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

JOURNAL OF GLOBAL OPTIMIZATION 2007年第3期37卷 357-379页

作者： Ng, Chi-Kong Li, Duan Zhang, Lian-Sheng Chinese Univ Hong Kong Dept Syst Engn & Engn Management Shatin Hong Kong Peoples R China Shanghai Univ Dept Math Shanghai 200436 Peoples R China

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 x 10(104) feasible points have demonstrated the applicability and efficiency of the proposed method.

关键词： discrete global descent method discrete global optimization nonlinear integer programming integer programming

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nonlinear integer programming for optimal allocation in stratified sampling

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1999年第3期116卷 667-680页

作者： Bretthauer, KM Ross, A Shetty, B Texas A&M Univ Dept Business Analysis & Res College Stn TX 77843 USA

A stratified random sampling plan is one in which the elements of the population are first divided into nonoverlapping groups, and then a simple random sample is selected from each group. In this paper, we focus on determining the optimal sample size of each group. We show that various versions of this problem can be transformed into a particular nonlinear program with a convex objective function, a single linear constraint, and bounded variables. Two branch and bound algorithms are presented for solving the problem. The first algorithm solves the transformed subproblems in the branch and bound tree using a variable pegging procedure. The second algorithm solves the subproblems by performing a search to identify the optimal Lagrange multiplier of the single constraint. We also present linearization and dynamic programming methods that can be used for solving the stratified sampling problem. Computational testing indicates that the pegging branch and bound algorithm is fastest for some classes of problems, and the linearization method is fastest for other classes of problems. (C) 1999 Elsevier Science B.V. All rights reserved.

关键词： nonlinear integer programming sampling

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A nonlinear integer programming Approach to Solving the Robust Parameter Design Optimization Problem

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QUALITY AND RELIABILITY ENGINEERING INTERNATIONAL 2016年第8期32卷 2859-2870页

作者： Ozdemir, Akin Cho, Byung Rae Clemson Univ Dept Ind Engn Clemson SC 29631 USA Clemson Univ Ind Engn Clemson SC 29631 USA

Robust parameter design is a widely implemented design methodology for continuous quality improvement by identifying optimal factor level settings with minimum product variation. However, apparent flaws surrounding the original version of robust parameter design have resulted in alternative approaches, of which response surface methodology using the central composite design, in particular, has drawn a great deal of attention. There is a large number of practical situations in which some or all of variables must be integers;however, the design space associated with the traditional central composite design is typically a bounded convex feasible set involving real numbers. The purpose of this paper is twofold. First, we discuss why the Box-Behnken design may be preferred over the central composite design and other three-level designs when maintaining constant or nearly constant prediction variance associated with a second-order model is crucial to integer-valued robust parameter design problems. Second, we lay out the foundation to show how the Box-Behnken design is transformed into a nonlinear integer programming framework. In this paper, we develop Box-Behnken design embedded nonlinear integer programming models, using the sequential quadratic integer programming and the Karush-Khun-Tucker conditions. Comparison studies of the proposed models and traditional counterparts are also conducted. It is believed that the proposed models have the potential to impact a wide range of engineering problems, ultimately leading to process and quality improvement. Copyright (C) 2016 John Wiley & Sons, Ltd.

关键词： Box-Behnken design response surface methodology nonlinear integer programming optimization

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Computing exact solution to nonlinear integer programming: Convergent Lagrangian and objective level cut method

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JOURNAL OF GLOBAL OPTIMIZATION 2007年第1期39卷 127-154页

作者： Li, D. Wang, J. Sun, X. L. Chinese Univ Hong Kong Dept Syst Engn & Engn Management Shatin Hong Kong Peoples R China Qingdao Univ Int Business Coll Dept Management Sci & Engn Qingdao 266071 Peoples R China Fudan Univ Dept Management Sci Sch Management Shanghai 200433 Peoples R China

In this paper, we propose a convergent Lagrangian and objective level cut method for computing exact solution to two classes of nonlinear integer programming problems: separable nonlinear integer programming and polynomial zero-one programming. The method exposes an optimal solution to the convex hull of a revised perturbation function by successively reshaping or re-confining the perturbation function. The objective level cut is used to eliminate the duality gap and thus to guarantee the convergence of the Lagrangian method on a revised domain. Computational results are reported for a variety of nonlinear integer programming problems and demonstrate that the proposed method is promising in solving medium-size nonlinear integer programming problems.

关键词： nonlinear integer programming Lagrangian duality object level cut separable nonlinear integer programming polynomial zero-one programming

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A Novel TRUST-TECH Guided Branch-and-Bound Method for nonlinear integer programming

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IEEE TRANSACTIONS ON SYSTEMS MAN CYBERNETICS-SYSTEMS 2015年第10期45卷 1361-1372页

作者： Chiang, Hsiao-Dong Wang, Tao Cornell Univ Sch Elect & Comp Engn Ithaca NY 14853 USA

nonlinear integer programming has not reached the same level of maturity as linear programming, and is still difficult to solve, especially for large-scale systems. Branch-and-bound (B&B) and its variants are widely used methods for integer programming, and numerical solutions obtained by them still can be far away from the global optimum. In this paper, we propose a novel approach to guide the deterministic/heuristic methods and the commercial solvers for nonlinear integer programming, and aim at improving the solution quality by taking advantage of transformation under stability-retraining equilibrium characterization (TRUST-TECH) method. Moreover, we examine the effectiveness by developing and simulating TRUST-TECH guided B&B and TRUST-TECH guided commercial solver(s), and compare their performance with that of the original methods/solvers (e.g., GAMS (General Algebraic Modeling System)/BARON, GAMS/SCIP, and LINDO (Linear, INteractive, Discrete Optimizer)/MINLP) and also with that of recentlyreported evolutionary-algorithm (EA)-based methods. Simulation results provide evidence that, the solution quality is substantially improved, and the global-optimal solutions are usually obtained after the application of TRUST-TECH. The proposed approach can be immediately utilized to guide other EA-based methods and commercial solvers which incorporate intelligent searching components.

关键词： Branch-and-bound (B&B) nonlinear integer programming transformation under stability-retraining equilibrium characterization (TRUST-TECH)

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Convergent Lagrangian and contour cut method for nonlinear integer programming with a quadratic objective function

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SIAM JOURNAL ON OPTIMIZATION 2006年第2期17卷 372-400页

作者： Li, D. Sun, X. L. Wang, F. L. Chinese Univ Hong Kong Dept Syst Engn & Engn Management Shatin Hong Kong Peoples R China Shanghai Univ Dept Math Shanghai 200444 Peoples R China

In this paper we present an efficient exact solution method for solving nonlinear separable integer programming problems with a quadratic objective function. The proposed method combines the Lagrangian dual method with a duality reduction scheme using contour cut. At each iteration of the algorithm, lower and upper bounds of the problem are determined by the Lagrangian dual search. To eliminate the duality gap, a novel cut-and-partition scheme is derived by exploring the special structure of the quadratic contour. The method finds an exact solution of the problem in a finite number of iterations. Computational results are reported for problems with up to 2000 integer variables. Comparison results with other methods are also presented.

关键词： nonlinear integer programming quadratic integer programming Lagrangian relaxation duality theory objective contour cut

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Exact Penalty Functions for nonlinear integer programming Problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2010年第3期145卷 479-488页

作者： Lucidi, S. Rinaldi, F. Univ Roma La Sapienza Dipartimento Informat & Sistemist Antonio Ruberti Rome Italy

In this work, we study exact continuous reformulations of nonlinear integer programming problems. To this aim, we preliminarily state conditions to guarantee the equivalence between pairs of general nonlinear problems. Then, we prove that optimal solutions of a nonlinear integer programming problem can be obtained by using various exact penalty formulations of the original problem in a continuous space.

关键词： nonlinear integer programming Continuous programming Exact penalty functions

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An enumerative algorithm framework for a class of nonlinear integer programming problems

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1997年第1期101卷 104-121页

作者： Djerdjour, M School of Commerce and Administration Laurentian University Ramsey Lake Road Sudbury Ontario P3E 2C6 Canada

This paper presents a framework for a branch and search algorithm for solving a class of general integer restricted, linearly constrained, quadratic integer programming problems where the objective function is a nonse... 详细信息

关键词： mathematical programming nonlinear integer programming dual surrogate problem enumeration tree eigenvalue

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A note on "A Continuous Approach to nonlinear integer programming"

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APPLIED MATHEMATICS AND COMPUTATION 2009年第6期215卷 2388-2389页

作者： Zhang, Guoqing Univ Windsor Dept Ind & Mfg Syst Engn Windsor ON N9B 3P4 Canada

Ge and Huang (1989) proposed an approach to transform nonlinear integer programming problems into nonlinear global optimization problems, which are then solved by the filled function transformation method. The approach has recently attracted much attention. This note indicates that the formulae to determine a penalty parameter in two fundamental theorems are incorrect, and presents the corrected formulae and revised theorems. (C) 2009 Elsevier Inc. All rights reserved.

关键词： nonlinear integer programming Global optimization Filled function method Penalty function

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A genetic algorithm for interval nonlinear integer programming problem

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COMPUTERS & INDUSTRIAL ENGINEERING 1996年第3-4期31卷 913-917页

作者： Yokota, T Gen, M Li, YX Kim, CE Dept. of Indust. and Syst. Eng. Ashikaga Institute of Technology Ashikaga 326 Japan Myongji University South Korea

In this paper, we formulate an optimal design of system reliability problem as a nonlinear integer programming problem with interval coefficients, transform it into a single objective nonlinear integer programming problem without interval coefficients, and solve it directly with keeping nonlinearity of the objective function by using Genetic Algorithms (GA). Also, we demonstrate the efficiency of this method with incomplete Fault Detecting and Switching (FDS) for allocating redundant units.

关键词： nonlinear integer programming interval coefficient genetic algorithms incomplete FDS

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