Sequence Of Unconstrained Problems Research Articles

On Necessary Conditions for a Minimum

We discuss a general approach to necessary optimality conditions based on the so-called “optimality alternative,” which reduces a problem with constraints to an unconstrained problem or a sequence of unconstrained problems. The power of the approach is demonstrated by a proof of a necessary optimality condition in an abstract problem with mixed (convex vs. nonconvex) structure and a new proof of Clarke’s “stratified” maximum principle for optimal control of differential inclusions.

TU‐G‐BRB‐09: A Bregman Iteration Method for IMRT Optimization with Dose‐Volume Histogram Constraints

Purpose: Dose‐volume histogram (DVH) is a clinically relevant criterion to evaluate treatment plan quality. It is hence desirable to incorporate the associated DVH constraints in treatment plan optimization. Yet, these constraints usually lead to difficulties due to their non‐convex nature. This project develops an algorithm to solve the intensity modulated radiation therapy (IMRT) optimization problem with DVH constraints considered via a Bregman iteration. Methods: We consider an objective function of a quadratic form with different overdose and underdose penalties, subject to DVH constraints. The constrained optimization problem is converted into a sequence of unconstrained problems via Bregman iteration approach, which are then solved sequentially using gradient‐descent algorithm with inexact line search. The objective functions in those unconstrained optimization problems contains terms that requires the computation of DVH values. In practice, this is achieved by approximating the Heaviside step‐functions in the expression of a DVH curve by a smoothed arctangent function. This approach also allows for the computation of gradient of the objective functions during optimization process. Results: We have tested our algorithm in the context of 7‐field IMRT treatment planning for prostate cancer in 4 patient cases. Clinically relevant DVH constraints are considered for PTV, rectum, and bladder. In all the cases, the algorithm is able to find the solutions that satisfy all the DVH constraints, while as DVH constraints may be violated when the Bregman algorithm is not applied. The local‐minima problem caused by the non‐convex constraints is not observed, which may be ascribed to the good quality of the initial dose distribution obtained after a standard IMRT optimization problem without DVH constraints. Conclusions: We have developed an algorithm to solve the IMRT optimization problem with DVH constraints using Bregman iteration approach. Tests conducted in prostate cancer cases have demonstrated the validity of our algorithm and its effectiveness.

Matrix filter design for passive sonar interference suppression

The performance of passive acoustic signal-processing techniques can become severely degraded when the acoustic source of interest is obscured by strong interference. The application of matrix filters to suppress interference while passing a signal of interest with minimal distortion is presented. An algorithm for single-frequency matrix filter design is developed by converting a constrained convex optimization problem into a sequence of unconstrained problems. The approach is extended to broadband data by incoherently combining the responses of matrix filters designed at frequencies across a band of interest. The responses of single-frequency and multifrequency matrix filters are shown. Examples are given which demonstrate the effectiveness of matrix filtering applied to matched-field localization of a weak source in the presence of a strong interferer and noise. These examples show the matrix filter effectively suppressing the interference, thereby enabling the localization of the weak source. Standard matched-field processing, without matrix filtering, is not effective in localizing the weak source.

Using multicriterion optimization for strength design of composite laminates

Using multicriterion optimization methods for weight minimization of failure strength controlled composite structures is described. A composite lay-up design problem for minimizing the number of layers under strength constraints with respect to multiple loading conditions is formulated. The constrained problem is transferred into a sequence of unconstrained problems and solved with an interactive descent method. A typical design cycle that comprises the finite element mesh generator and solver as well as the laminate analysis and optimization modules is illustrated with a numerical example.

NONLINEAR DISCRETE STRUCTURAL OPTIMIZATION

Structural design optimization will be more convenient to formulate the design problem with discrete variable than it would be if the variables were assumed to be continuous. In order to solve a structural design problem with discrete variable only, two completely different techniques, 1) Integer gradient direction, which is later supported by subsequential search interval technique and 2) Modified Rosenbrocks orthognalization techniques have hybridized. Rosenbrocks original procedure is a well established method to solve continuous variable optimization problem; but to suit to discrete variable problem solution some modifications are needed and reported here. By this hybridizing most of the practical difficulties, usually, encountered in the discrete optimization can be overcome. Details of the techniques are discussed and by their combination a solution code has been generated. A constrained problem is first converted into a sequence of unconstrained problem by use of interior penalty function and then solved by the generated code. The efficiency of the generated code is revealed by solving several test problems.

Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects

Two general algorithms for solving constrained minimization problems are presented and discussed in the context of analysis and assimilation of meteorological observations. In both algorithms, the original constrained problem is transformed by appropriate modifications into one unconstrained problem, or into a sequence of unconstrained problems. The main advantage of proceeding in this way is that the new unconstrained problems can be solved by classical descent algorithms, thus avoiding the need of directly solving the Euler-Lagrange equations of the original constrained problem. The first algorithm presented in the augmented lagrangian algorithm. It generalizes the more classical penalty and duality algorithms. The second algorithm, inspired from optimal control techniques, is based on an appropriate use of an adjoint dynamical equation, and seems to be particularly well adapted to the assimilation of observations distributed in time. Simple numerical examples show the ability of these algorithms to solve non-linear minimization problems of the type encountered in meteorology. Their possible use in more complex situations is discussed, in particular in terms of their computational cost. DOI: 10.1111/j.1600-0870.1986.tb00459.x

Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects

Two general algorithms for solving constrained minimization problems are presented and discussed in the context of analysis and assimilation of meteorological observations. In both algorithms, the original constrained problem is transformed by appropriate modifications into one unconstrained problem, or into a sequence of unconstrained problems. The main advantage of proceeding in this way is that the new unconstrained problems can be solved by classical descent algorithms, thus avoiding the need of directly solving the Euler-Lagrange equations of the original constrained problem. The first algorithm presented in the augmented lagrangian algorithm. It generalizes the more classical penalty and duality algorithms. The second algorithm, inspired from optimal control techniques, is based on an appropriate use of an adjoint dynamical equation, and seems to be particularly well adapted to the assimilation of observations distributed in time. Simple numerical examples show the ability of these algorithms to solve non-linear minimization problems of the type encountered in meteorology. Their possible use in more complex situations is discussed, in particular in terms of their computational cost. DOI: 10.1111/j.1600-0870.1986.tb00459.x

An extended identifiability in aquifer parameter identification and optimal pumping test design

An extended identifiability, called “δ identifiability”, is developed for groundwater modeling and management. Some fundamental concepts are developed for establishing a criterion in connection with the problem of optimal experimental design, such as the design of an optimum pumping test to assist aquifer parameter identification. In practice, the quantity and quality of data prohibits a unique solution of the inverse problem of parameter identification. The proposed “δ identifiability” is based upon the concept of weak uniqueness. The determination of the admissibility of a given design is formulated as a nonlinear programming problem. The original constrained problem is transformed into solving a sequence of unconstrained problems by a penalty function method. Numerical experiments are conducted to illustrate the proposed concept and algorithm.

Optimal solutions to differential inclusions in presence of state constraints

Necessary conditions in terms of the Hamiltonian are given for optimal solutions to the differential inclusion problem when state constraints are present. This result extends a result of Clarke for the unconstrained problem. The data are nonsmooth, nonlinear, nonconvex. The method incorporates the state constraint in the cost functional as a penalty term for a sequence of unconstrained problems that approximate our problem. An application of Ekeland's variational principle, the known necessary conditions for the auxiliary problems, and a limiting process provide the necessary conditions.

Structural optimization using interior penalty function

Abstract Mathematical programming methods are among the most powerful optimization techniques. They may be classified into direct or indirect methods. In the indirect methods, the constrained design problem is converted into a sequence of unconstrained problems using penalty functions. In this way, the optimal solution of a constrained problem may be obtained using one of the unconstrained search techniques. The interior penalty function appears to be the most reliable uncon strained method while the variable metric method seems to be an extremely powerful algorithm. This paper presents the use of the interior penalty function coupled with the variable metric method for the solution of structural design optimization problems.

An improved variable penalty algorithm for automated structural design

The paper presents refinements in a variable penalty algorithm proposed earlier [1]. The algorithm is an extension of the SUMT procedure where the design requirements are incorporated using a flexible penalty function which changes shape with the pattern of the constraint spectrum encountered during an unconstrained optimization process. The sequence of unconstrained problem is solved using a modified Newton method requiring only the first derivatives. The algorithm possesses inherent mechanism to control the quality of the second derivative matrix of the penalty function (Hessian) which is formulated in terms of the first derivatives of the constraints. The effectiveness of this algorithm is demonstrated on two structural problems: a p-member truss and a plane stress p-node plate, each having a relatively large set of test cases. The efficiency shows up in obtaining a converged minimum design in a small number of iterations, irrespective of the initial starting points or the number of design variables present.

On the relationship between a continuous update method of multipliers and the generalized reduced gradient method

AbstractThe method of multipliers1–3 (MOM) is a transformation technique which has enjoyed considerable popularity in recent years. The algorithmic philosophy is similar to conventional penalty function methods in that a constrained nonlinear programming problem is transformed into a sequence of unconstrained problems. In the standard MOM approach, the multipliers are updated after each unconstrained search. In this paper we investigate methods which involve continuous updating of the penalty parameters and design variables. We demonstrate that this continuous updating scheme is equivalent to the generalized reduced gradient method4,5 applied to a certain dual problem. Computational results are given which suggest that the continuous updating MOM is not as efficient as one might reasonably hope.

An improved penalty function method for solving constrained parameter optimization problems

An effective algorithm is described for solving the general constrained parameter optimization problem. The method is quasi-second-order and requires only function and gradient information. An exterior point penalty function method is used to transform the constrained problem into a sequence of unconstrained problems. The penalty weightr is chosen as a function of the pointx such that the sequence of optimization problems is computationally easy. A rank-one optimization algorithm is developed that takes advantage of the special properties of the augmented performance index. The optimization algorithm accounts for the usual difficulties associated with discontinuous second derivatives of the augmented index. Finite convergence is exhibited for a quadratic performance index with linear constraints; accelerated convergence is demonstrated for nonquadratic indices and nonlinear constraints. A computer program has been written to implement the algorithm and its performance is illustrated in fourteen test problems.

Zur zentrenmethode in der nichtlinearen optimierung

One possible way of solving a constrained optimization problem consists in its transformation into a sequence of unconstrained problems. In this paper is presented the connection between three of this methods; namely the method of centers (Huard), the SUMT (sequential unconstrained minimization technique) without parameters (Fiacco and Mccormick) and the method of barycenters (Lommatzsch). For the metyhod of centers is presented a simple proof of convergence without convexity assumptions, developed from an idea of Huard Further are given relations between some regularity conditions, which were used for this methods.

A Note on the Sequential Unconstrained Minimization Technique for Non-Linear Programming

This note considers a method due to Carroll for replacing a constrained minimization problem by a sequence of unconstrained problems. Mathematical validity or convergence of this technique is shown under fairly general conditions. No computability is asserted beyond that obtained by other authors under more restrictive assumptions.