Large Optimization Problems Research Articles

05/01598 The impact of random failures on nodal price andnodal reliability in restructured power systems

This paper studies the problem of optimal transmission grid expansion planning. Deterministic formulations are developed for a single time period planning problem, while the operation costs (including total energy, reserve procurement, and load and wind curtailment costs) and N-1 security constraints are taken into consideration. Benders decomposition approach is employed to decompose the large optimization problem into smaller subproblems. New and more efficient Benders cuts are proposed, which are specifically tailored to the binary decision variables of transmission planning problem. The advantages of the proposed Benders cuts to the regular cuts used in the literature are demonstrated, in terms of the increase in convergence speed, decrease in the computational time and required number of iterations, and the scalability of the solution approach. Besides, the proposed Benders cuts are extended to include multiple operation scenarios (similar to stochastic formulations), where the optimization problem considers multiple power demand and wind production scenarios. The proposed optimization models are investigated using Graver's 6-bus, IEEE 24-bus, IEEE 73-bus, and IEEE 118-bus test systems and compared to ones reported in the literature.

Scheduling Interdependent Waterway Projects through Simulation and Genetic Optimization

The evaluation, selection, sequencing, and scheduling of lock reconstruction and rehabilitation projects is a major concern for inland waterway systems. As traffic increases and existing lock conditions deteriorate, users suffer substantially longer processing times and delays. Lock projects tend to be interdependent as improvements at one lock may shift bottlenecks elsewhere. A generalized waterway simulation model that is independent of network geometry was developed to evaluate the system over a multiyear planning horizon. A genetic algorithm was also developed in order to efficiently explore the solution space of this large investment optimization problem. The proposed combination of two stochastic models can solve selection, sequencing, and scheduling problems for interdependent projects in large waterway networks but requires long computation times. Due to lock interdependency, the benefits of multiple projects are not directly obtainable by adding individual project benefits; they are measured from the marginal increment of total system delays. The results show how interdependent projects should be scheduled and how operational lock control improvements can sometimes improve delays significantly without requiring major construction investments.

Numerical methods for large-scale nonlinear optimization

Recent developments in numerical methods for solving large differentiable nonlinear optimization problems are reviewed. State-of-the-art algorithms for solving unconstrained, bound-constrained, linearly constrained and non-linearly constrained problems are discussed. As well as important conceptual advances and theoretical aspects, emphasis is also placed on more practical issues, such as software availability.

Mathematical programming models and algorithms for engineering design optimization

Mathematical programming provides general tools for engineering design optimization. We present numerical models for simultaneous analysis and design optimization (SAND) and multidisciplinary design optimization (MDO) represented by mathematical programs. These models are solved with numerical techniques based on the feasible arc interior point algorithm (FAIPA) for nonlinear constrained optimization. Even if MDO is a very large optimization problem, our approach reduces considerably the computer effort. Several tools for very large problems are also presented. The present approach is very strong and efficient for real industrial applications and can easily interact with existing simulation engineering codes.

A robust strategy for optimizing complex distillation columns

This work introduces a strategy for the optimal design of distillation systems based on continuous optimization. The approach is similar to the one proposed earlier by [Lang, Y. -D., & Biegler, L. T. (2002). A distributed stream method for tray optimization. AIChE Journal, 48, 582], avoiding the need of solving extremely large and non-linear discrete optimization problems. When used with complex distillation units, it can identify interesting design configurations not considered by other continuous formulations, and also relieve some of the numerical difficulties associated with the use of distribution functions for the optimal location of feed and side-streams. The method considers a relaxation of the original problem, where the streams are initially split to several trays in the column, not necessarily adjacent. The optimal location of each stream is converged by constraining the optimization problem, using adjustable parameters that control the minimum amount of aggregation allowed. The methodology is illustrated with the application to several industrial case studies, including sets of distillation columns. Models up to 17,000 variables/equations were solved, revealing large economic benefits in the design of new units and optimization of sets of existing ones.

On the asymptotic behaviour of some new gradient methods

The Barzilai-Borwein (BB) gradient method, and some other new gradient methods have shown themselves to be competitive with conjugate gradient methods for solving large dimension nonlinear unconstrained optimization problems. Little is known about the asymptotic behaviour, even when applied to n−dimensional quadratic functions, except in the case that n=2. We show in the quadratic case how it is possible to compute this asymptotic behaviour, and observe that as n increases there is a transition from superlinear to linear convergence at some value of n≥4, depending on the method. By neglecting certain terms in the recurrence relations we define simplified versions of the methods, which are able to predict this transition. The simplified methods also predict that for larger values of n, the eigencomponents of the gradient vectors converge in modulus to a common value, which is a similar to a property observed to hold in the real methods. Some unusual and interesting recurrence relations are analysed in the course of the study.

A NEW NONLINEAR CONJUGATE GRADIENT METHOD FOR UNCONSTRAINED OPTIMIZATION

Conjugate gradient methods are widely used for large scale unconstrained optimization problems. Most of conjugate gradient methods don’t always generate a descent search direction, so the descent condition is usually assumed in the analysis and implementations. Dai and Yuan (1999) proposed a conjugate gradient method which generates a descent search direction at every iteration and converges globally to the solution if the Wolfe conditions are satisfied within the line search strategy. In this paper, we give a new conjugate gradient method based on the study of Dai and Yuan, and show that our method always produces a descent search direction and converges globally if the Wolfe conditions are satisfied. Moreover our method has the second-order curvature information with a higher precision which uses the modified secant condition proposed by Zhang, Deng and Chen (1999) and Zhang and Xu (2001). Our numerical results show that our method is very efficient for given standard test problems, if we make a good choice of a parameter included in our method.

Ridge regression based hybrid genetic algorithms for multi-locus quantitative trait mapping

Genetic algorithms (GAs) are increasingly used in large and complex optimisation problems. Here we use GAs to optimise fitness functions related to ridge regression, which is a classical statistical procedure for dealing with a large number of features in a multivariable, linear regression setting. The algorithm avoids overfitting, gracefully handles collinearity and leads to easily interpretable results. We use the method to model the relationship between a quantitative trait and genetic markers in a mouse cross involving 69 F2 mice. The approach will be useful in the context of many genomic data sets where the number of features far exceeds the number of observations and where features can be highly correlated.

Intelligent Evolutionary Algorithms for Large Parameter Optimization Problems

This work proposes two intelligent evolutionary algorithms IEA and IMOEA using a novel intelligent gene collector (IGC) to solve single and multiobjective large parameter optimization problems, respectively. IGC is the main phase in an intelligent recombination operator of IEA and IMOEA. Based on orthogonal experimental design, IGC uses a divide-and-conquer approach, which consists of adaptively dividing two individuals of parents into N pairs of gene segments, economically identifying the potentially better one of two gene segments of each pair, and systematically obtaining a potentially good approximation to the best one of all combinations using at most 2N fitness evaluations. IMOEA utilizes a novel generalized Pareto-based scale-independent fitness function for efficiently finding a set of Pareto-optimal solutions to a multiobjective optimization problem. The advantages of IEA and IMOEA are their simplicity, efficiency, and flexibility. It is shown empirically that IEA and IMOEA have high performance in solving benchmark functions comprising many parameters, as compared with some existing EAs.

Computational formulation of shakedown analysis as a conic quadratic optimization problem

In the analysis of shakedown problems that have been discretized by means of the finite element method, large and sparse optimization problems are generated. The purpose of this paper is to provide the details of how such a problem can be cast in the form of a conic quadratic optimization problem, making use of Melan’s static theorem. An effective algorithm for the solution of the optimization problem is proposed.

Global Convergence Properties of Nonlinear Conjugate Gradient Methods with Modified Secant Condition

Conjugate gradient methods are appealing for large scale nonlinear optimization problems. Recently, expecting the fast convergence of the methods, Dai and Liao (2001) used secant condition of quasi...

Multipoint cubic surrogate function for sequential approximate optimization

Multipoint cubic approximations are investigated as surrogate functions for nonlinear objective and constraint functions in the context of sequential approximate optimization. The proposed surrogate functions match actual function and gradient values, including the current expansion point, thus satisfying the zero and first-order necessary conditions for global convergence to a local minimum of the original problem. Function and gradient information accumulated from multiple design points during the iteration history is used in estimating a reduced Hessian matrix and selected cubic terms in a design subspace appropriate for problems with many design variables. The resulting approximate response surface promises to accelerate convergence to an optimal design within the framework of a trust region algorithm. The hope is to realize computational savings in solving large numerical optimization problems. Numerical examples demonstrate the effectiveness of the new multipoint surrogate function in reducing errors over large changes in design variables.

Optimal planning model for rehabilitation of water networks

An optimal planning model for rehabilitation of water networks is presented. Capital costs (replacement, rehabilitation and repairing costs), benefits (by the reduction of pumping cost and leakage cost), and hydraulic reliability are used for making an optimal decision for the rehabilitation plan of a water pipe network. KYPIPE is used for checking the hydraulic reliability. A multi-objective optimization model is successfully developed in this study. And the task is tackled using a new meta-heuristic algorithm, Harmony Search, for solving a large optimization problem to which conventional optimization techniques are poorly suited. Five different models with different objective functions are developed and tested according to various conditions considered in this study. These models provide more options for the rehabilitation of pipe network systems compared to previously suggested models in the literature.

Optimal capacitor placement for distribution systems using genetic algorithms

In this paper we describe a new technique for placing fixed capacitors in radial distribution systems based on genetic algorithms (GA). Actually, current optimization models of capacitor placement only consider losses reduction and voltage profile simultaneously, but the compensation cost and the load changes are not taking into account as part of the objective function. Also, the result may be not the best choice because this is a very large optimization problem and there are too many combinations. We present here a general approach for the optimal solution of this problem considering all the parameters of the distribution system involved: capacitor cost, voltage, angle and load changes per hour. An exhaustive search through all possible solutions is needed. Therefore, GA is an ideal candidate to solve this situation. Working with this algorithm, a Microsoft Excel Program has been developed in order to answer the capacitor placement problem for any kind of radial distribution system. The test results are presented along with the discussion of the algorithm.

Adaptive Approximation of Young Measure Solutions in Scalar Nonconvex Variational Problems

This paper addresses the numerical approximation of Young measures appearing as generalized solutions to scalar nonconvex variational problems. We prove a priori and a posteriori error estimates for a macroscopic quantity, the stress. For a scalar three-well problem we show convergence of other quantities such as Young measure support and microstructure region. Numerical experiments indicate that the computational effort in the solution of the large optimization problem is significantly reduced by using an adaptive mesh refinement strategy based on a posteriori error estimates in combination with an active set strategy due to Carstensen and Roubicek [Numer. Math., 84 (2000), pp. 395--414].

Simultaneous solution approaches for large optimization problems

In this paper, efficient simultaneous strategies are presented for the optimization of practical problems involving PDE-models. In particular, reduced sequential quadratic programming methods for problems with only few influence variables and simultaneous quadratic programming iterations are discussed. As a result we obtain algorithms whose overall computational complexity is reduced considerably in comparison to a black-box approach.

Some variants of projection methods for large nonlinear optimization problems

Some variants of projection methods for large nonlinear optimization problems

Force allocation in a large-scale distributed active surface

This paper investigates the problem of closed-loop control of a small number of parameters by allocating actuation in a system with many binary degrees of freedom, using an actual large-scale air-jet table as an example. In this system, the desired force and torque are produced by a large number of spatially distributed binary air jets directing individual forces on an object. Various algorithms for solving the force allocation problem-determining the appropriate valve states in this hyper-redundant system-are investigated. The algorithms range from discrete optimal search to continuous constrained optimization to a hybrid hierarchical approach that can be distributed. The latter consists of using the continuous optimal solutions to recursively break the large optimization problems into smaller problems that can be solved using optimal search methods or precomputed lookup tables. A tradeoff between computation time and allocation error was found. The optimal algorithms yield low errors but the time is exponential in the number of actuators, while the continuous solutions execute quickly but yield larger errors. The hybrid hierarchical optimal algorithms give the best compromise between these conflicting goals, and their applicability spans the full range of degrees of freedom from a few to many thousands. These hierarchical algorithms are useful in many such highly redundant systems.

Application of simulated annealing and tabu search for loss minimization in distribution systems

This article presents an efficient algorithm for the loss minimization by automatic sectionalizing switch operation in large-scale distribution systems. Simulated annealing is particularly well suited for large combinational optimization problems, but the use of this algorithm also requires excessive computation time. Tabu search attempts to determine a better solution in the manner of a greatest-descent algorithm, but it cannot give any guarantee for the convergence property. The hybrid algorithm of these two methods with some adaptations is applied to improve the computation time and convergence property. Numerical examples demonstrate the validity and effectiveness of the proposed methodology using a Korea Electric Power Corporation's distribution system.

Simulation and optimization approaches to scenario tree generation

In this paper, three approaches are presented for generating scenario trees for financial portfolio problems. These are based on simulation, optimization and hybrid simulation/optimization. In the simulation approach, the price scenarios at each time period are generated as the centroids of random scenario simulations generated sequentially or in parallel. The optimization method generates a number of discrete outcomes which satisfy specified statistical properties by solving either a sequence of non-linear optimization models (one at each node of the scenario tree) or one large optimization problem. In the hybrid approach, the optimization problem is reduced in size by fixing price variables to values obtained by simulation. These procedures are backtested using historical data and computational results are presented.