic algorithm approach is the introduction of a selective-smart method developed in order to improve the convergence rate of large optimization problems. This selective genetic algorithm is based on a preliminary sensitivity analysis performed over each variable, in order to reduce the search space of the evolutionary process. In order to account for the optimization of the total weight, the displacement (of a specific section) and the internal stresses distribution of the structure a multiobjective optimization function was proposed. Findings – The numerical results presented in this paper show a significant improvement in the convergence rate as well as an important reduction in the relative error, compared to the exact solution. Originality/value – The variables sensitivity analysis put forward in this approach introduces a significant improvement in the convergence rate of the genetic algorithm proposed in this paper.

A parallel improved IWO algorithm on GPU for solving large scale global optimization problems

Considering the problems of slow convergence and easily getting into local optimum of invasive weed optimization (IWO) algorithm in finding the optimal solution to large scale global optimization (LSGO) problems, we have proposed an improved IWO (IIWO) algorithm on the basis of the basic IWO algorithm. Concrete adjustments include setting the newborn weed seeds per plant to a fixed number of parameters, changing the initial step and final step to adaptive step, and re-initializing the solution which exceeds the limit value. Meanwhile, through applying the IIWO algorithm to the GPU platform, a parallel IIWO (PIIWO) based on GPU is obtained. The algorithm not only improves the convergence rate, but also strikes a balance between the global and local search capabilities. The simulation results of solving on the LSGO problems (CEC' 2010 high-dimensional functions), have shown that, compared with other algorithms, our designed IIWO can yield better performance, faster convergence speed and higher accuracy; whilst the PIIWO has fewer iterations, higher computing accuracy and significant speedup than the serial algorithm IIWO.

A fitness approximation assisted competitive swarm optimizer for large scale expensive optimization problems

Surrogate assisted meta-heuristic algorithms have received increasing attention over the past years due to the fact that many real-world optimization problems are computationally expensive. However, most existing surrogate assisted meta-heuristic algorithms are designed for small or medium scale problems. In this paper, a fitness approximation assisted competitive swarm optimizer is proposed for optimization of large scale expensive problems. Different from most surrogate assisted evolutionary algorithms that use a computational model for approximating the fitness, we estimate the fitness based on the positional relationship between individuals in the competitive swarm optimizer. Empirical study on seven widely used benchmark problems with 100 and 500 decision variables show that the proposed fitness approximation assisted competitive swarm optimizer is able to achieve competitive performance on a limited computational budget.

A self-adaptive binary differential evolution algorithm for large scale binary optimization problems

This study proposes a new self-adaptive binary variant of a differential evolution algorithm, based on measure of dissimilarity and named SabDE. It uses an adaptive mechanism for selecting how new trial solutions are generated, and a chaotic process for adapting parameter values. SabDE is compared against a number of existing state of the art algorithms, on a set of benchmark problems including high dimensional knapsack problems with up to 10,000 dimensions as well as on the 15 learning based problems of the Congress on Evolutionary Computation (CEC 2015). Experimental results reveal that the proposed algorithm performs competitively and in some cases is superior to the existing algorithms.

Quadratic reformulations of nonlinear binary optimization problems

Very large nonlinear unconstrained binary optimization problems arise in a broad array of applications. Several exact or heuristic techniques have proved quite successful for solving many of these problems when the objective function is a quadratic polynomial. However, no similarly efficient methods are available for the higher degree case. Since high degree objectives are becoming increasingly important in certain application areas, such as computer vision, various techniques have been recently developed to reduce the general case to the quadratic one, at the cost of increasing the number of variables by introducing additional auxiliary variables. In this paper we initiate a systematic study of these quadratization approaches. We provide tight lower and upper bounds on the number of auxiliary variables needed in the worst-case for general objective functions, for bounded-degree functions, and for a restricted class of quadratizations. Our upper bounds are constructive, thus yielding new quadratization procedures. Finally, we completely characterize all minimal quadratizations of negative monomials.

TH‐EF‐BRB‐05: 4pi Non‐Coplanar IMRT Beam Angle Selection by Convex Optimization with Group Sparsity Penalty

Purpose:Integrated beam orientation and fluence map optimization is expected to be the foundation of robust automated planning but existing heuristic methods do not promise global optimality. We aim to develop a new method for beam angle selection in 4π non‐coplanar IMRT systems based on solving (globally) a single convex optimization problem, and to demonstrate the effectiveness of the method by comparison with a state of the art column generation method for 4π beam angle selection.Methods:The beam angle selection problem is formulated as a large scale convex fluence map optimization problem with an additional group sparsity term that encourages most candidate beams to be inactive. The optimization problem is solved using an accelerated first‐order method, the Fast Iterative Shrinkage‐Thresholding Algorithm (FISTA). The beam angle selection and fluence map optimization algorithm is used to create non‐coplanar 4π treatment plans for several cases (including head and neck, lung, and prostate cases) and the resulting treatment plans are compared with 4π treatment plans created using the column generation algorithm.Results:In our experiments the treatment plans created using the group sparsity method meet or exceed the dosimetric quality of plans created using the column generation algorithm, which was shown superior to clinical plans. Moreover, the group sparsity approach converges in about 3 minutes in these cases, as compared with runtimes of a few hours for the column generation method.Conclusion:This work demonstrates the first non‐greedy approach to non‐coplanar beam angle selection, based on convex optimization, for 4π IMRT systems. The method given here improves both treatment plan quality and runtime as compared with a state of the art column generation algorithm. When the group sparsity term is set to zero, we obtain an excellent method for fluence map optimization, useful when beam angles have already been selected.NIH R43CA183390, NIH R01CA188300, Varian Medical Systems; Part of this research took place while D. O'Connor was a summer intern at RefleXion Medical.

Large Scale Reservoirs System Operation Optimization: the Interior Search Algorithm (ISA) Approach

Reservoirs are built to provide a powerful tool to control and manage surface water resources in order to cover inconsistency between water resources and demands. Due to finite available water and the increasing demands for water especially in arid and semi-arid regions like Iran, reservoirs must be optimally operated in order to use water in the most efficient way. This study applies the Interior Search Algorithm (ISA) to solve large scale reservoirs system operation optimization problems. The ISA is a meta-heuristic algorithm inspired from a systematic methodology of architecture process and mirror work utilized by Persian designers for decoration. Unlike other meta-heuristic algorithms, the ISA just have one parameter to tune which is a great advantage. In this study the parameter of the ISA tuned automatically using a linear equation. A real-world one-reservoir operation problem (i.e. Karun-4) and two large scale benchmark problems (i.e. four-reservoir and ten-reservoir operation problem) were employed to show the effectiveness of the ISA. The results shows the high ability of the ISA to solve reservoirs system operation problems as it achieved solutions 99.97, 99.99 and 99.95 % of global optimum for Karun-4 reservoir, four-reservoir and ten-reservoir system operation problems, respectively. These results are the best results reported so far in the studied problems. Comparing results of the ISA with those of non-linear programming (NLP), linear programming (LP), genetic algorithm (GA) and other meta-heuristic algorithms indicates fast convergence to global optimum. Considering the results, it can be stated that the ISA is a powerful tool to optimize complex large scale reservoir system operation problems.