A new discrete filled function algorithm for discrete global optimization

A definition of the discrete filled function is given in this paper. Based on the definition, a discrete filled function is proposed. Theoretical properties of the proposed discrete filled function are investigated, and an algorithm for discrete global optimization is developed from the new discrete filled function. The implementation of the algorithms on several test problems is reported with satisfactory numerical results.     

Supernode processing of mixed-integer models

This paper discusses processing software for large scale mixed-integer optimization models. The software is part of the Mathematical OPtimization System MOPS [18] which contains algorithms for solving large-scale LP and mixed-integer programs. The processing techniques are implemented in such a way that they can be applied not only initially but also during the branch-and-bound algorithm.This paper discusses only a subset of the processing techniques included in MOPS. Algorithmic and software design aspects of the branch-and-bound process are not part of this paper.Dedicated to Professor George B. Dantzig on the occasion of his eightieth birthday.     

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.     

Discrete global descent method for discrete global optimization and nonlinear integer programming

A novel method, entitled the discrete global descent method, is developed in this paper to solve discrete global optimization problems and nonlinear integer programming problems. This method moves from one discrete minimizer of the objective function f to another better one at each iteration with the help of an auxiliary function, entitled the discrete global descent function. The discrete global descent function guarantees that its discrete minimizers coincide with the better discrete minimizers of f under some standard assumptions. This property also ensures that a better discrete minimizer of f can be found by some classical local search methods. Numerical experiments on several test problems with up to 100 integer variables and up to 1.38 × 10¹⁰⁴ feasible points have demonstrated the applicability and efficiency of the proposed method.     

Facility location models for distribution planning

A strategic issue which is of interest to distribution planners is where to best site warehouses. Model formulations, and solution approaches, which address the issue vary widely in terms of mathematical and computational complexity. This paper reviews some of the significant contributions which have been made to the relevant and current state of knowledge.     

On stopped decision processes with discrete time parameter

A non-stationary stopped decision process is investigated under rather weak convergence assumptions on the expected total rewards. Sufficient conditions are given for the approximation of the maximal conditional expected rewards from infinite stage play by the maximal conditional expected rewards from finite stage play. General criteria of optimality are derived. The results are essentially based on two lemmas given in this paper. The existence of optimal plans is established using results of non-stationary dynamic programming.     

Square-root variable-metric methods for minimization

Variable-metric methods are presented which do not need an accurate one-dimensional search and eliminate roundoff error problems which can occur in updating the metric for large-dimension systems. The methods are based on updating the square root of the metric, so that a positive-definite metric always results. The disadvantage of intentionally relaxing the accuracy of the one-dimensional search is that the number of iterations (and hence, gradient evaluations) increases. For problems involving a large number of variables, the square-root method is presented in a triangular form to reduce the amount of computation. Also, for usual optimization problems, the square-root procedure can be carried out entirely in terms of the metric, eliminating storage and computer time associated with computations of the square root of the metric.     

Decision support systems for water management: The Lake Como case study

We review an application-oriented study on the management of Lake Como, a natural multipurpose reservoir in Northern Italy. The emphasis is on the Decision Support System that resulted from this study and which is now being used by the manager to take his decisions on the amount of water to be released each day from the lake. The decision support system is based on the optimal solutions of complex multiobjective mathematical programming problems.     

A novel approach based on recurrent neural networks applied to nonlinear systems optimization

This paper presents an efficient approach based on recurrent neural network for solving nonlinear optimization. More specifically, a modified Hopfield network is developed and its internal parameters are computed using the valid subspace technique. These parameters guarantee the convergence of the network to the equilibrium points that represent an optimal feasible solution. The main advantage of the developed network is that it treats optimization and constraint terms in different stages with no interference with each other. Moreover, the proposed approach does not require specification of penalty and weighting parameters for its initialization. A study of the modified Hopfield model is also developed to analyze its stability and convergence. Simulation results are provided to demonstrate the performance of the proposed neural network.     

On consistency of bounding operations in deterministic global optimization

This technical comment refers to the discussion of strong consistency of several bounding procedures in Lemma 2.1 and Proposition 2.1 of Ref. 1. A necessary clarification is given of the notion of convergence _q in Lemma 2.1, and a derivation of Proposition 2.1 is presented that includes a new and simple consistency proof of the classical bounding by convex envelopes used in many branch-and-bound procedures.     

Localization for discrete one-dimensional random word models

We consider Schrödinger operators in whose potentials are obtained by randomly concatenating words from an underlying set according to some probability measure ν on . Our assumptions allow us to consider models with local correlations, such as the random dimer model or, more generally, random polymer models. We prove spectral localization and, away from a finite set of exceptional energies, dynamical localization for such models. These results are obtained by employing scattering theoretic methods together with Furstenberg's theorem to verify the necessary input to perform a multiscale analysis.     

A piecewise linearization framework for retail shelf space management models

Managing shelf space is critical for retailers to attract customers and optimize profits. This article develops a shelf-space allocation optimization model that explicitly incorporates essential in-store costs and considers space- and cross-elasticities. A piecewise linearization technique is used to approximate the complicated nonlinear space-allocation model. The approximation reformulates the non-convex optimization problem into a linear mixed integer programming (MIP) problem. The MIP solution not only generates near-optimal solutions for large scale optimization problems, but also provides an error bound to evaluate the solution quality. Consequently, the proposed approach can solve single category-shelf space management problems with as many products as are typically encountered in practice and with more complicated cost and profit structures than currently possible by existing methods. Numerical experiments show the competitive accuracy of the proposed method compared with the mixed integer nonlinear programming shelf-space model. Several extensions of the main model are discussed to illustrate the flexibility of the proposed methodology.     

Demurrage costs optimization in ports using binary mathematical programming models

Summary SODA is a system by which the schedule of ships in a port can be found in order to minimize the demurrage and operating costs associated with shipping operations. SODA constructs and resolves binary programming models using shipping data (arrival dates, demurrage costs, berths to be used, etc.) and available berths in a planning horizon. The model is of an adjustable precision: major accuracy implies greater calculation time. SODA provides an optimal result to a complex problem in a resolution time which is generally better than that needed for a manual solution.     

Stochastic quasigradient methods for optimization of discrete event systems

In this paper, stochastic programming techniques are adapted and further developed for applications to discrete event systems. We consider cases where the sample path of the system depends discontinuously on control parameters (e.g. modeling of failures, several competing processes), which could make the computation of estimates of the gradient difficult. Methods which use only samples of the performance criterion are developed, in particular finite differences with reduced variance and concurrent approximation and optimization algorithms. Optimization of the stationary behavior is also considered. Results of numerical experiments and convergence results are reported.     

Solving the nonlinear least square problem: Application of a general method

An algorithm for solving the general nonlinear least-square problem is developed. An estimate for the Hessian matrix is constructed as the sum of two matrices. The first matrix is the usual first-order estimate used by the Gauss method, while the second matrix is generated recursively using a rank-one formula. Test results indicate that the method is superior to the standard Gauss method and compares favorably with other methods, especially for problems with nonzero residuals at the solution.This work was supported by the US Air Force under Contract No. F04701-73-C-0074.The author expresses his appreciation to Dr. H. E. Pickett and Dr. J. L. Searcy for their continuing support in the theoretical and practical development of the algorithm. The recursive method for generating the estimate of the Hessian matrix was developed jointly with Drs. Pickett and Searcy and is included here with their permission. The author would also like to acknowledge the contribution made by the stimulating environment of an optimal control seminar held at The Aerospace Corporation since 1970. Principle members of the seminar have been H. E. Pickett, J. L. Searcy, R. W. Reid, and the author.     

一种基于进化算法的非线性两层规划的快速全局优化方法

由于非线性两层规划具有非凸性、NP-难等计算困难，高效的算法并不多见。本文设计了一种新的进化算法，基于此进化算法提出了求解带有一重或多重下层的非线性两层规划的高效算法。该算法充分利用两层规划的结构特点。最后，给出了六个不同类型的算例，数值结果表明，本算法是快速和有效的。     

Optimization using spreadsheets on a microcomputer

This paper discusses the advantages of using spreadsheets for problem specification and report generation in optimization projects. It summarizes some of the mathematical programming software which is compatible with popular spreadsheets. A small production planning problem is used to illustrate the steps in input and processing of the results. Two programs are compared in detail.     

Optimization using spreadsheets on a microcomputer

This paper discusses the advantages of using spreadsheets for problem specification and report generation in optimization projects. It summarizes some of the mathematical programming software which is compatible with popular spreadsheets. A small production planning problem is used to illustrate the steps in input and processing of the results. Two programs are compared in detail.     

Accelerations for global optimization covering methods using second derivatives

Two improvements for the algorithm of Breiman and Cutler are presented. Better envelopes can be built up using positive quadratic forms. Better utilization of first and second derivative information is attained by combining both global aspects of curvature and local aspects near the global optimum. The basis of the results is the geometric viewpoint developed by the first author and can be applied to a number of covering type methods. Improvements in convergence rates are demonstrated empirically on standard test functions.Partially supported by an University of Canterbury Erskine grant.