The unconstrained optimization of a function of several variables is considered. An algorithm is constructed using the notion of generalized conjugate directions. It is proved that this method will find the minimum of...
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The unconstrained optimization of a function of several variables is considered. An algorithm is constructed using the notion of generalized conjugate directions. It is proved that this method will find the minimum of a quadratic function in a finite number of steps. Some well-known conjugate direction methods are shown to be special cases of the generalized method.
In 1952, Hestenes and Stiefel first established, along with the conjugate-gradient algorithm, fundamental relations which exist between conjugate direction methods for function minimization on the one hand and Gram-Sc...
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In 1952, Hestenes and Stiefel first established, along with the conjugate-gradient algorithm, fundamental relations which exist between conjugate direction methods for function minimization on the one hand and Gram-Schmidt processes relative to a given positive-definite, symmetric matrix on the other. This paper is based on a recent reformulation of these relations by Hestenes which yield the conjugate Gram-Schmidt (CGS) algorithm. CGS includes a variety of function minimization routines, one of which is the conjugate-gradient routine. This paper gives the basic equations of CGS, including the form applicable to minimizing general nonquadratic functions ofn variables. Results of numerical experiments of one form of CGS on five standard test functions are presented. These results show that this version of CGS is very effective.
Real-time failure detection for systems having linear stochastic dynamical truth models is posed in terms of two confidence region sheaths. One confidence region sheath is about the expected no-failure trajectory; the...
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Real-time failure detection for systems having linear stochastic dynamical truth models is posed in terms of two confidence region sheaths. One confidence region sheath is about the expected no-failure trajectory; the other is about the Kalman estimate. If these two confidence regions of ellipsoidal cross section are disjoint at any time instant, a failure is declared. A test for two-ellipsoid overlap is developed which involves finding a single pointx* whose presence in both ellipsoids is necessary and sufficient for overlap. Thus, the overlap test is contorted into a search forx*, shown to be the solution of a nonlinear optimization problem that is easily solved once an associated scalar Lagrange multiplier is known. A successive approximations iteration equation for λ is obtained and is shown to converge as a contraction mapping. The method was developed to detect failures in an inertial navigation system that appear as uncompensated gyroscopic drift rate. For simulated gyroscopic failures, the iterations converged very quickly, easily allowing real-time failure detection.
An iterative technique is developed to solve the problem of minimizing a functionf(y) subject to certain nonlinear constraintsg(y)=0. The variables are separated into the basic variablesx and the independent variables...
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An iterative technique is developed to solve the problem of minimizing a functionf(y) subject to certain nonlinear constraintsg(y)=0. The variables are separated into the basic variablesx and the independent variablesu. Each iteration consists of a gradient phase and a restoration phase. The gradient phase involves a movement (on a surface that is linear in the basic variables and nonlinear in the independent variables) from a feasible point to a varied point in a direction based on the reduced gradient. The restoration phase involves a movement (in a hyperplane parallel tox-space) from the nonfeasible varied point to a new feasible point. The basic scheme is further modified to implement the method of conjugate gradients. The work required in the restoration phase is considerably reduced when compared with the existing methods.
This paper is a sequel to a previous article by the author, concerned with a certain canonical problem in optimal control involving constraints of the typeψ α(t, x)⩽0, α=1,...,m. In that article, a s...
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This paper is a sequel to a previous article by the author, concerned with a certain canonical problem in optimal control involving constraints of the typeψ
α(t, x)⩽0, α=1,...,m. In that article, a set of second-order conditions necessary for a solution arc was obtained. In this paper, those results are extended to a general control problem involving the above type of constraints.
This paper deals with the estimation of the size of clinical trials for comparing two binomial proportions in both fixed and group-sequential designs. In the fixed size approach, it focuses on 1.) equal sample size de...
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This paper deals with the estimation of the size of clinical trials for comparing two binomial proportions in both fixed and group-sequential designs. In the fixed size approach, it focuses on 1.) equal sample size design; 2.) unequal allocation designs that either maximize the test power subject to fixed total cost or minimize the total expenditure subject to prespecified power using the simplex procedure for function minimization. Emphasis is also placed on group-sequential designs, based upon closed stopping rules, multiple testing and range of clinical equivalence. The efficiency of different allocation designs is assessed by computing the power of the exact conditional Fisher-Irwin test. Designs were applied in planning a clinical trial in which the drug Pancuronium Bromide was compared with a standard treatment to reduce intraventricular haemorrhage in preterm infants.
A Newton-like method is presented for minimizing a function ofn variables. It uses only function and gradient values and is a variant of the discrete Newton algorithm. This variant requires fewer operations than the s...
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A Newton-like method is presented for minimizing a function ofn variables. It uses only function and gradient values and is a variant of the discrete Newton algorithm. This variant requires fewer operations than the standard method whenn > 39, and storage is proportional ton rather thann
2.
The CGS (conjugate Gram-Schmidt) algorithms of Hestenes and Stiefel are formulated so as to obtain least-square solutions of a system of equationsg(x)=0 inn independent variables. Both the linear caseg(x)=Ax−h and the...
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The CGS (conjugate Gram-Schmidt) algorithms of Hestenes and Stiefel are formulated so as to obtain least-square solutions of a system of equationsg(x)=0 inn independent variables. Both the linear caseg(x)=Ax−h and the nonlinear case are discussed. In the linear case, a least-square solution is obtained in no more thann steps, and a method of obtaining the least-square solution of minimum length is given. In the nonlinear case, the CGS algorithm is combined with the Gauss-Newton process to minimize sums of squares of nonlinear functions. Results of numerical experiments with several versions of CGS on test functions indicate that the algorithms are effective.
Abstract: Quasi-Newton methods accelerate gradient methods for minimizing a function by approximating the inverse Hessian matrix of the function. Several papers in recent literature have dealt with the generat...
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Abstract: Quasi-Newton methods accelerate gradient methods for minimizing a function by approximating the inverse Hessian matrix of the function. Several papers in recent literature have dealt with the generation of classes of approximating matrices as a function of a scalar parameter. This paper derives necessary and sufficient conditions on the range of one such parameter to guarantee stability of the method. It further shows that the parameter effects only the length, not the direction, of the search vector at each step, and uses this result to derive several computational algorithms. The algorithms are evaluated on a series of test problems.
This paper is concerned with the problem of investigating the properties and comparing the methods of nonlinear programming. The steepest-descent method, the method of Davidon, the method of conjugate gradients, and o...
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This paper is concerned with the problem of investigating the properties and comparing the methods of nonlinear programming. The steepest-descent method, the method of Davidon, the method of conjugate gradients, and other methods are investigated for the class of essentially nonlinear valley functions.
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