Large Optimization Problems Research Articles

On the solution of Large Scale Bi-Level Linear Vector Optimization Problems through TOPSIS

In this paper, we extend TOPSIS (Technique for Order Preference by Similarity Ideal Solution) for solving Large Scale Bi-level Linear Vector Optimization Problems (LS-BL-LVOP). In order to obtain a compromise ( satisfactory) solution to the LS-BL-LVOP problems using the proposed TOPSIS approach, a modified formulas for the distance function from the positive ideal solution (PIS ) and the distance function from the negative ideal solution (NIS) are proposed and modeled to include all the objective functions of both the first and the second levels. An interactive decision making algorithm for generating a compromise ( satisfactory) solution through TOPSIS approach is provided where the first level decision maker (FLDM) is asked to specify the relative importance of the objectives. Finally, a numerical example is given to clarify the main results developed in the paper.

Optimal integrated management of groundwater resources and irrigated agriculture in arid coastal regions

Abstract. Groundwater systems in arid coastal regions are particularly at risk due to limited potential for groundwater replenishment and increasing water demand, caused by a continuously growing population. For ensuring a sustainable management of those regions, we developed a new simulation-based integrated water management system. The management system unites process modelling with artificial intelligence tools and evolutionary optimisation techniques for managing both water quality and water quantity of a strongly coupled groundwater–agriculture system. Due to the large number of decision variables, a decomposition approach is applied to separate the original large optimisation problem into smaller, independent optimisation problems which finally allow for faster and more reliable solutions. It consists of an analytical inner optimisation loop to achieve a most profitable agricultural production for a given amount of water and an outer simulation-based optimisation loop to find the optimal groundwater abstraction pattern. Thereby, the behaviour of farms is described by crop-water-production functions and the aquifer response, including the seawater interface, is simulated by an artificial neural network. The methodology is applied exemplarily for the south Batinah re-gion/Oman, which is affected by saltwater intrusion into a coastal aquifer system due to excessive groundwater withdrawal for irrigated agriculture. Due to contradicting objectives like profit-oriented agriculture vs aquifer sustainability, a multi-objective optimisation is performed which can provide sustainable solutions for water and agricultural management over long-term periods at farm and regional scales in respect of water resources, environment, and socio-economic development.

Lagrangian relaxation hybrid with evolutionary algorithm for short-term generation scheduling

Short-term generation scheduling is an important function in daily operational planning of power systems. It is defined as optimal scheduling of power generators over a scheduling period while respecting various generator constraints and system constraints. Objective of the problem includes costs associated with energy production, start-up cost and shut-down cost along with profits. The resulting problem is a large scale nonlinear mixed-integer optimization problem for which there is no exact solution technique available. The solution to the problem can be obtained only by complete enumeration, often at the cost of a prohibitively computation time requirement for realistic power systems. This paper presents a hybrid algorithm which combines Lagrangian Relaxation (LR) together with Evolutionary Algorithm (EA) to solve the problem in cooperative and competitive energy environments. Simulation studies were carried out on different systems containing various numbers of units. The outcomes from different algorithms are compared with that from the proposed hybrid algorithm and the advantages of the proposed algorithm are briefly discussed.

Solving difficult mixed integer and disjunctive non-linear problems on single and parallel processors

Difficult non-linear, mixed integer non-linear and general disjunctive programming (NLP/MINLP/GDP) problems are considered in the present study. We show using random relaxation procedures that sampling over a subset of the integer variables and parallel processing have potential to simplify large scale MINLP/GDP problems and hence to solve them faster and more reliably than conventional approaches on single processors. Gray coding can be utilized to assign problems to the local processors. Some efficient non-linear solvers have been connected to the Genetic Hybrid Algorithm (GHA) through a switch board minlp_machine(), monitoring a vector of caller functions. The switch board is tested on single processor computers and massively parallel supercomputers in a sample of optimization problems. A new line search algorithm based on multi-sectioning, i.e. repeated bi-sectioning of parallel threads is applied to a set of irregular NLP-problems. Simulations indicate that step search based on multi-sectioning and re-centering is robust. Time recording indicates that the step search procedure is fast also in large optimization problems. Parallel processing with Gray coding and random sampling over a subset of the integer variables improves the performance of the sequential quadratic programming algorithm with multi-sectioning in large-scale problems. An early mesh interrupt guarantees load balancing in regular problems. General disjunctive programming (GDP) problems can be simplified through parallel processing. Certain problems are solved in larger scale than reported previously. Function pointer logic and intelligent switching procedures provide a fruitful basis for solving high-order GDP problems, where the computational challenge in direct MINLP-formulations would be insurmountable.

Algorithm tuners for PSO methods and genetic programming techniques for learning tuning rules

In recent years, improvements in processing power have allowed the application of optimization methods to complicated large optimization problems. Among these methods, heuristic optimization techniques such as particle swarm optimization (PSO) have been a particular focus of attention because of their simplicity, performance, and easy software implementation. However, there is no solid theoretical foundation for analyzing the convergence of these algorithms, and in practice, their rate of convergence is often determined by the choice of parameters. For this reason, the algorithm's parameters must be tuned appropriately for each new optimization problem we want to solve, and in some cases the parameters must be varied as the algorithm is updated. In this paper, we combine a feedback element as an algorithm tuner with an original algorithm; the resulting algorithm is applied to the optimization problem in question, and we use genetic programming (GP) to generate tuning rules to automatically tune the algorithm, according to its current state, as the algorithm is updated. More specifically, we adopt PSO as a heuristic optimization method, and we augment PSO by using GP as a meta-algorithm to solve the learning problem of automatically generating tuning rules for the parameters in the PSO algorithm. This leads to the proposed method for generating parameter tuning rules to solve optimization problems more efficiently. © 2014 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Hybrid Simulated Annealing and Nelder-Mead Algorithm for Solving Large-Scale Global Optimization Problems

This paper presents a new algorithm for solving large scale global optimization problems based on hybridization of simulated annealing and Nelder-Mead algorithm. The new algorithm is called simulated Nelder-Mead algorithm with random variables updating (SNMRVU). SNMRVU starts with an initial solution, which is generated randomly and then the solution is divided into partitions. The neighborhood zone is generated, random number of partitions are selected and variables updating process is starting in order to generate a trail neighbor solutions. This process helps the SNMRVU algorithm to explore the region around a current iterate solution. The Nelder- Mead algorithm is used in the final stage in order to improve the best solution found so far and accelerates the convergence in the final stage. The performance of the SNMRVU algorithm is evaluated using 27 scalable benchmark functions and compared with four algorithms. The results show that the SNMRVU algorithm is promising and produces high quality solutions with low computational costs..

Nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization

We propose a nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization. Unlike traditional trust region methods, the subproblem in our method is a simple conic model, where the Hessian of the objective function is approximated by a scalar matrix. The trust region radius is adjusted with a new self-adaptive adjustment strategy which makes use of the information of the previous iteration and current iteration. The new method needs less memory and computational efforts. The global convergence and Q-superlinear convergence of the algorithm are established under the mild conditions. Numerical results on a series of standard test problems are reported to show that the new method is effective and attractive for large scale unconstrained optimization problems.

Scheduling short-term hydrothermal generation using predator prey optimization technique

Paper presents predator–prey based optimization (PPO) technique to obtain optimal generation scheduling of short-term hydrothermal system. PPO is a part of the swarm intelligence family and is capable to solve large scale non-linear optimization problems. PPO based algorithm combines the idea of particle swarm optimization concept with predator effect that helps to, maintain diversity in the swarm and preventing premature convergence to local sub-optimal. In this paper first of all feasible solution is obtained through random heuristic search and then thermal and hydro power generations are searched for optimum hydrothermal scheduling problem using PPO. Variable elimination method is implemented to handle the equality constraint by eliminating variable explicitly. These eliminated variables are considered by penalty method restricts slack units with in limits. Slack thermal generating unit for each sub-interval handles power balance equality constraint and slack hydro units handle water equality constraint. The performance of the proposed approach is illustrated on, fixed-head and variable-head hydrothermal power systems. The results obtained from the proposed technique are compared with the existing technique. From the numerical results, it is experienced that the PPO based approach is able to provide a better solution.

Parallel ant colony optimization for resource constrained job scheduling

In mining supply chains, large combinatorial optimization problems arise. These are NP-hard and typically require a large number of computing resources to solve them. In particular, the run-time overheads can become increasingly prohibitive with increasing problem sizes. Parallel methods provide a way to manage such run-time issues by utilising several processors in independent or shared memory architectures. However it is not obvious how to adapt serial optimisation algorithms to perform best in a parallel environment. Here, we consider a resource constrained scheduling problem which is motivated in mining supply chains and present two popular meta-heuristics, ant colony optimization (ACO) and simulated annealing and investigate how best to parallelize these methods on a shared memory architecture consisting of several cores. ACO’s solution construction framework is inherently parallel allowing a relatively straightforward parallel implementation. However, for best performance, ACO needs an element of local search. This significantly complicates the paralellization. Several alternative schemes for parallel ACO with elements of local search are considered and evaluated empirically. We find that ACO with local search is the most effective single-threaded algorithm. The best parallel implementation can obtain similar quality results to the serial method in significantly less elapsed time.

Soccer league competition algorithm: A novel meta-heuristic algorithm for optimal design of water distribution networks

Water distribution networks are one of the most important elements in the urban infrastructure system and require huge investment for construction. Optimal design of water systems is classified as a large combinatorial discrete non-linear optimization problem. The main concern associated with optimization of water distribution networks is related to the nonlinearity of discharge-head loss equation, availability of the discrete nature of pipe sizes, and constraints, such as conservation of mass and energy equations. This paper introduces an efficient technique, entitled Soccer League Competition (SLC) algorithm, which yields optimal solutions for design of water distribution networks. Fundamental theories of the method are inspired from soccer leagues and based on the competitions among teams and players. Like other meta-heuristic methods, the proposed technique starts with an initial population. Population individuals (players) are in two types: fixed players and substitutes that all together form some teams. The competition among teams to take the possession of the top ranked positions in the league table and the internal competitions between players in each team for personal improvements are used for simulation purpose and convergence of the population individuals to the global optimum. Results of applying the proposed algorithm in three benchmark pipe networks show that SLC converges to the global optimum more reliably and rapidly in comparison with other meta-heuristic methods.

Self Organization of Tilts in Relay Enhanced Networks: A Distributed Solution

Despite years of physical-layer research, the capacity enhancement potential of relays is limited by the additional spectrum required for Base Station (BS)-Relay Station (RS) links. This paper presents a novel distributed solution by exploiting a system level perspective instead. Building on a realistic system model with impromptu RS deployments, we develop an analytical framework for tilt optimization that can dynamically maximize spectral efficiency of both the BS-RS and BS-user links in an online manner. To obtain a distributed self-organizing solution, the large scale system-wide optimization problem is decomposed into local small scale subproblems by applying the design principles of self-organization in biological systems. The local subproblems are non-convex, but having a very small scale, can be solved via standard nonlinear optimization techniques such as sequential quadratic programming. The performance of the developed solution is evaluated through extensive simulations for an LTE-A type system and compared against a number of benchmarks including a centralized solution obtained via brute force, that also gives an upper bound to assess the optimality gap. Results show that the proposed solution can enhance average spectral efficiency by up to 50% compared to fixed tilting, with negligible signaling overheads. The key advantage of the proposed solution is its potential for autonomous and distributed implementation.

A Two-Level Genetic Algorithm for Large Optimization Problems

Many local two-level algorithms have been proposed for accelerating the electromagnetic optimization by stochastic algorithms. These algorithms use a combination of a coarse and a fine model in the optimization procedure. Despite the good results, the global convergence properties represent an important drawback of these approaches. A global two-level algorithm had been proposed to deal with the convergence problems, but the requirement to refine the global surrogate model in each step can demand high computational time. This paper introduces a global two-level genetic algorithm that uses single predefined coarse and fine surrogate models, which are defined as an artificial neural network nonlinear regression of a preliminary set of finite element simulations. The benchmark test problem, Hartmann 6, and the problem dealing with the eight-parameter design of superconducting magnetic energy storage have been analyzed..

Production Scheduling of Open Pit Mines Using Particle Swarm Optimization Algorithm

Determining an optimum long term production schedule is an important part of the planning process of any open pit mine; however, the associated optimization problem is demanding and hard to deal with, as it involves large datasets and multiple hard and soft constraints which makes it a large combinatorial optimization problem. In this paper a procedure has been proposed to apply a relatively new and computationally less expensive metaheuristic technique known as particle swarm optimization (PSO) algorithm to this computationally challenging problem of the open pit mines. The performance of different variants of the PSO algorithm has been studied and the results are presented.

Global convergence properties of a new class of conjugate gradient method for unconstrained optimization

Nonlinear conjugate gradient (CG) methods are widely used for solving large scale unconstrained optimization problems. Many studies have been devoted to modified and improve this method. In this paper, a new parameter of CG method that possesses global convergence properties using exact line search is proposed. Numerical results show that the new formula is best and more efficient when compared with the other classical CG methods

The Hybrid BFGS-CG Method in Solving Unconstrained Optimization Problems

In solving large scale problems, the quasi-Newton method is known as the most efficient method in solving unconstrained optimization problems. Hence, a new hybrid method, known as the BFGS-CG method, has been created based on these properties, combining the search direction between conjugate gradient methods and quasi-Newton methods. In comparison to standard BFGS methods and conjugate gradient methods, the BFGS-CG method shows significant improvement in the total number of iterations and CPU time required to solve large scale unconstrained optimization problems. We also prove that the hybrid method is globally convergent.

The convergence properties of a new type of conjugate gradient methods

Conjugate gradient (CG) methods are widely used for solving large scale unconstrained optimization problems. Many studies have been devoted to develop and improve these methods. In this paper, we compare our new CG coefficient

Efficient Robust Economic Nonlinear Model Predictive Control of an Industrial Batch Reactor

In this paper we present a systematic and efficient approach to deal with uncertainty in Nonlinear Model Predictive Control (NMPC). The main idea of the approach is to represent the NMPC setting as a real-time decision problem under uncertainty that is formulated as a multi-stage stochastic problem with recourse, based on a description of the uncertainty by a scenario tree. This formulation explicitly takes into account the fact that new information will be available in the future and thus reduces the conservativeness compared to open-loop worst-case approaches. We show that the proposed multistage NMPC formulation can deal with significant plant-model mismatch as it is usually encountered in the process industry and still satisfies tight constraints for the different values of the uncertain parameters, in contrast to standard NMPC. The use of an economic cost function leads to a superior performance compared to the standard tracking formulation. The potential of the approach is demonstrated for an industrial case study provided by BASF SE in the context of the European Project EMBOCON. The numerical solution of the resulting large optimization problems is implemented using the optimization framework CasADi.

Optimal CD-DY Conjugate Gradient Methods with Sufficient Descent

Conjugate Gradient (CG) methods are widely used for large scale unconstrained optimization problems. Most of CG-methods don’t always generate a descent search direction, so the descent condition is usually assumed in the analysis and implementations. In this paper, we have studied several modified CG-methods based on the famous CD (CG-method), and show that our new proposed CG-methods produces sufficient descent and converges globally if the Wolfe conditions are satisfied. Moreover, they produces the original version of the CD (CG-method), if the line searches are exact. The numerical results show that the new methods are more effective and promising by comparing with the standard CD and DY (CG-methods).

Decomposition Algorithms for Market Clearing With Large-Scale Demand Response

This paper is concerned with large-scale integration of demand response (DR) from small loads such as residential smart appliances into modern electricity systems. These appliances have intertemporal consumption constraints, while the satisfaction of the end-user from operating them is captured through utility functions. Incorporation of the appliance scheduling flexibility and user satisfaction in the system optimization points to maximizing the social welfare. In order to solve the resultant very large optimization problem with manageable complexity, dual decomposition is pursued. The problem decouples into separate subproblems for the market operator and each aggregator. Each aggregator addresses its local optimization aided by the end-users' smart meters. The subproblems are coordinated with carefully designed information exchanges between the market operator and the aggregators so that user privacy is preserved. Numerical tests illustrate the benefits of large-scale DR incorporation.

Optimal Transport with Proximal Splitting

This article reviews the use of first order convex optimization schemes to solve the discretized dynamic optimal transport problem, initially proposed by Benamou and Brenier. We develop a staggered grid discretization that is well adapted to the computation of the $L^2$ optimal transport geodesic between distributions defined on a uniform spatial grid. We show how proximal splitting schemes can be used to solve the resulting large scale convex optimization problem. A specific instantiation of this method on a centered grid corresponds to the initial algorithm developed by Benamou and Brenier. We also show how more general cost functions can be taken into account and how to extend the method to perform optimal transport on a Riemannian manifold.