Showing papers on "Nonlinear programming published in 1982"

Generalized equations and their solutions, part II: Applications to nonlinear programming

Abstract: We prove that if the second-order sufficient condition and constraint regularity hold at a local minimizer of a nonlinear programming problem, then for sufficiently smooth perturbations of the constraints and objective function the set of local stationary points is nonempty and continuous; further, if certain polyhedrality assumptions hold (as is usually the case in applications), then the local minimizers, the stationary points and the multipliers all obey a type of Lipschitz condition.

Quantitative Analysis for Management

Abstract: CHAPTER 1 Introduction to Quantitative Analysis 1 CHAPTER 2 Probability Concepts and Applications 23 CHAPTER 3 Decision Analysis 69 CHAPTER 4 Regression Models 117 CHAPTER 5 Forecasting 157 CHAPTER 6 Inventory Control Models 199 CHAPTER 7 Linear Programming Models: Graphical and Computer Methods 255 CHAPTER 8 Linear Programming Modeling Applications:With Computer Analyses in Excel and QM for Windows 311 CHAPTER 9 Linear Programming: The Simplex Method 351 CHAPTER 10 Transportation and Assignment Models 409 CHAPTER 11 Integer Programming, Goal Programming, and Nonlinear Programming 469 CHAPTER 12 Network Models 515 CHAPTER 13 Project Management 543 CHAPTER 14 Waiting Lines and Queuing Theory Models 585 CHAPTER 15 Simulation Modeling 625 CHAPTER 16 Markov Analysis 669 CHAPTER 17 Statistical Quality Control 699 CD-ROM MODULES 1 Analytic Hierarchy Process M1-1 2 Dynamic Programming M2-1 3 Decision Theory and the Normal Distribution M3-1 4 Game Theory M4-1 5 Mathematical Tools: Determinants and Matrices M5-1 6 Calculus-Based Optimization M6-1

Introduction to Management Science

Abstract: 1. Management Science The Management Science Approach to Problem Solving Model Building: Break-Even Analysis Computer Solution Management Science Modeling Techniques Business Usage of Management Science Techniques Management Science Models in Descision Support Systems 2. Linear Programming: Model Formulation and Graphical Solution Model Formulation A Maximization Model Example Graphical Solutions of Linear Programming Methods A Minimization Model Example Irregular Types of Linear Programming Problems Characteristics of Linear Programming Problems 3. Linear Programming: Computer Solution and Sensitivity Analysis Computer Solution Sensitivity Analysis 4. Linear Programming: Modeling Examples A Product Mix Example A Diet Example An Investment Example A Marketing Example A Transportation Example A Blend Example A Multiperiod Scheduling Example A Data Envelopment Analysis Example 5. Integer Programming Integer Programming Models Integer Programming Graphical Solution Computer Solution of Integer Programming Problems with Excel and QM for Windows 0-1 Integer Programming Modeling Examples 6. Transportation, Transshipment, and Assignment Problems The Transportation Model Computer Solution of a Transportation Problem The Transshipment Model The Assignment Model Computer Solution of the Assignment Problem 7. Network Flow Models Network Components The Shortest Route Problem The Minimal Spanning Tree Problem The Maximal Flow Problem 8. Project Management The Elements of Project Management CPM/PERT Probabilistic Activity Times Microsoft Project Project Crashing and Time-Cost Trade-Off Formulating the CPM/PERT Network as a Linear Programming Model 9. Multicriteria Decision Making Goal Programming Graphical Interpretation of Goal Programming Computer Solution of Goal Programming Problems with QM for Windows and Excel The Analytical Hierarchy Process Scoring Model 10. Nonlinear Programming Nonlinear Profit Analysis Constrained Optimization Solution of Nonlinear Programming Problems with Excel A Nonlinear Programming Model with Multiple Constraints Nonlinear Model Examples 11. Probability and Statistics Types of Probability Fundamentals of Probability Statistical Independence and Dependence Expected Value The Normal Distribution 12. Decision Analysis Components of Decision Making Decision Making without Probabilities Decision Making with Probabilities Decision Analysis with Additional Information Utility 13. Queuing Analysis Elements of Waiting Line Analysis The Single-Server Waiting Line System Undefined and Constant Service times Finite Queue Length Finite Calling Population The Multiple-Server Waiting Line Additional Types of Queuing Systems 14. Simulation The Monte Carlo Process Computer Simulation with Excel Spreadsheets Simulation of a Queuing System Continuous Probability Distributions Statistical Analysis of Simulation Results Crystal Ball Verification of the Simulation Model Areas of Simulation Application 15. Forecasting Forecasting Components Time Series Methods Forecast Accuracy Time Series Forecasting Using Excel Time Series Forecasting Using QM for Windows Regression Methods 16. Inventory Management Elements of Inventory Management Inventory Control Systems Economic Order Quantity Models The Basic EOQ Model The EOQ Model with Noninstantaneous Receipt The EOQ Model with Shortages EOQ Analysis with QM for Windows EOQ Analysis with Excel and Excel QM Quantity Discounts Reorder Point Determining Safety Stocks Using Service Levels Order Quantity for a Periodic Inventory System Appendix A Normal Table Chi-Square Table Appendix B Setting Up and Editing a Spreadsheet Appendix C The Poisson and Exponential Distributions Solutions to Selected Odd-Numbered Problems Glossary Index Photo Credits CD-ROM Modules

A projected Lagrangian algorithm and its implementation for sparse nonlinear constraints

Abstract: An algorithm is described for solving large-scale nonlinear programs whose objective and constraint functions are smooth and continuously differentiable The algorithm is of the projected Lagrangian type, involving a sequence of sparse, linearly constrained subproblems whose objective functions include a modified Lagrangian term and a modified quadratic penalty function

The nonlinear programming method of Wilson, Han, and Powell with an augmented Lagrangian type line search function

Abstract: The paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms. This study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively, belong to the most efficient and reliable nonlinear programming algorithms available at present. The purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail. In Part 1, the exactL 1-penalty function of Han and Powell is replaced by a differentiable augmented Lagrange function for the line search computation to the able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution. In Part 2, the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency, and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblems.

On the Convergence of a Sequential Quadratic Programming Method with an Augmented Lagrangian Line Search Functions.

Abstract: Sequential quadratic programming methods as developed by Wilson, Han, and Powell have gained considerable attention in the last few years mainly because of their outstanding numerical performance. Although the theoretical convergence aspects of this method and its various modifications have been investigated in the literature, there still remain some open questions which will be treated in this paper. The convergence theory to be presented, takes into account the additional variable introduced in the quadratic programming subproblem to avoid inconsistency, the one-dimensional minimization procedure, and, in particular, an “ active set” strategy to avoid the recalculation of unnecessary gradients. This paper also contains a detailed mathematical description of a nonlinear programming algorithm which has been implemented by the author. the usage of the code and detailed numerical test results are presented in [5].

Differential properties of the marginal function in mathematical programming

Abstract: This paper consists in a study of the differential properties of the marginal or perturbation function of a mathematical programming problem where a parameter or perturbation vector is present. Bounds for the Dini directional derivatives and estimates for the Clarke generalized gradient are obtained for the marginal function of the mathematical program neither assumed convex in its variables or in its parameters. This study generalizes some previously published results on this subject for the special case of right-hand side parameters or perturbations.

Nonlinear Optimization by Successive Linear Programming

Abstract: Successive Linear Programming SLP, which is also known as the Method of Approximation Programming, solves nonlinear optimization problems via a sequence of linear programs. This paper reports on promising computational results with SLP that contrast with the poor performance indicated by previously published comparative tests. The paper provides a detailed description of an efficient, reliable SLP algorithm along with a convergence theorem for linearly constrained problems and extensive computational results. It also discusses several alternative strategies for implementing SLP. The computational results show that SLP compares favorably with the Generalized Reduced Gradient Code GRG2 and with MINOS/GRG. It appears that SLP will be most successful when applied to large problems with low degrees of freedom.

Nonlinear programming: a historical view

Abstract: A historical survey of the origins of nonlinear programming is presented with emphasis placed on necessary conditions for optimality. The mathematical sources for the work of Karush, John, Kuhn, and Tucker are traced and compared. Their results are illustrated by duality theorems for nonlinear programs that antedate the modern development of the subject.

A model algorithm for composite nondifferentiable optimization problems

Abstract: Composite functions ϕ(x)=f(x)+h(c(x)), where f and c are smooth and h is convex, encompass many nondifferentiable optimization problems of interest including exact penalty functions in nonlinear programming, nonlinear min-max problems, best nonlinear L 1, L 2 and L ∞ approximation and finding feasible points of nonlinear inequalities. The idea is used of making a linear approximation to c(x) whilst including second order terms in a quadratic approximation to f(x). This is used to determine a composite function ψ which approximates ϕ(x) and a basic algorithm is proposed in which ψ is minimized on each iteration. If the technique of a step restriction (or trust region) is incorporated into the algorithm, then it is shown that global convergence can be proved. It is also described briefly how the above approximations ensure that a second order rate of convergence is achieved by the basic algorithm.

Nonlinear programming via an exact penalty function: Asymptotic analysis

Abstract: In this paper we consider the final stage of a ‘global’ method to solve the nonlinear programming problem. We prove 2-step superlinear convergence. In the process of analyzing this asymptotic behavior, we compare our method (theoretically) to the popular successive quadratic programming approach.

On regularity conditions in mathematical programming

Abstract: Constraint qualifications are revisited, once again. These conditions are shown to be reminiscent of transversality theory. They are used as a useful tool for computing tangent cones, by the means of generalized inverse function theorems. The finite dimensional case is given a special treatment as the results are nicer and simpler in this case. Some remarks on the nondifferentiable case are also presented.

Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming

Abstract: For finite-dimensional optimization problems with locally Lipschitzian equality and inequality constraints and also an abstract constraint described by a closed set, a Lagrange multiplier rule is derived that is sharper is in some respects than the ones of Clarke and Hiriart-Urruty. The multiplier vectors provided by this rule are given meaning in terms of the generalized subgradient set of the optimal value function in the problem with respect to perturbational parameters. Bounds on subderivatives of the optimal value function are thereby obtained and in certain cases the existence of ordinary directional derivatives.

The convergence of diagonalization algorithms for asymmetric network equilibrium problems

Abstract: The purpose of this paper is to provide sufficient conditions for the convergence of a certain class of algorithms (diagonalization algorithms) for equilibrium traffic assignment problems with link user cost functions that may depend on the flows of several modes on all the links of the network and have asymmetric Jacobian matrices. These problems do not have equivalent convex cost minimization formulations and may not be solved with the adaptation of a suitable nonlinear programming method. (TRRL)

Submodular functions and convexity.

Abstract: In “continuous” optimization convex functions play a central role. Besides elementary tools like differentiation, various methods for finding the minimum of a convex function constitute the main body of nonlinear optimization. But even linear programming may be viewed as the optimization of very special (linear) objective functions over very special convex domains (polyhedra). There are several reasons for this popularity of convex functions: Convex functions occur in many mathematical models in economy, engineering, and other sciencies. Convexity is a very natural property of various functions and domains occuring in such models; quite often the only non-trivial property which can be stated in general.

The nonlinear programming method of Wilson, Han, and Powell with an augmented Lagrangian type line search function: Part 2: An efficient implementation with linear least squares subproblems

Abstract: SummaryThe paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms. This study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively, belong to the most efficient and reliable nonlinear programming algorithms available at present. The purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail. In Part 1, the exactL1-penalty function of Han and Powell is replaced by a differentiable augmented Lagrange function for the line search computation to the able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution. In Part 2, the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency, and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblems.

Solution Algorithms for Network Equilibrium Models with Asymmetric User Costs

Abstract: We consider algorithms proposed for solving the fixed demand user optimized network equilibrium problem with asymmetric user costs. Because the Jacobian matrix for the costs is asymmetric, no known equivalent convex optimization problem exists and alternative solution methods to nonlinear programming techniques must be sought. The purpose of this paper is to compare the performance of several existing methods for determining the equilibrium network flows of a small realistic network model in which intersection controls are incorporated which lead to particularly asymmetric cost functions.

Obtaining the Layout of Water Distribution Systems

Abstract: An essential first step in the design of a municipal water distribution system is the determination of the locational placement or layout of the links of pipe that will form the system. A two-level hierarchically integrated system of models is developed for the layout of both single and multiple source water distribution systems. The first level, a nonlinear programming model, selects an economical tree layout for the major pipe links. The second level, an integer programming model, chooses the loop- forming links to add to the first level tree layout in order to minimize the cost of providing a specified level of reliability in case of failure of the larger first level links. The system of models is applied to an example two-source water distribution system layout problem.

Properties and Solution Methods for Large Location—Allocation Problems

Abstract: Location-allocation with lp distances is studied. It is shown that this structure can be expressed as a concave minimization programming problem. Since concave minimization algorithms are not yet well developed, five solution methods are developed which utilize the special properties of the location-allocation problem. Using the rectilinear distance measure, two of these algorithms achieved optimal solutions in all 102 test problems for which solutions were known. The algorithms can be applied to much larger problems than any existing exact methods.

Nonlinear programming via an exact penalty function : Global analysis

Abstract: In this paper we motivate and describe an algorithm to solve the nonlinear programming problem. The method is based on an exact penalty function and possesses both global and superlinear convergence properties. We establish the global qualities here (the superlinear nature is proven in [7]). The numerical implementation techniques are briefly discussed and preliminary numerical results are given.

Bibliography in fractional programming

Abstract: A bibliography in fractional programming is provided which contains 551 references. It was attempted to include all publications in this area of nonlinear programming as they have appeared in more than 45 years now.

Decomposition strategy for designing flexible chemical plants

Abstract: One of the main computational problems faced in the optimal design of flexible chemical plants with multi-period operation is the large number of decision variables that are involved in the corresponding nonlinear programming formulation. To overcome this difficulty, a decomposition technique based on a projectionrestriction strategy is suggested to exploit the block-diagonal structure in the constraints. Successful application of this strategy requires an efficient method to find an initial feasible point, and the extension of current equation ordering algorithms for adding systematically inequality constraints that become active. General trends in the performance of the proposed decomposition technique are presented through an example.

A linear finite element approach to the solution of the variational inequalities arising in contact problems of structural dynamics

Abstract: The present paper deals with the theoretical and numerical treatment of dynamic unilateral problems. The governing equations are formulated as an equivalent variational inequality expressing D' Alembert's principle in its inequality form. The discretization with respect to time and space leads to a static nonlinear programming problem which is solved by an appropriate algorithm. Some properties of dynamic unilateral problems are outlined and the influence of several parameters on the solution is investigated by means of numerical examples.

A new augmented Lagrangian function for inequality constraints in nonlinear programming problems

Abstract: In this paper, a new augmented Lagrangian function is introduced for solving nonlinear programming problems with inequality constraints. The relevant feature of the proposed approach is that, under suitable assumptions, it enables one to obtain the solution of the constrained problem by a single unconstrained minimization of a continuously differentiable function, so that standard unconstrained minimization techniques can be employed. Numerical examples are reported.

Reduced-Order Optimal Feedback Control Law Synthesis for Flutter Suppression

Abstract: A method of synthesizing reduced-order optimal feedback control laws for a high-order system is developed. A nonlinear programming algorithm is employed to search for the control law design variables that minimize a performance index defined by a weighted sum of mean-square steady-state responses and control inputs. An analogy with the linear quadratic Gaussian solution is utilized to select a set of design variables and their initial values. An input-noise adjustment procedure is used in the design algorithm to improve the stability margins of the system. The method is applied to the synthesis of an active flutter-suppression control law for an aeroelastic wind tunnel wing model. The reduced-order control law is compared with the corresponding full-order control law. The study indicates that by using the present algorithm, near optimal low-order flutter suppression control laws with good stability margins can be synthesized.

Minimal representation of convex polyhedral sets

Abstract: A system of linear inequality and equality constraints determines a convex polyhedral set of feasible solutionsS. We consider the relation of all individual constraints toS, paying special attention to redundancy and implicit equalities. The main theorem derived here states that the total number of constraints together determiningS is minimal if and only if the system contains no redundant constraints and/or implicit equalities. It is shown that the existing theory on the representation of convex polyhedral sets is a special case of the theory developed here.

An Efficient Method for the Multi-Depot Location-Allocation Problem

Abstract: This paper describes a procedure based on large-scale nonlinear programming for solving the multidepot location-allocation problem. Both the location of depots and the allocation of customers are allowed to vary simultaneously. Numerical experience on a 5 x 50 and a 10 x 50 example is described, and possible extensions to the basic model are discussed.

An efficient method for the multidepot location allocation problem

Abstract: This paper describes a procedure based on large scale non linear programming for solving the multi depot location allocation problem. Both the location of depots and the allocation of customers are allowed to vary simultaneously. Numerical experience on a 5 multiplied by 50 and a 10 multiplied by 50 example is described, and possible extensions to the basic model are discussed.