A critical review of discrete filled function methods in solving nonlinear discrete optimization problems

Many real life problems can be modeled as nonlinear discrete optimization problems. Such problems often have multiple local minima and thus require global optimization methods. Due to high complexity of these problems, heuristic based global optimization techniques are usually required when solving large scale discrete optimization or mixed discrete optimization problems. One of the more recent global optimization tools is known as the discrete filled function method. Nine variations of the discrete filled function method in literature are identified and a review on theoretical properties of each method is given. Some of the most promising filled functions are tested on various benchmark problems. Numerical results are given for comparison.     

线性与非线性规划算法与理论

线性规划与非线性规划是数学规划中经典而重要的研究方向. 主要介绍该研究方向的背景知识,并介绍线性规划、无约束优化和约束优化的最新算法与理论以及一些前沿与热点问题. 交替方向乘子法是一类求解带结构的约束优化问题的方法,近年来倍受重视. 全局优化是一个对于应用优化领域非常重要的研究方向. 因此也试图介绍这两个方面的一些最新研究进展和问题.     

基于卷积神经网络气动力降阶模型的翼型优化方法

针对非线性大扰动翼型气动力优化问题,提出了基于卷积神经网络气动力降阶模型的优化方法.该方法用不同形状参数下翼型的气动力数据作为训练信号,训练卷积神经网络翼型气动力降阶模型.采用该气动力降阶模型,以最大升阻比为目标,对翼型进行优化,结果表明该方法可用于大扰动下翼型气动力的预测和优化.该文同时还讨论了池化法和径向基法的训练...     

Mixed Integer Models for the Stationary Case of Gas Network Optimization

A gas network basically consists of a set of compressors and valves that are connected by pipes. The problem of gas network optimization deals with the question of how to optimize the flow of the gas and to use the compressors cost-efficiently such that all demands of the gas network are satisfied. This problem leads to a complex mixed integer nonlinear optimization problem. We describe techniques for a piece-wise linear approximation of the nonlinearities in this model resulting in a large mixed integer linear program. We study sub-polyhedra linking these piece-wise linear approximations and show that the number of vertices is computationally tractable yielding exact separation algorithms. Suitable branching strategies complementing the separation algorithms are also presented. Our computational results demonstrate the success of this approach. Received: April, 2004     

Global optimization of multilevel electricity market models including network design and graph partitioning

We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal network expansion and the optimal zonal configuration of zonal pricing electricity markets, which is an extension of the model discussed in Grimm et al. (2019) that does not include a network design problem. The two classical discrete optimization problems of network design and graph partitioning together with nonlinearities due to economic modeling yield extremely challenging mixed-integer nonlinear multilevel models for which we develop two problem-tailored solution techniques. The first approach relies on an equivalent bilevel formulation and a standard KKT transformation thereof including novel primal-dual bound tightening techniques, whereas the second is a tailored generalized Benders decomposition. For the latter, we strengthen the Benders cuts of Grimm et al. (2019) by using the structure of the newly introduced network design subproblem. We prove for both methods that they yield global optimal solutions. Afterward, we compare the approaches in a numerical study and show that the tailored Benders approach clearly outperforms the standard KKT transformation. Finally, we present a case study that illustrates the economic effects that are captured in our model.     

A sparse nonlinear optimization algorithm

One of the most effective numerical techniques for solving nonlinear programming problems is the sequential quadratic programming approach. Many large nonlinear programming problems arise naturally in data fitting and when discretization techniques are applied to systems described by ordinary or partial differential equations. Problems of this type are characterized by matrices which are large and sparse. This paper describes a nonlinear programming algorithm which exploits the matrix sparsity produced by these applications. Numerical experience is reported for a collection of trajectory optimization problems with nonlinear equality and inequality constraints.The authors wish to acknowledge the insightful contributions of Dr. William Huffman.     

Solving nonlinear integer programs with large-scale optimization software

This paper describes recent experience in tackling large nonlinear integer programming problems using the MINOS large-scale optimization software. A technique is presented for extending the constrained search approach used in MINOS to exploring integer-feasible solutions once a continuous optimal solution is obtained. Computational experience with this approach is described for two classes of problems: quadratic assignment problems and pipeline network design problems.     

Solving nonlinear integer programs with large-scale optimization software

This paper describes recent experience in tackling large nonlinear integer programming problems using the MINOS large-scale optimization software. A technique is presented for extending the constrained search approach used in MINOS to exploring integer-feasible solutions once a continuous optimal solution is obtained. Computational experience with this approach is described for two classes of problems: quadratic assignment problems and pipeline network design problems.     

Scope of stationary multi-objective evolutionary optimization: a case study on a hydro-thermal power dispatch problem

Many engineering design and developmental activities finally resort to an optimization task which must be solved to get an efficient and often an intelligent solution. Due to various complexities involved with objective functions, constraints, and decision variables, optimization problems are often not adequately suitable to be solved using classical point-by-point methodologies. Evolutionary optimization procedures use a population of solutions and stochastic update operators in an iteration in a manner so as to constitute a flexible search procedure thereby demonstrating promise to such difficult and practical problem-solving tasks. In this paper, we illustrate the power of evolutionary optimization algorithms in handling different kinds of optimization tasks on a hydro-thermal power dispatch optimization problem: (i) dealing with non-linear, non-differentiable objectives and constraints, (ii) dealing with more than one objectives and constraints, (iii) dealing with uncertainties in decision variables and other problem parameters, and (iv) dealing with a large number (more than 1,000) variables. The results on the static power dispatch optimization problem are compared with that reported in an existing simulated annealing based optimization procedure on a 24-variable version of the problem and new solutions are found to dominate the solutions of the existing study. Importantly, solutions found by our approach are found to satisfy theoretical Kuhn–Tucker optimality conditions by using the subdifferentials to handle non-differentiable objectives. This systematic and detail study demonstrates that evolutionary optimization procedures are not only flexible and scalable to large-scale optimization problems, but are also potentially efficient in finding theoretical optimal solutions for difficult real-world optimization problems. Kalyanmoy Deb, Deva Raj Chair Professor. Currently a Finland Distinguished Professor, Department of Business Technology, Helsinki School of Economics, 00101 Helsinki, Finland.     

天然气稳态运行优化的混合整数模型及其算法

天然气稳态运行优化问题的难点在于网络结构复杂、规模大、目标函数及约束高度非线性.针对其混合整数非线性规划模型,基于网络约简和线性化技术,建立了线性近似模型,并提出一种新的求解算法.将新算法用于优化我国西部天然气管网系统,结果表明所提算法是有效的.     

Solving Planning and Design Problems in the Process Industry Using Mixed Integer and Global Optimization

This contribution gives an overview on the state-of-the-art and recent advances in mixed integer optimization to solve planning and design problems in the process industry. In some case studies specific aspects are stressed and the typical difficulties of real world problems are addressed. Mixed integer linear optimization is widely used to solve supply chain planning problems. Some of the complicating features such as origin tracing and shelf life constraints are discussed in more detail. If properly done the planning models can also be used to do product and customer portfolio analysis. We also stress the importance of multi-criteria optimization and correct modeling for optimization under uncertainty. Stochastic programming for continuous LP problems is now part of most optimization packages, and there is encouraging progress in the field of stochastic MILP and robust MILP. Process and network design problems often lead to nonconvex mixed integer nonlinear programming models. If the time to compute the solution is not bounded, there are already a commercial solvers available which can compute the global optima of such problems within hours. If time is more restricted, then tailored solution techniques are required.     

Sequential Semidefinite Program for Maximum Robustness Design of Structures under Load Uncertainty

A robust structural optimization scheme as well as an optimization algorithm are presented based on the robustness function. Under the uncertainties of the external forces based on the info-gap model, the maximization of the robustness function is formulated as an optimization problem with infinitely many constraints. By using the quadratic embedding technique of uncertainty and the S-procedure, we reformulate the problem into a nonlinear semidefinite programming problem. A sequential semidefinite programming method is proposed which has a global convergent property. It is shown through numerical examples that optimum designs of various linear elastic structures can be found without difficulty.The authors are grateful to the Associate Editor and two anonymous referees for handling the paper efficiently as well as for helpful comments and suggestions.     

Case Studies for a First-Order Robust Nonlinear Programming Formulation

In this paper, we conduct three case studies to assess the effectiveness of a recently proposed first-order method for robust nonlinear programming [Zhang, Y.: J. Optim. Theory Appl. 132, 111–124 (2007)]. Three robust nonlinear programming problems were chosen from the literature using the criteria that results calculated using other methods must be available and the problems should be realistic, but fairly simple. Our studies show that the first-order method produced reasonable solutions when the level of uncertainty was small to moderate. In addition, we demonstrate a method for leveraging a theoretical result to eliminate constraint violations. Since the first-order method is relatively inexpensive in comparison to other robust optimization techniques, our studies indicate that, under moderate uncertainty, the first-order approach may be more suitable than other methods for large problems. The authors recognize funding from NSF Grants DMS-0405831 and DMS-0240058.     

A conjugate-gradient optimization method invariant to nonlinear scaling

A conjugate-gradient optimization method which is invariant to nonlinear scaling of a quadratic form is introduced. The technique has the property that the search directions generated are identical to those produced by the classical Fletcher-Reeves algorithm applied to the quadratic form. The approach enables certain nonquadratic functions to be minimized in a finite number of steps. Several examples which illustrate the efficacy of the method are included.     

An optimization-based approach for the design of Bayesian networks

Bayesian networks model conditional dependencies among the domain variables, and provide a way to deduce their interrelationships as well as a method for the classification of new instances. One of the most challenging problems in using Bayesian networks, in the absence of a domain expert who can dictate the model, is inducing the structure of the network from a large, multivariate data set. We propose a new methodology for the design of the structure of a Bayesian network based on concepts of graph theory and nonlinear integer optimization techniques.     

混沌粒子群神经网络在上证指数预测中的应用

上证指数预测是一个非常复杂的非线性问题,为了提高对上证指数预测的准确性,本文采用基于混沌粒子群(CPSO)算法对BP神经网络算法改进的方法来进行预测.BP神经网络算法目前已经应用到预测、聚类、分类等许多领域,取得了不少的成果.但自身也有明显的缺点,比如易陷入局部极小值、收敛速度慢等.用混沌粒子群算法改进BP神经网络算法的基本思想是用混沌粒子群算法优化BP神经网络算法的权值和阈值,在粒子群算法中加入混沌元素,提高粒子群算法的全局搜索能力.对上证指数预测的结果表明改进后的预测方法,具有更好的准确性.     

Smoothed penalty algorithms for optimization of nonlinear models

We introduce an algorithm for solving nonlinear optimization problems with general equality and box constraints. The proposed algorithm is based on smoothing of the exact l ₁-penalty function and solving the resulting problem by any box-constraint optimization method. We introduce a general algorithm and present theoretical results for updating the penalty and smoothing parameter. We apply the algorithm to optimization problems for nonlinear traffic network models and report on numerical results for a variety of network problems and different solvers for the subproblems.     

Steepest descent preconditioning for nonlinear GMRES optimization

Steepest descent preconditioning is considered for the recently proposed nonlinear generalized minimal residual (N‐GMRES) optimization algorithm for unconstrained nonlinear optimization. Two steepest descent preconditioning variants are proposed. The first employs a line search, whereas the second employs a predefined small step. A simple global convergence proof is provided for the N‐GMRES optimization algorithm with the first steepest descent preconditioner (with line search), under mild standard conditions on the objective function and the line search processes. Steepest descent preconditioning for N‐GMRES optimization is also motivated by relating it to standard non‐preconditioned GMRES for linear systems in the case of a standard quadratic optimization problem with symmetric positive definite operator. Numerical tests on a variety of model problems show that the N‐GMRES optimization algorithm is able to very significantly accelerate convergence of stand‐alone steepest descent optimization. Moreover, performance of steepest‐descent preconditioned N‐GMRES is shown to be competitive with standard nonlinear conjugate gradient and limited‐memory Broyden–Fletcher–Goldfarb–Shanno methods for the model problems considered. These results serve to theoretically and numerically establish steepest‐descent preconditioned N‐GMRES as a general optimization method for unconstrained nonlinear optimization, with performance that appears promising compared with established techniques. In addition, it is argued that the real potential of the N‐GMRES optimization framework lies in the fact that it can make use of problem‐dependent nonlinear preconditioners that are more powerful than steepest descent (or, equivalently, N‐GMRES can be used as a simple wrapper around any other iterative optimization process to seek acceleration of that process), and this potential is illustrated with a further application example. Copyright © 2012 John Wiley & Sons, Ltd.     

New mutation schemes for differential evolution algorithm and their application to the optimization of directional over-current relay settings

Differential evolution is a novel evolutionary approach capable of handling non-differentiable, nonlinear and multimodal objective functions. It has been consistently ranked as one of the best search algorithm for solving global optimization problems in several case studies. In the present study we propose five new mutation schemes for the basic DE algorithm. The corresponding versions are termed as MDE1, MDE2, MDE3, MDE4 and MDE5. These new schemes make use of the absolute weighted difference between the two points and instead of using a fixed scaling factor F, use a scaling factor following the Laplace distribution. The performance of the proposed schemes is validated empirically on a suit of ten benchmark problems having box constraints. Numerical analysis of results shows that the proposed schemes improves the convergence rate of the DE algorithm and also maintains the quality of solution. Efficiency of the proposed schemes is further validated by applying it to a real life electrical engineering problem dealing with the optimization of directional over-current relay settings. It is a highly constrained nonlinear optimization problem. A constraint handling mechanism based on repair methods is used for handling the constraints. Once again the simulation results show the compatibility of the proposed schemes for solving the real life problem.