Showing papers on "Nonlinear programming published in 1985"

Branch and Bound Experiments in Convex Nonlinear Integer Programming

Abstract: The branch and bound principle has long been established as an effective computational tool for solving mixed integer linear programming problems. This paper investigates the computational feasibility of branch and bound methods in solving convex nonlinear integer programming problems. The efficiency of a branch and bound method often depends on the rules used for selecting the branching variables and branching nodes. Among others, the concepts of pseudo-costs and estimations are implemented in selecting these parameters. Since the efficiency of the algorithm also depends on how fast an upper bound on the objective minimum is attained, heuristic rules are developed to locate an integer feasible solution to provide an upper bound. The different criteria for selecting branching variables, branching nodes, and heuristics form a total of 27 branch and bound strategies. These strategies are computationally tested and the empirical results are presented.

CONOPT: A GRG code for large sparse dynamic nonlinear optimization problems

Abstract: The paper presents CONOPT, an optimization system for static and dynamic large scale nonlinearly constrained optimization problems The system is based on the GRG algorithm All computations involving the Jacobian of the constraints use sparse matrix algorithm from linear programming modified to take optimal advantage of the nonlinearity The paper presents the main features of the system, especially the inversion routines and their data structures, the dynamic setting of tolerances in Newton's algorithm, and the user features in the overall packaging Computational experience with some medium to large models is presented

On Converting Optimal Control Problems into Nonlinear Programming Problems

Abstract: Direct solutions of the optimal control problem are considered. Two discretization schemes are proposed which are based on the parameterization of the control functions and on the parameterization of the control and the state functions, leading to direct shooting and direct collocation algorithms, respectively. The former is advantageous for problems with unspecified final state, the latter for prescribed final state and especially for stiff problems. The sparsity of the Jacobian matrix of the constraints and the Hessian matrix of the Lagrangian must be exploited in the direct collocation method in order to be efficient. The great advantage of the collocation approach lies in the availability of analytical gradients.

Optimal simultaneous structural and control design of maneuvering flexible spacecraft

Abstract: An optimization problem for maneuvering flexible spacecraft is discussed wherein both structural parameters and active control forces are to be determined so that a specific cost functional is minimized. The problem is an application of the general theory of optimal control of parametric systems. For simplicity, only maneuvers from a specified initial state to a specified final state in a specified time interval are considered. Numerical examples are presented for single-axis slew maneuvers of a symmetric four-boom flexible structure. The mass and stiffness distributions of the booms are determined as part of the optimization problem.

Finite Algorithms in Optimization and Data Analysis

Abstract: Preface Table of Notation Some Results from Convex analysis Linear Programming Applications of linear Programming in Discrete Approximation Polyhedral Convex Functions Least Squares and Related Methods Some Applications to Non-Convex Problems Some Questions of Complexity and Performance Appendices References Index.

Optimal Load Balancing in a Multiple Processor System with Many Job Classes

Abstract: A loosely coupled multiprocessor system contains multiple processors which have their own local memories To balance the load among multiple processors is of fundamental importance in enhancing the performance of such a multiple processor system Probabilistic load balancing in a heterogeneous multiple processor system with many job classes is considered in this study The load balancing scheme is formulated as a nonlinear programming problem with linear constraints An optimal probabilistic load balancing algorithm is proposed to solve this nonlinear programming problem The proposed load balancing method is proven globally optimum in the sense that it results in a minimum overall average job response time on a probabilistic basis

Semi-Definite Matrix Constraints in Optimization

Abstract: Positive semi-definite matrix constraints arise in a number of optimization problems in which some or all of the elements of a matrix are variables, such as the educational testing and matrix modification problems. The structure of such constraints is developed, including expressions for the normal cone, feasible directions and their application to optimality conditions. A computational framework is given within which these concepts can be exploited and which permits the quantification of second order effects. The matrix of Lagrange multipliers in this formulation is shown to have an important relationship to the characterization of the normal cone. Modifications of various iterative schemes in nonlinear programming are considered in order to develop an effective algorithm for the educational testing problem, and comparative numerical experiments are described. It is shown that a particular choice of the penalty parameter for an $l_1 $ exact penalty function is appropriate for this type of problem. The be...

An Improved Successive Linear Programming Algorithm

Abstract: Successive Linear Programming SLP algorithms solve nonlinear optimization problems via a sequence of linear programs. They have been widely used, particularly in the oil and chemical industries, beginning with their introduction by Griffith and Stewart of Shell Development Company in 1961. Since then, several applications and variants of SLP have appeared, the most recent being the SLPR algorithm described in this journal in 1982 Palacios-Gomez et al.. SLP procedures are attractive because they are fairly easy to implement if an efficient, flexible LP code is available, can solve nonseparable as well as separable problems, can be applied to as large a problem as the LP code can handle often thousands of constraints and variables, and have been successful in many practical applications. This paper describes a new SLP algorithm called PSLP Penalty SLP. PSLP represents a significant strengthening and refinement of the SLPR procedure described in Palacios-Gomez et al. Palacios-Gomez, F., L. Lasdon, M. Engquist. 1982. Nonlinear optimization by successive linear programming. Management Sci.28 1106-1120.. We give a convergence proof for PSLP-the first SLP convergence proof for nonlinearly constrained problems of general form. This theory is supported by computational performance-in our tests, PSLP is significantly more robust than SLPR, and at least as efficient. A Fortran computer implementation is described. A simplified version of PSLP has already solved several "real world" NLP problems at Exxon Baker and Lasdon [Baker, T. E., L. S. Lasdon. 1985. Successive linear programming at Exxon. Management Sci.31 March 264-274., including nonlinear refinery models of up to 1000 rows. As with other SLP algorithms, PSLP is especially efficient on problems which are highly constrained, i.e., which have nearly as many active constraints as there are variables. For problems with vertex optima at least as many active constraints as variables, it is quadratically convergent. Nonlinear refinery models often have vertex optima, since they are large and mostly linear, and on line process unit optimization problems are likely to possess highly constrained solutions as well. PSLP has great potential for accurate, efficient solution of such problems.

Risk Modeling Using Direct Solution of Nonlinear Approximations of the Utility Function

Abstract: A risk model is developed which involves direct solution of the expected utility maximization problem utilizing nonlinear programming. The model permits the use of utility functions exhibiting increasing, constant, and decreasing absolute risk aversion. Demonstrations are done using functions exhibiting such properties over normal, uniform, and triangular data sets. The desire to reflect uncertainty of future events within decision-making problems has led to a number of risk models. Many of these risk models attempt to reflect the decision maker's expectations of possible outcomes and their probabilities, along with the decision maker's attitude toward assuming risk. Methodological approaches to risky decision problems range from using conservative estimates for the uncertain elements (e.g., a pessimistic price), through methods which explicitly include probability density functions for the uncertain parameters. A class of the latter models has arisen within the mathemat

An interactive fuzzy satisficing method using augmented minimax problems and its application to environmental systems

Abstract: A new interactive fuzzy satisficing method for multiobjective nonlinear programming is presented which considers that the decision-maker (DM) has fuzzy goals for each of the objective functions. Through the interaction with the DM, the fuzzy goals of the DM are quantified by eliciting corresponding membership functions. In order to generate a candidate for the satisficing solution (Pareto optimal) after determining the membership functions, if the DM specifies his/her reference membership values, the augmented minimax problem is solved. The DM is thus supplied with the corresponding Pareto optimal solution together with the tradeoff rates between the membership functions. Then by considering the current values of the membership functions as well as the tradeoff rates, the DM acts on this solution by updating his/her reference membership values. A time-sharing computer program is written to implement man-machine interactive procedures based on this method. An application to an industrial pollution control problem is demonstrated.

Eigenvalue optimization algorithms for structure/controller design iterations

Abstract: An eigenspace optimization approach is proposed and demonstrated for the design of feedback controllers for the maneuvers/vibration arrests of flexible structures. The algorithm developed is shown to be equally useful in sequential or simultaneous design iterations that modify the structural parameters, sensor/actuator locations, and control feedback gains. The approach is demonstrated using a differential equation model for the "Draper/RPL configuration." This model corresponds to the hardware used for experimental verification of large flexible spacecraft maneuver controls. A number of sensor/actuat or configuration s are studied vis-a-vis the degree of controllabili ty. Linear output feedback gains are determined using a novel optimization strategy. The feasibility of the approach is established, but more research and numerical studies are required to extend these ideas to truly high-dimensioned systems. Parameterization of the Controlled System's Eigenvalues and Eigenvectors C ONSIDER a linear structure (modeled by a finite element or similar discretization scheme) in which the configuration vector jc is governed by the system of differential equations

A simplex algorithm for piecewise-linear programming I: Derivation and proof

Abstract: The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. This three-part paper develops and analyzes a general, computationally practical simplex algorithm for piecewiselinear programming. Part I derives and justifies the essential steps of the algorithm, by extension from the simplex method for linear programming in bounded variables. The proof employs familiar finite-termination arguments and established piecewise-linear duality theory. Part II considers the relaxation of technical assumptions pertaining to finiteness, feasibility and nondegeneracy of piecewise-linear programs. Degeneracy is found to have broader consequences than in the linear case, and the standard techniques for prevention of cycling are extended accordingly. Part III analyzes the computational requirements of piecewise-linear programming. The direct approach embodied in the piecewise-linear simplex algorithm is shown to be inherently more efficient than indirect approaches that rely on transformation of piecewise-linear programs to equivalent linear programs. A concluding section surveys the many applications of piecewise-linear programming in linear programming,l 1 estimation, goal programming, interval programming, and nonlinear optimization.

Lower subdifferentiable functions and their minimization by cutting planes

Abstract: This paper introduces lower subgradients as a generalization of subgradients. The properties and characterization of boundedly lower subdifferentiable functions are explored. A cutting plane algorithm is introduced for the minimization of a boundedly lower subdifferentiable function subject to linear constraints. Its convergence is proven and the relation is discussed with the well-known Kelley method for convex programming problems. As an example of application, the minimization of the maximum of a finite number of concave-convex composite functions is outlined.

A Condition Number for Differentiable Convex Inequalities

Abstract: For a system of differentiable convex inequalities, a new bound is given for the absolute error in an infeasible point in terms of the absolute residual. By using this bound a condition number is defined for the system of inequalities which gives a bound for the relative error in an infeasible point in terms of the relative residual.

A Superlinearly Convergent Minimax Algorithm for Microwave Circuit Design

Abstract: A new and highly efficient algorithm for nonlinear minimax optimization is presented. The algorithm, based on the work of Hald and Madsen, combines linear programming methods with quasi-Newton methods and has sure convergence properties. A critical review of the existing minimax algorithms is given, and the relation of the quasi-Newton iteration of the algorithm to the Powell method for nonlinear programming is discussed. To demonstrate the superiority of this algorithm over the, existing ones, the classical three-section transmission-line transformer problem is used. A novel approach to worst-case design of microwave circuits using the present algorithm is proposed. The robustness of the algorithm is proved by using it for practical design of contiguous and noncontiguous-band multiplexer, involving up to 75 design variables.

On topology optimization of trusses

Abstract: The selection of a minimum weight truss, out of a large set of candidate trusses, is treated. An initial configuration is generated by connecting all the n nodal points with n(n − 1)\2 truss members. A feasible displacement field, satisfying displacement and stress constraints, is obtained from a static analysis of this truss, followed by a uniform scaling of all truss dimensions. The finite element formulation for the initial configuration is then reformulated to yield a linear programming problem. The solution to this leads to a new configuration which is further optimized by solving a small nonlinear programming problem. With the method proposed, trusses subject to one loading condition and subject to stress and displacement constraints can be selected and optimized using a modest computational effort. Three examples are given to demonstrate the usefulness of the proposed method.

Second order sensitivity analysis and asymptotic theory of parametrized nonlinear programs

Abstract: In this paper we study second-order differential properties of an optimal-value functionź(x). It is shown that under certain conditionsź(x) possesses second-order directional derivatives, which can be calculated by solving corresponding quadratic programs. Also upper and lower bounds on these derivatives are introduced under weaker assumptions. In particular we show that the second-order directional derivative is infinite if the corresponding quadratic program is unbounded. Finally sensitivity results are applied to investigate asymptotics of estimators in parametrized nonlinear programs.

A Continuously Differentiable Exact Penalty Function for Nonlinear Programming Problems with Inequality Constraints

Abstract: In this paper it is shown that, given a nonlinear programming problem with inequality constraints, it is possible to construct a continuously differentiable exact penalty function whose global or local unconstrained minimizers correspond to global or local solutions of the constrained problem.

Alternative approximation concepts for space frame synthesis

Abstract: A structural synthesis methodology for the minimum mass design of 3-dimensionall frame-truss structures under multiple static loading conditions and subject to limits on displacements, rotations, stresses, local buckling, and element cross-sectional dimensions is presented. A variety of approximation concept options are employed to yield near optimum designs after no more than 10 structural analyses. Available options include: (A) formulation of the nonlinear mathematcal programming problem in either reciprocal section property (RSP) or cross-sectional dimension (CSD) space; (B) two alternative approximate problem structures in each design space; and (C) three distinct assumptions about element end-force variations. Fixed element, design element linking, and temporary constraint deletion features are also included. The solution of each approximate problem, in either its primal or dual form, is obtained using CONMIN, a feasible directions program. The frame-truss synthesis methodology is implemented in the COMPASS computer program and is used to solve a variety of problems. These problems were chosen so that, in addition to exercising the various approximation concepts options, the results could be compared with previously published work.

Optimal Control of Nonlinear Groundwater Hydraulics: Theoretical Development and Numerical Experiments

Abstract: Nonlinear optimization models are presented for the optimal operation of an unconfined aquifer system. The aquifer's response equations are developed using finite difference methods, quasilinearization, and matrix calculus. The optimization model, which is structured as a discrete time optimal control problem, identifies the optimal pumping pattern necessary to satisfy an exogenous water demand. A quasilinearization optimization algorithm and projected Lagrangian methods are used for the solution of the planning model. Example problems are presented which demonstrate the viability of the approach for nonlinear, nonconvex groundwater management problems.

Optimal Dynamic Dispatch Owing to Spinning-Reserve and Power-Rate Limits

Abstract: This paper deals with the formulation and solution of the optimal dynamic dispatch problem owing to spinning-reserve and power-rate limits. The power production of a thermal unit is considered as a dynamic system, which limits the maximum increase and decrease of power. The solution is obtained with a special projection method having conjugate search directions that quickly and accurately solves the associated non-linear programming problem with up to 2400 variables and up to 9600 constraints.

The Entropic Penalty Approach to Stochastic Programming

Abstract: A penalty-type decision-theoretic approach to Nonlinear Programming Problems with stochastic constraints is introduced. The Stochastic Program SP is replaced by a Deterministic Program DP by adding a term to the objective function to penalize solutions which are not “feasible in the mean.” The special feature of our approach is the choice of the penalty function PE, which is given in terms of the relative entropy functional, and is accordingly called entropic penalty. It is shown that PE has properties which make it suitable to treat stochastic programs. Some of these properties are derived via a dual representation of the entropic penalty which also enable one to compute PE more easily, in particular if the constraints in SP are stochastically independent. The dual representation is also used to express the Deterministic Problem DP as a saddle function problem. For problems in which the randomness occurs in the rhs of the constraints, it is shown that the dual problem of DP is equivalent to Expected Utility Maximization of the classical Lagrangian dual function of SP, with the utility being of the constant-risk-aversion type. Finally, mean-variance approximations of PE and the induced Approximating Deterministic Program are considered.

A Computational Comparison of the Ellipsoid Algorithm with Several Nonlinear Programming Algorithms

Abstract: A computational comparison of several general purpose nonlinear programming algorithms is presented. This study was motivated by the preliminary results in [12] which show that the recently developed ellipsoid algorithm is competitive with a widely used augmented Lagrangian algorithm. To provide a better perspective on the value of ellipsoid algorithms in nonlinear programming, the present study includes some of the most highly regarded nonlinear programming algorithms and is a much more comprehensive study than [12]. The algorithms considered here are chosen from four distinct classes and 50 well-known test problems are used. The algorithms used represent augmented Lagrangian, ellipsoid, generalized reduced gradient, and iterative quadratic programming methods. Results regarding robustness and relative efficiency are presented.

A Linearization Algorithm for Nonsmooth Minimization

Abstract: We present a readily implementable algorithm for finding stationary points for locally Lipschitzian functions that are not necessarily convex or differentiable. The algorithm is an extension to the nonconvex case of the aggregate subgradient method. The user can impose an upper bound on storage and work per iteration, whereas no such uniform bound exists for the algorithms due to Lemarechal and Mifflin on which our method is based. The algorithm is globally convergent in the sense that all its accumulation points are stationary. Moreover, the algorithm converges when the objective function happens to be convex. This seems to be the first such result for descent methods for nonsmooth minimization.

Algorithms for Nonlinear Least Squares with Linear Inequality Constraints

Abstract: Two algorithms for solving nonlinear least squares problems with general linear inequality constraints are described. At each step, the problem is reduced to an unconstrained linear least squares problem in the subspace defined by the active constraints, which is solved using the Levenberg–Marquardt method. The desirability of leaving an active constraint is evaluated at each step, using a different technique for each of the two algorithms. Each step is constrained to be within a circular region of trust about the current approximate minimizes, whose radius is updated according to the quality of the step after each iteration. Comparisons of the relative performance of the two algorithms on small problems and on a larger exponential data-fitting problem are presented.