General Constrained Optimization Problems Research Articles

Adaptive Algorithm for Constrained Least-Squares Problems

This paper is concerned with the implementation and testing of an algorithm for solving constrained least-squares problems. The algorithm is an adaptation to the least-squares case of sequential quadratic programming (SQP) trust-region methods for solving general constrained optimization problems. At each iteration, our local quadratic subproblem includes the use of the Gauss–Newton approximation but also encompasses a structured secant approximation along with tests of when to use this approximation. This method has been tested on a selection of standard problems. The results indicate that, for least-squares problems, the approach taken here is a viable alternative to standard general optimization methods such as the Byrd–Omojokun trust-region method and the Powell damped BFGS line search method.

Solving a generalized constrained optimization problem with both logic AND and OR relationships by a mathematical transformation and its application to robot motion planning

The logic relationship among the equality and inequality constraints in a standard constrained optimization problem (SCOP) is the logical AND. Various efficient, convergent and robust algorithms have been developed for such a SCOP. Motivated by a practical application, a more general constrained optimization problem (GCOP) with not only logic AND but also OR relationships is proposed in this paper. In order to solve such a generalized problem, a mathematical transformation which can transfer a set of inequalities with logic OR into one inequality is developed. This transformation provides a necessary and sufficient condition which enables us to use the algorithms developed for SCOPs to solve the generalized optimization problems. The research is motivated by the requirements of developing an efficient, robust, and reliable navigation algorithm for a mobile robot such as an autonomous underwater vehicle (AUV). The original contributions of the paper include threefold: first, from the viewpoint of optimization theory, this paper, to the authors' best knowledge, is the first one to propose such a GCOP. Second, a method is developed to solve such a GCOP. Third, from the viewpoint of robot path planning, this paper presents a new way of using classical optimization approach to solve robot path planning.

Hopfield neuron based nonlinear constrained programming to fuzzy structural engineering optimization

Using the continuous Hopfield network model as the basis to solve the general crisp and fuzzy constrained optimization problem is presented and examined. The model lies in its transformation to a parallel algorithm which distributes the work of numerical optimization to several simultaneously computing processors. The method is applied to different structural engineering design problems that demonstrate this usefulness, satisfaction or potential. The computing algorithm has been given and discussed for a designer who can program it without difficulty.

An ODE method of solving nonlinear programming

This paper proposes a new method to solve general constrained optimization problem. The problem of finding the local optimal points of a nonlinear programming problem with equality and inequality constraints is considered by solving the ODE (i.e., ordinary differential equation) with an appropriate numerical procedure. Moreover, the rate of convergence to optimal points is quadratic. Some numerical result is given to show the efficiency of the proposed method.

Evolutionary programming techniques for constrained optimization problems

Two evolutionary programming (EP) methods are proposed for handling nonlinear constrained optimization problems. The first, a hybrid EP, is useful when addressing heavily constrained optimization problems both in terms of computational efficiency and solution accuracy. But this method offers an exact solution only if both the mathematical form of the objective function to be minimized/maximized and its gradient are known. The second method, a two-phase EP (TPEP) removes these restrictions. The first phase uses the standard EP, while an EP formulation of the augmented Lagrangian method is employed in the second phase. Through the use of Lagrange multipliers and by gradually placing emphasis on violated constraints in the objective function whenever the best solution does not fulfill the constraints, the trial solutions are driven to the optimal point where all constraints are satisfied. Simulations indicate that the TPEP achieves an exact global solution without gradient information, with less computation time than the other optimization methods studied here, for general constrained optimization problems.

Superlinear and quadratic convergence of some primal-dual interior point methods for constrained optimization

This paper proves local convergence rates of primal-dual interior point methods for general nonlinearly constrained optimization problems. Conditions to be satisfied at a solution are those given by the usual Jacobian uniqueness conditions. Proofs about convergence rates are given for three kinds of step size rules. They are: (i) the step size rule adopted by Zhang et al. in their convergence analysis of a primal-dual interior point method for linear programs, in which they used single step size for primal and dual variables; (ii) the step size rule used in the software package OB1, which uses different step sizes for primal and dual variables; and (iii) the step size rule used by Yamashita for his globally convergent primal-dual interior point method for general constrained optimization problems, which also uses different step sizes for primal and dual variables. Conditions to the barrier parameter and parameters in step size rules are given for each case. For these step size rules, local and quadratic convergence of the Newton method and local and superlinear convergence of the quasi-Newton method are proved.

Optimization over analytic functions whose founrier coefficients are constrained

This article gives necessary and sufficient conditions for local solutions to several very general constrained optimization problems over spaces of analytic functions.

A SEQUENTIAL QUADRATIC PROGRAMMING ALGORITHM UTILIZING QR MATRIX FACTORIZATION

Abstract Some recent studies indicate that the sequential quadratic programming (SQP) approach has a sound theoretical basis and promising empirical results for solving general constrained optimization problems. This paper presents a variant of the SQP method which utilizes QR matrix factorization to solve the quadratic programming subproblem which result from taking a quadratic approximation of the original problem. Theoretically, the QR factorization method is more robust and computationally efficient in solving quadratic programs. To demonstrate the validity of this variant, a computer program named SQR is coded in Fortran to solve twenty-eight test problems. By comparing with three other algorithms: one multiplier method, one GRG-type method, and another SQP-type method, the numerical results show that, in general, SQR as devised in this paper is the best method as far as robustness and speed of convergence are concerned in solving general constrained optimization problems.

Optimal door fitting with systematic fixture adjustment

A systematic approach is presented to obtain the best door gap quality through optimal door fitting in automobile body manufacturing. First, three indexes of gap quality are defined; they are: (1) door gap width deviation relative to design nominal; (2) door gap parallelism; and (3) car-to-car gap consistency. Then the door-fitting problem is formulated into a general constrained optimization problem. The effects of optimal door fitting on the three quality indexes are evaluated through computer simulation. These results provide a lower bound on the design of nominal door gap by considering process capability. Finally, a computer-aided fixture adjustment scheme is developed to orient a door in a body side opening to achieve the optimal fitting. The amount of adjustment, with the desired orientation obtained from optimization, is calculated based on parametrically modeled local surface features of the fixture and the door. The adequacy of door feature modeling is verified through a door-fitting experiment.

On a single-machine optimal constant due-date assignment and sequencing problem

This paper considers the problem of optimal constant due-date assignment and sequencing of jobs in a single-machine shop. We formulate the problem as a general constrained optimization problem and apply the Kuhn-Tucker conditions to find the optimal solution which is shown to be independent of the job sequence.

Decomposability in fixed point computation with applications and acceleration techniques

In the past decade, several complementary pivot algorithms have been developed to search for fixed points of certain functions and point to set maps on unbounded regions. This paper develops a structure (called decomposability), which, when present, enables one to work in a lower dimensional space when solving these problems. Several examples of where this structure arises in applications are presented. It is shown that under suitable circumstances, the general constrained optimization problem (that of optimizing an objective function subject to both equality and inequality constraints) may be formulated as a decomposèble fixed point problem. At the same time, an approximation technique is developed to potentially improve the efficiency of the complementary pivot algorithms.

A Robust Conjugate-Gradient-Penalty Function Algorithm for Solving General Constrained Optimization Problems with Application to Computer-Aided Design

An exact penalty function type of algorithm is proposed to solve a general class of constrained parameter optimization problems. Some standard test functions are included to show the performance of the algorithm, and an application of the algorithm is made to solve some computer-aided design problems occurring in the area of control system synthesis.