Showing papers on "Nonlinear programming published in 1983"

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.

A numerically stable dual method for solving strictly convex quadratic programs

Abstract: An efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point. Its implementation utilizes the Cholesky and QR factorizations and procedures for updating them. The performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code (the principal motivation for the development of the algorithm). These computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available. The algorithm is also compared theoretically to the modified-simplex type dual methods of Lemke and Van de Panne and Whinston and it is illustrated by a numerical example.

Optimal process design under uncertainty

Abstract: A rigorous mathematical formulation is presented for the problem of optimal design under uncertainty. This formulation involves a nonlinear infinite programming problem in which an optimization is performed on the set of design and control variables, such that the inequality constraints of the chemical plant are satisfied for every parameter value that belongs to a specified polyhedral region. To circumvent the problem of infinite dimensionality in the constraints, an equivalence for the feasibility condition is established which leads to a max-min-max constraint. It is shown that if the inequalities are convex, only the vertices in the polyhedron need to be considered to satisfy this constraint. Based on this feature, an algorithm is proposed which uses only a small subset of the vertices in an iterative multiperiod design formulation. Examples are presented to illustrate the application to flexible design problems.

MINOS 5.0 User's Guide.

Abstract: : MINOS is a large-scale optimization system, for the solution of sparse linear and nonlinear programs. The objective function and constraists may be linear or nonlinear, or a mixture of both. The nonlinear functions must be smooth. Stable numerical methods are employed throughout. Features include a new basis package(for maintaining sparse LU factors of the basis matrix), automatic scaling of linear constraints, and automatic estimation of some or all gradients. Upper and lower bounds on the variables are handled efficiently. File formats for constraint and basis data are compatible with the industry MPS format. The source code is suitable for machines with a Fortran 66 or 77 compiler and at least 500K bytes of storage. (Author)

Recent Developments in Algorithms and Software for Trust Region Methods

Abstract: Trust region methods are an important class of iterative methods for the solution of nonlinear optimization problems. Algorithms in this class have been proposed for the solution of systems of nonlinear equations, nonlinear estimation problems, unconstrained and constrained optimization, nondifferentiable optimization, and large scale optimization. Interest in trust region methods derives, in part, from the availability of strong convergence results and from the development of software for these methods which is reliable, efficient, and amazingly free of ad-hoc decisions. In this paper we survey the theoretical and practical results available for trust region methods and discuss the relevance of these results to the implementation of trust region methods.

User's Guide for SOL/NPSOL: A Fortran Package for Nonlinear Programming.

Abstract: : This report forms the user's guide of SOL/NPSOL, a set of Fortran subroutines designed to minimize an arbitrary smooth function subject to constraints, which may include simple bounds on the variables, linear constraints and smooth nonlinear constraints (NPSOL may also be used for uncontrained, bound-constrained and linearly constrained optimization.) The user must provide subroutines that define the objective and constraint functions and their gradients. All matrices are treated as dense, and hence NPSOL is not intended for large sparse problems. NPSOL uses a sequential quadratic programming (SQP) algorithm, in which the search direction is the solution of a quadratic programming (QP) subproblem. The algorithm treats bounds, linear constraints and nonlinear constraints separately. The Hessian of each QP subproblem is a positive-definite quasi-Newton approximation to the Hessian of an augmented Lagrangian function. The steplength at each iteration is required to produce a sufficient decrease in an augmented Lagrangian merit function. Each QP subproblem is solved using a quadratic programming package with several features that improve the efficiency of an SQP algorithm.

A Comparative Performance Evaluation of 27 Nonlinear Programming Codes

Abstract: The numerical performance of 27 computer programs, which are all designed to solve the general constrained nonlinear optimization problem, is to be evaluated. In contrast to Schittkowski [34], where besides of one exception, the same codes are compared on randomly generated test problems, the test examples are now given by the 115 hand-selected and real life problems published in Hock and Schittkowski [19]. The different type of the test examples requires the development of a special evaluation system based on priority theory. Detailed numerical results are presented allowing a quantitative comparison of the performance criteria efficiency and reliability.

Some Guidelines for Constructing Exact D-Optimal Designs on Convex Design Spaces

Abstract: Some problems unique to the construction of N-point D-optimal designs on convex design spaces are considered. Multiple-point augmentation and exchange algorithms are shown to be more costly and less efficient than the analogous single-point procedures. Moreover, some recommendations for improving single-point methods are given. Finally, empirical evidence is found that supports the Galil and Kiefer option for constructing initial designs and Powell's optimization method for design augmentation.

An Algorithm for Solving the General Bilevel Programming Problem

Abstract: The conflict that naturally arises in a hierarchical system can often be modeled as a multistage optimization problem. This paper considers the sequential uncooperative problem in which two decision makers wish to maximize their own objective functions over a feasible region defined by interactive strategy sets. The resultant problem is known as the bilevel program. Such programs are inherently nonconvex and resistant to standard nonlinear programming solution techniques such as piecewise linearization and convex underestimating envelopes. Alternatively, a grid search algorithm is offered which exhibits the desirable property of monotonicity. The algorithm is based on two sets of necessary conditions previously developed and combined here to provide an operational check for stationarity and local optimality. Potential solutions are obtained from a parameterized master program whose feasible region approximates that of the original problem. As the one-dimensional parameter is varied over the unit interval ...

Optimum sensitivity derivatives of objective functions in nonlinear programming

Abstract: The feasibility of eliminating second derivatives from the input of optimum sensitivity analyses of optimization problems is demonstrated. This elimination restricts the sensitivity analysis to the first-order sensitivity derivatives of the objective function. It is also shown that when a complete first-order sensitivity analysis is performed, second-order sensitivity derivatives of the objective function are available at little additional cost. An expression is derived whose application to linear programming is presented.

Computing Forward-Difference Intervals for Numerical Optimization

Abstract: When minimizing a smooth nonlinear function whose derivatives are not available, a popular approach is to use a gradient method with a finite-difference approximation substituted for the exact gradient In order for such a method to be effective, it must be possible to compute “good” derivative approximations without requiring a large number of function evaluations Certain “standard” choices for the finite-difference interval may lead to poor derivative approximations for badly scaled problems We present an algorithm for computing a set of intervals to be used in a forward-difference approximation of the gradient

Solution of a statistical optimization problem by rearrangement methods

Abstract: Inequalities for the rearrangement of functions are applied to obtain a solution of a statistical optimization problem. This optimization problem arises in situations where one wants to describe the influence of stochastic dependence on a statistical problem.

ε-Optimal solutions in nondifferentiable convex programming and some related questions

Abstract: In this paper we present e-optimality conditions of the Kuhn-Tucker type for points which are within e of being optimal to the problem of minimizing a nondifferentiable convex objective function subject to nondifferentiable convex inequality constraints, linear equality constraints and abstract constraints. Such e-optimality conditions are of interest for theoretical consideration as well as from the computational point of view. Some illustrative applications are made. Thus we derive an expression for the e-subdifferential of a general convex ‘max function’. We also show how the e-optimality conditions given in this paper can be mechanized into a bundle algorithm for solving nondifferentiable convex programming problems with linear inequality constraints.

An Approach to the Optimal Positioning of a New Product

Abstract: This study examines the problem faced by a firm which wishes to position a new choice object in an existing product class. It is assumed that both the consumer and the firm are involved in a two-stage decision process. The consumer first decides on his budget for the product class. He then evaluates, within the product class, that subset of competing objects which have prices approximately equal to his budget constraint. This evaluation is performed through a weighted multi-attribute utility model. The product classes considered here are ones for which the consumer has a finite ideal level on each attribute. The consumer is hypothesized to choose, without error, that object which is closest to his ideal. Different individuals are assumed to be heterogeneous in both attribute weights and ideal levels. The firm is assumed to first identify, in the attribute space which contains ideal points and competing objects, promising product positions which would attract a large number of consumers. It then evaluates these positions in terms of costs and resulting profits. The problem of identifying the promising positions, given information on a sample of individuals, is formulated as a Mixed Integer Nonlinear Program. Due to the inability of such a program to solve even small sample problems, the spatial properties of the problem are examined. An exact solution algorithm which is computationally feasible for small samples is developed. It is based on an examination of intersections of indifference hyperellipsoids. For larger sample problems an efficient heuristic which is an extension of the random point search used in nonlinear programming is provided. It involves random line search procedures for our noncontinuous-type problem. The positioning approach and the heuristic are illustrated in a simulated positioning problem in the small car market.

Interactive computer programs for fuzzy linear programming with multiple objectives

Abstract: In this article, we present interactive computer programs fof solving fuzzy linear programming problems with multiple objectives. Through the use of five types of membership functions including non-linear functions, the fuzzy of imprecise goals of the decision maker are quantified. Although the formulated problem becomes a nonlinear programming problem, it can be reduced to a set of linear inequalities if some variable is fixed. Based on this idea, we propose a new method by combined use of bisection method and linear programming method. On the basis of the proposed method, FORTRAN programs that run in conversational mode are developed to implement man-machine interactive procedures. The commands in our programs and major prompt messages are also explained. An illustrative numerical example for the interactive processes is demonstrated together with the computer outputs

Generalized Convexity and Concavity Properties of the Optimal Value Function in Parametric Nonlinear Programming.

Abstract: : This paper considers generalized convexity and concavity properties of the optimal value function f* for the general parametric optimization problem P(e) of the form min x sub f (x,e) s.t. x epsilon R(e). Many results on convexity and concavity characterizations of f* were presented by the authors in a previous paper. Such properties of f* and the solution map S* form an important part of the theoretical basis for sensitivity, stability and parametric analysis in mathematical optimization. The authors give sufficient conditions for several types of generalized convexity and concavity of f*, in terms of respective generalized convexity and concavity assumptions on f and convexity and concavity assumptions on the feasible region point-to-set map R. Specializations of these results to the general parametric inequality-equality constrained nonlinear programming problem and its right-hand-side version are provided. (Author)

Optimal Design of Linear and Nonlinear Vibration Absorbers for Damped Systems

Abstract: Nonlinear programming techniques are applied to obtain optimal tuning and damping parameters for dynamic absorbers The optimization has been carried out for damped as well as undamped primary systems It is found that optimal tuning parameters, obtained with the goal of minimizing the main mass maximum displacement, undergo small changes as damping is introduced into the main system The use of other objective functions, such as minimizing maximum velocity, or mean-squared motion to white noise excitation lead to more significant changes in optimal parameter values It is shown, on the basis of approximate solutions to the nonlinear absorber problem, that only small improvements in steady state response are obtained using hardening or softening coupling springs

Identifying Redundant Constraints and Implicit Equalities in Systems of Linear Constraints

Abstract: Redundant constraints are constraints that can be omitted from a system of linear constraints without changing the feasible region. Implicit equalities are inequality constraints that can be replaced by equalities without changing the feasible region. We prove some theorems concerning the identification of both kinds of constraints. Based on these theorems two methods are developed that allow identification of all redundant constraints and all implicit equalities. Computational experience on small problems indicates a behavior at least competitive with other methods in this field.

Sufficiency of exact penalty minimization

Abstract: By employing a recently obtained error bound for differentiable convex inequalities, it is shown that, under appropriate constraint qualifications, a minimum solution of an exact penalty function for a single value of the penalty parameter which exceeds a certain threshold, is also a solution of the convex program associated with the penalty function. No a priori assumption is made regarding the solvability of the convex program. If such a solvability assumption is made, then we show that a threshold value of the penalty parameter can be used which is smaller than both the above-mentioned value and that of Zangwill. These various threshold values of the penalty parameter also apply to the well-known big-M method of linear programming.

Nonlinear 0-1 Programming: II. Dominance Relations and Algorithms. Revision.

Abstract: : A nonlinear 0-1 program can be restated as a multilinear 0-1 program, which in turn is known to be equivalent to a linear 0-1 program with generalized covering (g.c.) inequalities. In a companion paper 6 we have defined a family of linear inequalities that contains more compact (smaller cardinality) linearizations of a multilinear 0-1 program than the one based on the g.c. inequalities. In this paper we analyze the dominance relations between inequalities of the above family. In particular, we give a criterion that can be checked in linear time, for deciding whether a g.c. inequality can be strengthened by extending the cover from which it was derived. We then describe a class of algorithms based on these results and discuss our computational experience. We conclude that the g.c. inequalities can be strengthened most of the time an extent that increases with problem density. In particular, the algorithm using the strengthening procedure outperforms the one using only g.c. inequalities whenever the number of nonlinear terms per constraint exceeds about 12-15, and the difference in their performance grows with the number of such terms. (Author)

An ellipsoid algorithm for nonlinear programming

Abstract: We investigate an ellipsoid algorithm for nonlinear programming. After describing the basic steps of the algorithm, we discuss its computer implementation and present a method for measuring computational efficiency. The computational results obtained from experimenting with the algorithm are discussed and the algorithm's performance is compared with that of a widely used commercial code.

Optimal Operations Planning in a Large Hydro-Thermal Power System

Abstract: In this paper a realistic model of seasonal production planning in the Swedish State Power Board (SSPB) System is described. This model properly accounts for the nonlinearities introduced by variable head losses and state dependent control constraints in the hydro system. The seasonal hydro-thermal problem exhibits certain complex features that prevent the use of existing nonlinear programming theory for its solution. The paper presents the development of a method of solution which recast the problem into a network flow formulation and employs network flow algorithms. This paper presents the latest developments in the area of network flow modelling for operations planning at the SSPB. For implementation of this scheme, to facilitate its use by operational engineers an information system is set up which will enable efficient handling of data bases and computational routines.

Revisions of constraint approximations in the successive QP method for nonlinear programming problems

Abstract: In the last few years the successive quadratic programming methods proposed by Han and Powell have been widely recognized as excellent means for solving nonlinea programming problems. However, there remain some questions about their linear approximations to the constraints from both theoretical and empirical points of view. In this paper, we propose two revisions of the linear approximation to the constraints and show that the directions generated by the revisions are also descent directions of exact penalty functions of nonlinear programming problems. The new technique can cope better with bad starting points than the usual one.

Feasible specifications in azeotropic distillation

Abstract: Feasible operating conditions are obtained for an azeotropic distillation tower using a nonlinear programming algorithm. The boil-up rate, fractional recovery of product, and bottoms purities of entrainer and by-product are adjusted to locate an overhead vapor stream that condenses into two liquid phases, but is in equilibrium with a single liquid phase on the top tray. A new objective function is introduced and minimized, subject to inequality constraints, using Powell's algorithm (1977). Results are obtained for dehydration of alcohol with benzene.

Multicriteria spatial price equilibrium network design: theory and computational results

Abstract: In this paper we consider the problem of determining the optimal design of a transportation network using a vector valued criterion function when the flow pattern is assumed to correspond to a spatial price equilibrium. This problem arises in rail and freight network design, where the spatial price equilibrium is a better behavioral description than the Wardropian user equilibrium characteristic of urban transportation applications. We describe two alternative heuristic solution techniques for the multicriteria spatial price equilibrium network design problem. The first is based on iteration between a pure spatial price equilibrium model and a vector optimization model with only nonnegativity constraints. The second solution technique employs the Hooke and Jeeves algorithm for nonlinear programming to solve a vector optimization model with implicit constraints guaranteeing a spatial price equilibrium flow pattern. In these solution procedures, rather than represent the equilibrium problem as a mathematical program, as is normally done for the Wardropian traffic assignment problems used in urban applications, we employ an original nonlinear complementarity formulation of the spatial price equilibrium problem written entirely in terms of nodal and arc variables and solved extremely efficiently through the iterative application of a linear complementarity algorithm. The nonlinear complementarity formulation allows us to address problems with asymmetric transportation cost, commodity demand and commodity supply functions without the specialized diagonalization/relaxation algorithms required by other approaches.