Nonlinear Constrained Problem Research Articles

Optimality conditions for state constrained nonlinear control problems

The problem of optimal control of nonlinear control and state constrained control problems, where the state constraint may involve differential operators and the cost functionals may be nonsmooth, is studied. For this class of problems, necessary optimality conditions using techniques from infinite dimensional optimization theory adapted to the framework of control problems are derived. It is shown that the underlying structure admits a considerable relaxation of the classical constraint qualifications. The theory then is applied to examples of various nonlinear elliptic equations and state constraints.

Near minimum-time trajectory for two coordinated manipulators

SummaryThis paper presents a method to obtain near minimum-time trajectories for two coordinated manipulators handling a rigid object. A piece-wise constant function is used to approximate the second derivatives of the generalised coordinates of the manipulators of the system. This transforms the time optimal control problem into a non-linearly constrained optimisation problem. The transformed problem is then solved by the sequential quadratic programming technique. A numerical example involving two SCARA type manipulators handling a long beam is used to illustrate the proposed scheme.

冗長双腕マニピュレータ系の準最短時間軌道計画法

This paper describes a method for sub-minimal-time trajectory planning of a dual manipulator system required to move a rigid task object from one spatial point to another. In this method, in order to transform the time optimal control problem into a non-linearly constrained optimisation problem, a piece-wise constant function is used to approximate the third derivatives of the generalised coordinates of the active joints of the dual manipulator system. However, the system is underspecified due to the redundancy, which is inherent to the system. A technique to overcome the problem, in which an excessive torque is used as optimisation variables, is proposed accordingly. A numerical example involving two SCARA type manipulators handling a long beam was used to illustrate the proposed scheme and results showed the effectiveness of the proposed technique.

Optimal population stabilization and control usingthe Leslie matrix model

We consider the problem of optimal stabilization and control of populations which follow the Leslie model dynamics, within state space and control systems theory and methodology. Various types of culling strategies are formulated and introduced into the Leslie model as control inputs, and their effect on global asymptotic stability is investigated. Our new approach provides answers to several unexplored problems. We show that in general it is possible to achieve a desired stable equilibrium population level, through the design of a class ofshifted-proportional stabilizing culling policies. Further, we formulate general non-linear constrained opitmization problems, for obtaining the cost-optimal policy among this generally infinite class of such stabilizing policies. The theoretical findings are illustrated through the solution of the problem over an infinite planning horizon for a numerical example. A comparative study of the costs and dynamic effects of various culling strategies also supports the mathematical results.

Constrained Optimization Via Genetic Algorithms

This paper presents an application of genetic algorithms (GAs) to nonlinear constrained optimization. GAs are general purpose optimization algorithms which apply the rules of natural genetics to explore a given search space. When GAs are applied to nonlinear constrained problems, constraint handling becomes an important issue. The proposed search algorithm is realized by GAs which utilize a penalty function in the objective function to account for violation. This extension is based on systematic multi-stage assignments of weights in the penalty method as opposed to single-stage assignments in sequential unconstrained minimization. The experimental results are satisfactory and agree well with those of the gradient type methods.

Convergence of Tikhonov regularization for constrained ill-posed inverse problems

In this paper convergence and rate-of-convergence results for nonlinear constrained ill-posed inverse problems formulated as regularized least-squares problems are given.

Recent developments in constrained optimization

Constrained optimization problems occur in many applications of science, engineering, social science and medicine, and hence it is useful for practitioners in these areas to be informed about numerical methods for solving such problems. However, improvements in optimization typically do not spread quickly to the non-specialist literature. This paper contains a brief summary of two recent developments in the solution of nonlinearly constrained problems: direct use of the Lagrangian function, and linearization of nonlinear constraints. In order to highlight the new approaches, we also discuss techniques used in older methods. To illustrate the capabilities of modern optimization methods, a brief description is given of the development of methods to solve large nonlinearly constrained problems that arise in the electrical power industry. The general nonlinear programming problem may be stated in the following form:

Optimisation de procédés chimiques complexes: Etude d'un atelier de monochloration du benzène

Several procedures for the computer aided design of complex plants involving chemical reactors, rectification columns and recycle streams are presented. In the first part, these cyclic processes are represented by a sequential, two-hierarchical-level tree. A “branch and bound” method is then used to find the optimal structure for given operating conditions. The synthesis of a benzene chlorination process with one or several reactors is used to illustrate the implementation of the algorithm. The structure and the operating conditions are both optimized in the second part of this work. By using structural parameters to represent physical unit connections within a superstructure, the problem becomes one of large-scale nonlinear programming (NLP). A projected reduced gradient algorithm, extended to nonlinear constrained problems, is used to solve the nonlinear program. The same example as was used in the first part shows the complementary nature of the two approaches, with the first one providing a good initial feasible point for the second.

KNOWLEDGE GATHERING FOR HEURISTIC PROGRAMMING IN DESIGN OPTIMIZATION

Abstract Presented is an approach to optimization which uses performance criteria from tests conducted on a large variety of different classes of nonlinearly constrained design problems, to improve the design optimization process. The knowledge gathering methods and evaluation techniques are discussed along with how heuristic procedures are integrated to enhance the program's optimization capabilities.

OPTIMUM DESIGN OF SMALL DISTRIBUTION TRANSFORMER USING ALUMINIUM CONDUCTORS

ABSTRACT Design Optimization of distribution transformer of small KVA rating using aluminium conductors is considered as a nonlinear multivariable constrained programming problem. For this purpose a set of five independent design variables is identified and suitable constraints are imposed to meet the thermal and other performance requirements of the transformer. The various objective functions are formed in terms of cost of active material, capitalized cost of losses as the operating cost and the overall cost as the sum of these two. This is done to select most effective and appropriate optimized design for these types of transformers. The optimization is achieved through Rosenbrock's method of direct search in conjunction with the sequential unconstrained minimization Technique (SUMT). The optimized design results for a 25 KVA distribution transformer using aluminium conductors are presented along with the design results using copper as the winding material for different objective functions.

A method for the identification of in-vivo segmental stiffness properties of the spine.

A numerical algorithm is used to estimate in-vivo segmental stiffness properties of individual spine segments based upon existing load-displacement data. A static nonlinear finite element model stimulates a pathological spine and corrective instrumentation system. A systematic procedure for establishing the model's stiffness parameters is described, in the form of a nonlinear constrained optimal design problem. The numerical method is demonstrated using as an example a case of adolescent idiopathic scoliosis requiring corrective surgery.

Linear Least Squares with Bounds and Linear Constraints

An algorithm is given for solving linear least squares systems of algebraic equations subject to simple bounds on the unknowns and (more general) linear equality and inequality constraints. The method used is a penalty function approach wherein the linear constraints are (effectively) heavily weighted. The resulting system is then solved as an ordinary bounded least squares system except for some important numerical and algorithmic details. This report is a revision of an earlier work. It reflects some hard-won experience gained while using the resulting software to solve nonlinear constrained least squares problems.

Gustaf: A quasi-newton nonlinear ADI FORTRAN IV program for solving the shallow-water equations with augmented lagrangians

A FORTRAN IV computer program is documented which implements the nonlinear alternating direction implicit (ADI) method of Gustafsson (1971) for a limited area finite-difference integration of a shallow-water equations model on a β-plane. In this method a computationally efficient quasi-Newton method is used to solve, at each time-step, the resulting nonlinear systems of algebraic equations. Large time-steps can be employed with this method, which is stable unconditionally for the linearized equations. Owing to its nonlinearity, the method is useful particularly where accuracy is important. An augmented Lagrangian method is applied to enforce conservation of the integral invariants of the shallow-water equations. This method approximates the nonlinearly constrained minimization problem by solving a series of unconstrained minimization problems. Program options include a line-printer plot of the height-field contour and determination, at each time-step, of the three integral invariants of the shallow-water equations. According to the number of nonlinear quasi-Newton (QN) iterations performed at each time-step, different QN methods are presented. Long-term runs have been performed using this program and, due to the enforcement of conservation of integral-invariants via the augmented Lagrangian method, no finite-time “blowing” was experienced.

A BRANCH AND BOUND ALGORITHM FOR TOPOLOGY OPTIMIZATION OF TRUSS STRUCTURES

An algorithm for the selection of a minimum weight truss, out of a set of possible candidate trusses, is presented. The trusses are subject to stress and deflection constraints. Multiple loading conditions are considered. Starling with a ground structure, a sequence of subtrusses called candidate trusses are generated and analyzed. Several criteria are used to rapidly discard non-optimal configurations. Sequential Quadratic Programming is used to solve a non-linearly constrained problem which is part of the algorithm. A technique, frequently used to keep second derivative approximations positive definite, is found to give numerical instabilities. Possible modifications to improve stability are discussed. Finally, three examples are given to demonstrate the algorithm.

Experience in Design Optimization of Induction Motor Using 'SUMT' Algorithm

Design optimization of an induction motor is considered as a nonlinear multivariable constrained programming problem. A set of nine basic variables is identified and suitable constraints are imposed to meet the thermal, starting and other performance requirements of the machine. Six different objective functions are considered to facilitate the selection of suitable optimized designs for any given application. The optimization is achieved through Rosenbrock's method of direct search in conjunction with the sequential unconstrained minimization technique (SUMT). For a typical 3.7kw cage motor, the optimized design results with different objective functions are presented.

Combined Penalty Multiplier Optimization Methods to Enforce Integral Invariants Conservation

Abstract An augmented Lagrangian multiplier-penalty method is applied for the first time to solving the problem of enforcing simultaneous conservation of the nonlinear integral invariants of the shallow water equations on a limited-area domain. The method approximates the nonlinearly constrained minimization problem by solving a series of unconstrained minimization problems. The computational efficiency and accuracy of the method is tested using two finite-difference solvers of the nonlinear shallow water equations on a β-plane. The method is also compared with a pure quadratic penalty approach. The updating of the Lagrangian multipliers and the penalty parameters is done using procedures suggested by Bertsekas. The method yielded satisfactory results in the conservation of the integral constraints while the additional CPU time required did not exceed 15% of the total CPU time spent on the numerical solution of the shallow water equations. The methods proved to be simple in their implementation and they h...

A Theory of the Optimal Policy of Oil Recovery by Secondary Displacement Processes

We consider the stability of the displacement of a viscous fluid in a porous medium by a less viscous fluid containing a solute. For the case in which the total amount of injected solute is fixed, we formulate a theory for the optimal injection policy, i.e., the policy which minimizes the growth constant of the Saffman–Taylor–Chouke instability. The result is a nonlinear constrained eigenvalue problem with the viscosity ratio, $\alpha $, of the two fluids and the total solute, N, as parameters. We develop perturbation solutions for large and small N and numerical solutions for all N. Several features of the optimal policy which have been noted or demonstrated empirically by previous workers are proven, and new features of the solution are discussed.

AN ALGORITHM FOR OPTIMIZING NONLINEAR CONSTRAINED ZERO-ONE PROBLEMS TO IMPROVE WASTEWATER TREATMENT

A method has been developed to optimize nonlinear constrained zero-one problems incorporating a modified Branch and Bound technique in combination with Complementary Geometric Programming. In the Branch and Bound phase the bivalent variable is introduced in a pseudo-Boolean formal and a gradient criterion is used to accelerate convergence. This procedure has been implemented in a computer program. The algorithm was developed as part of an overall model of selecting the appropriate wastewater treatment methods, and to control the biological biomass types in high rate algae ponds. An example of this application is presented.

A Boundary Tracking Optimization Algorithm for Constrained Nonlinear Problems

A new procedure for numerical optimization of constrained nonlinear problems is described. The method makes use of an efficient “Boundary Tracking” strategy to move on the constraint surfaces. In a comparison study it was found to be an effective method for treating nonlinear mathematical programming problems particularly those with difficult nonlinear constraints.

Efficiently Converging Minimization Methods Based on the Reduced Gradient

This paper presents three computational methods which extend to nonlinearly constrained minimization problems the efficient convergence properties of, respectively, the method of steepest descent, the variable metric method, and Newton’s method for unconstrained minimization. Development of the algorithms is based on use of the implicit function theorem to essentially convert the original constrained problem to an unconstrained one. This approach leads to practical and efficient algorithms in the framework of Abadie’s generalized reduced gradient method. To achieve efficiency, it is shown that it is necessary to construct a sequence of approximations to the Lagrange multipliers of the problem simultaneously with the approximations to the solution itself. In particular, the step size of each iteration must be determined by a linesearch for a minimum of an approximate Lagrangian function.