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This work considers the regulator problem for single input-output linear plants in which only the plant output is available for control. A method is presented to specify an optimal nonlinear controller characteristic, constrained to lie within a given Popov sector, that yields improved performance over that obtained by proportional controllers. System performance is measured by a quadratic performance index, and performance is compared with that of both proportional and optimum linear state regulators.

The analysis of time series data can be significantly eased by the use of orthogonal polynomials. This technique is based on traditional regression analysis, but the necessity for solving the associated normal equations is eliminated. Also, the orthogonal polynomial technique includes a means to statistically test the level of fit of the independent terms in a forecast model (e.g., constant, linear, or quadratic). This paper shows the application of orthogonal polynomial technique to the analysis of time series data, but more importantly develops a mathematical means for updating the forecast model from one period to the next without completely recomputing the model from stored historical data. This result considerably improves the ease of application of the orthogonal polynomial technique and increases the value of the technique in practice.

This paper gives counterexamples to: (1) Ritter's algorithm for the global maximization of a quadratic subject to linear inequality constraints, and (2) Tui's algorithm for the global minimization of a concave function subject to linear inequality constraints.

The situation in which two groups of people have conflicts of interest is considered as a two-team zero-sum game problem. Two special cases of this problem are solved to illustrate that communication among members of a team may not be worth-while and extra information need not always be desired by decision makers. In the appendix, it is shown that the optimal saddle-point solution exists and is still affine for the general problem with quadratic Gaussian performance index.

The capital budgeting problem is treated as a part of the general theory of choice, where utility is to be maximized subject to the opportunities and constraints. The utility function is assumed to be quadratic to take care of any risk-averse behavior of the shareholders and a quadratic mixed zero-one integer programming model is developed. Special solution techniques for the resulting mixed zero-one integer quadratic programming model are discussed.

A queueing problem is considered with one server, independent discrete inter-arrival times, and independent discrete service times. The steady state distribution of the waiting time is approximated by means of the solution of certain linear equations and applying a generalization of the Lin-Bairstow method for finding quadratic factors of polynomials.

A Bayesian decision theoretic approach is employed to compare sampling schemes designed to estimate the reliability of series and parallel systems by testing individual components. Quadratic loss is assumed and schemes are found which minimize Bayes risk plus sampling cost. Several kinds of initial information concerning the reliability of the individual components, all of which assume the components function independently, are considered. The case for which the initial information is in terms of the system's reliability is briefly considered and related to the aforementioned case

For a linear Nth-order time-invariant system with scalar control variable and quadratic performance index, a procedure is given for computing the optimal value of the feedback vector k. Instead of determining, as a first, step, the N(N+1)/2 elements of a symmetric matrix, the K values of k are computed directly, leading to substantial simplifications, and making the method well suited for higher-order systems. In contrast to other methods starting from the same premise, no restrictions are imposed on the form of matrix A. In addition, and based on the previous considerations, a solution of the inverse optimal control problem is also presented. The form of the weighting matrix in the performance index can be chosen arbitrarily (e.g. as diagonal). In contrast to other known procedures, the computations are straightforward, and again no restrictions are imposed on the form of the system.

A design approach is presented for systems operating over a relatively short period of time about various operating points with state variable constraints. This class of problems is especially relevant to certain flight control problems. The design approach is applied to a simplified model of longitudinal dynamics of the F-4 aircraft operating in three widely separated flight conditions. A linear model is assumed about each operating point. Control is achieved via constrained state feedback. The basic problem is then to minimize a suitable integral quadratic performance measure subject to state variable constraints. The main theoretical result is theorem 1 which supplies the inequality constraints required to guarantee short-time stability. The short-time optimization problem is ultimately reduced to a nonlinear programming problem with inequality constraints.

The state of a continuous system most naturally enters the performance index as a discrete sum if there is a finite set of observation times, or if there are only a few key times over which errors really matter. Such continuous-discrete problems are studied here for linear plants under a broad class of continuous metrics on control and summed metrics on trajectory error. The approach is through the dual optimization problem and alignment properties of the primal and dual solution. The dual problem, is often more attractive computationally than the primal, and is always finite-dimensional as opposed to the infinite-dimensional primal problem. Occasionally the dual problem has a straightforward analytical solution, as in the case of the quadratic continuous-discrete regulator. In any case, information on the structure of the primal control is available through the transformation even before the dual problem is solved. Three examples are detailed involving L1, L2 and L∞ measures on trajectory errors.

In an earlier paper, it was shown that the linear and quadratic programming problems in complex space could be unified in the “complex linear complementarity problem” (complex LCP). An existence theory for this problem was provided. The present paper addresses the problem of actually solving the complex LCP and hence complex linear and quadratic programs as well. Two solution procedures are described and an example is given.

A mean-variance portfolio selection model suitable for the small investor is formulated as a sequence of quadratic integer programming problems. The special structure of these quadratic problems is exploited in a partial enumeration algorithm which uses cutting planes to accelerate convergence. Computational experience is reported on problems ranging in size from fifteen to fifty variables.

Wheeler [6] proposes simple formulas for quickly determining the sample sizes necessary to achieve a prescribed power. We find Wheeler's evaluation lacking in accuracy because of his application of a linear approximation to quadratic formulas. Therefore, “portable power” is potentially harmful to the experimenter who may not be knowledgeable in the theory.

A principal pivoting algorithm is given for finding local minimizing points for general quadratic minimization problems. The method is a generalization of algorithms of Dantzig, and Van de Panne and Whinston for convex quadratic minimization problems.

An ordinary differential equation for the groundwater allocation problem is derived by variational techniques and solved by specifying either the length of a mining period, or the total volume to be mined. Specifying either of these uniquely determines the other in terms of the economic parameters entering through the benefit function. An infinite mining period sets an upper limit to the total volume to be optimally extracted, and provides for the greatest economic benefit. As the mining period is shortened, the one-time exploitable volume is reduced. Although most of the calculations have been carried out for a quadratic benefit function, the detected competition between marginal productivity and the rate of interest insofar as they influence the optimal storage volume and its allocation through use rates appears to be valid in general.

This paper provides a method for solving mixed integer quadratic programs with the help of cutting-plane technique.

The decision-theoretic approach to point estimation involves the choice of an estimate to minimize the expected loss associated with the estimate. The purpose of this paper is to indicate the influence of risk aversion on point estimates for classes of payoff functions including the piecewise linear and quadratic payoff functions. Increased risk aversion results in a point estimate closer to zero for a quadratic payoff function and a lower estimate with a piecewise linear payoff function, for example.

In this paper chance-constrained programming is used to model a system of linked multipurpose reservoirs. The objective of the model is to determine an optimal operating policy for a given time sequence of minimum and maximum reservoir levels. The unregulated inflows into the reservoirs are assumed to be stochastic with a known distribution for each time frame. The chance-constraints, based on material balance equations, are converted to an equivalent linear deterministic constraint set. The linked system model is a natural extension of the single reservoir model. The problem of stochastic demands as well as stochastic inflows is shown to be a straightforward generalization of the stochastic inflow problem. Since the resulting constraints for these models are linear and the decision variables are deterministic rather than random variables, linear, quadratic, and even general convex objective functions can be readily handled.

and IntroductionQuadratic programming means maximizing or minimizing a quadratic function of one or more variables subject to linear restrictions i.e. linear equations and/or inequalities.Among the numerous insurance problems which can be formulated as quadratic programs we shall only discuss four, namely the Credibility, Retention, IBNR and the Cost Distribution problems.Generally, there is no explicite solution to quadratic optimization problems, only statements about the existence of a solution can be made or some algorithm may be recommended in order to get exact or approximate numerical solutions. Restricting ourselves to typical problems of the above mentioned type, however, enables us to give an explicit solution in terms of general formulae for quite a number of cases, such as the onedimensional credibility problem, the retention problem and—under relatively week assumptions— for the IBNR-problem.The results given here are by no means new. The only goal of this paper is to describe a few fundamental insurance problems from a common mathematical standpoint, namely that of quadratic programming and at the same time, to draw attention to a few special aspects and open questions in this field.

A quadratic performance index together with a set of prescribed closed-loop eigenvalues are considered as criteria for designing discrete multi-variable control systems. The design method presented in the paper is an extension of an earlier solution of the equivalent problem for continuous systems.

This paper discusses the problem of determining which of a large set of possible but improbable malfunctions gave rise to a given set of measurements. The classes of systems under consideration generally lead to underdetermined sets of equations. Three methods of formulating and solving this class of problems are presented: 1) the pseudoinverse method: this leads to an easily-solved computational problem but it is not physically realistic and it tends to give poor results; 2) a pattern recognition approach based on a more realistic problem formulation: unfortunately, the computational problems associated with this formulation may be formoidable; and 3) a quadratic programming approack: this is based on minimization of a physically realistic objective function. A bmaosdification to eliminate discontinuities in the objetive function and a quasilinearization transform the original problem to an inequality-constrained quadratic minimization problem, which is readily solved by Lemke's complementary pivoting method. A sequence of successive quasilinearizations and estimations is defilned which is proved to converge to a minimum of the original objective function. In tests this convergence occurred very fast. Examples are given; very general classes of problems are discussed which can be handled in this way.

Optimal sampled-data controls for linear processes with quadratic criteria are determined through application of the discrete minimum principle. The effect of sampling on the closed-loop system's performance is investigated and the asymptotic behaviour of the optimal cost for large sampling periods is determined. The resulting design method is applicable to continuous, sampled-data and discrete regulators.

The problem of obtaining maximum likelihood estimates for the parameters involved in a stationary single-channel, Markovian queuing process is considered. A method of taking observations is presented which simplifies this problem to that of determining a root of a certain quadratic equation. A useful and even simpler rational approximation is also studied.

Necessary and sufficient conditions are developed for the existence and calculation of well-defined Riccati differential equation solutions associated with quadratic loss minimization problems. Of particular interest is the fact that a. covariance condition is involved. The disclosure of this condition not only extends the range of optimal control problems for which a solution, guaranteed to be well defined, may be calculated, but also introduces an approach for establishing the existence of well-defined solutions in other problems involving covariance conditions, as for example, in a time-varying spectral factorization procedure. This paper is concerned with finite time results, whilo a companion paper considers tho infinite time case.

The climate of a system is identified with the set of long-term statistical properties. Methods of deducing the climate from the equations which govern the system are enumerated. These methods are illustrated by choosing a first-order quadratic difference equation in one variable as a governing equation. The equation contains a single parameter. Particular attention is given to the climatic mean of the single variable. Analytic methods yield the climate in some cases where the system varies periodically, but generally fail when the system varies nonperiodically. Numerical integration yields a value of the climatic mean for any individual value of the parameter. Additional analytic reasoning is needed to determine the nature of the climatic mean as a function of the parameter. The progression from steady-state to periodic to nonperiodic behavior, as the parameter increases, is compared to the progression from steady-state to periodic to irregular flow in the rotating-basin experiments, as the rate of rotation increases. DOI: 10.1111/j.2153-3490.1964.tb00136.x

A linear time and state dependent approximation to a non-linear and non-stationary system is optimized with respect to a quadratic performance index by treating it as an instantaneously linear stationary system. By allowing for the variations of the model a realizable adaptive typo of controller can be devised.

A formal Bayesian framework is provided for the problem where some possibly contaminated data item can be collected periodically and where additional information about each period's true data value can be obtained at some cost. A decision is made each period with loss depending on the true data value and the decision. The special case with normal distributions and quadratic loss is formulated as a dynamic programming problem, and results include proof that the optimum policy structure calls for a more expensive form of data collection initially with less expensive data collection after the process parameters are better known.

This paper investigates the problem of scheduling production from one period to the next to minimize long-run expected costs where the only relevant considerations are the expense of revising production and the charges related to inventory. We first consider the case of constant sales known over time in order to compare exact and approximate solutions. The approach to these solutions is based on minimizing the costs of deviating from long-run equilibrium values of production and inventory. The same approach is then used to handle the more realistic possibility of uncertain future demands. Solutions are obtained for both linear and quadratic inventory and production cost functions. Because future demand is assumed to be specified by some probability distribution, inventory costs must be approximated in order to obtain a closed-form solution; however, the result should differ insignificantly from an exact optimal solution.

In this paper two computational techniques are presented, to solve non-linear differential games which have a value. It is assumed that there may be interaction in the dynamics of the players and that the conflicting index of performance of the game is a scalar function of the final state of the players of the miss-distance type. The first of these techniques is based on the Newton-Raphson method in function space in association with one of the two following non-iterative procedures to solve linear differential games with quadratic pay-off function : the generalized Riccati transformation or the transition matrix method. The second computational technique presented is based on the gradient of the Hamiltonian function with respect to the strategies. Some numerical examples are given to illustrate the methods.

A new approach for optimum control of stochastic non-linear systems is developed from practical engineering assumptions. Systems amenable to this new approach include optimum guidance and navigation systems for space and terrestrial vehicles, optimum closed-loop process controllers and optimum controllers for systems with unknown parameters. The classical quadratic synthesis approach-optimization of a deterministic cost and perturbation estimation and control about that solution-is shown to give 24 per cent more cost and 97 per cent more mean-squared terminal error than the combined optimization approach presented for a sample problem involving control of a first-order system with an unknown time constant. A similar improvement in terminal error is shown for an atmospheric entry problem. Furthermore, the optimum controller automatically designs the best controller to minimize the effects of the unknown parameter without artificial augmentation of the cost function as is done in the sensitivity theory approach. The solution is obtained by expansion of the cost function in a power series around a deterministic trajectory with the assumption of linear perturbation estimation and control about that trajectory. Optimization of the expanded cost function gives necessary conditions dependent on the covariance matrices and the deterministic portion of the cost. When the necessary conditions are solved, a set of open-loop controls, perturbation controller gains, and perturbation estimator gains are obtained that can be precomputed and implemented into the system.

Optical experimental designs are developed for the inverse estitator in the Linear Calibration problem using the crieterion of minimum intetrated or average mean squared error. Designs are developed for a linear approximation when the true model is Linear and when it is quadratic. In both cases, the optimal designs depend on unknown model parameters and are not realistically usable. Eowever, designs are shown to exist which are near optimal and do not depend on unknown model parameters

This paper investigates the proper selection of a performance index to satisfy a word statement of desired effect. A performance index formed as the time integral of Un is solved for various values of n. When viewed in this light, the classical quadratic index (n = 2) gives a seemingly arbitrary solution. From this investigation the true optimal control solution can be ascertained. For the different values of n, solutions for totally different performance can be obtained. Performance indices combining two effects of minimum control effort and minimum response error are also investigated. The differences of a summed combination and a multiplicative combination are studied.

In this paper, the problem of minimizing a functionf(x) subject to a constraint ϕ(x)=0 is considered, wheref is a scalar,x ann-vector, and ϕ aq-vector, withq

An important parameter in the design of countercurrent dialyzers is the permeability of the membrane which separates the blood and dialyzate. In this paper, methods of optimization theory are applied to specify membrane permeability along the length of a countercurrent dialyzer operating under quasi-static conditions. Two optimization problems are considered. In the first the performance functional is a quadratic measure of membrane complexity; in the second it includes also a quadratic measure of the deviation of the blood output concentration from a desired (low) level. For both situations it is shown that the optimal membrane is one of constant permeability along its length, a result which is intuitively appealing and has an obvious economic consequence. Formulas for computing the optimal permeabilities are given, and some problems for future research in this area are outlined. A generalization of the principal result is also included, and some deficiencies in the present analysis are discussed.

The numerical solution of nonlinear programming problems by first-order techniques, such as gradient projection techniques, is difficult when ridges are encountered. Ridges are described and analyzed. Second-order information about the augmented objective function, properly used, can yield a search direction that moves along a ridge, even in the constrained case. The analysis applies to situations in which the objective function (maximization) is not necessarily concave and the constraint functions (G j ⩽ 0) are not necessarily convex.

This note considers the question of why some convex quadratic programming algorithms fail and others succeed when applied to nonconvex quasiconvex quadratic programs. Several algorithms are identified as being capable of solving quasiconvex quadratic programs using only a finite number of arithmetic and logical operations. These algorithms are all primal feasible, pivot algorithms.

The calculus of variations is applied to the problem of finding the optimal control for linear systems with quadratic costs which are not necessarily positive definite. The conjugate point condition is transformed to an explicit condition upon the parameters of the problem, in the one-dimensional case for arbitrary costs and in the n -dimensional case for final costs only. The expressions for the optimal controls are given. A numerical example of a second-order system is solved.

The paper gives a comparison of Beale's method for quadratic programming and the Simplex Method for quadratic programming as developed by Dantzig and Van de Panne and Whinston. After summary expositions of both methods and a reformulation of Beale's method in the framework of quadratic Simplex tableaux, the relation between the two methods is analyzed. A numerical example is used as illustration.

The search for a linear Λ-minimax estimate (see, e.g., [19]) of a multivariate location parameter (with quadratic loss) is reduced to a problem of maximizing a continuous function on a compact, convex set—a problem which can be solved numerically with little difficulty. This reduction is accomplished by showing that a minimax theorem applies, thus changing a difficult minimax calculation to a simpler maximin problem. The technique is observed to be applicable to other, related problems. Also considered is the choice of experiment under the assumption that the decision maker can, at some cost, improve the specification of his prior distribution.

The role of linear control theory as an aid to the integral control of hydrologic systems is investigated for the case of a combined lake and aquifer storage system that supplies either a deterministic or stochastic water demand. Only lumped time-invariant systems are considered but both deterministic and stochastic inflows to storage are allowed. The computational example allows for recharge of lake water into the aquifer as well as for the subsequent diversion of pumped groundwater back to the lake. Stability criteria are presented for the closed-loop features of the overall control system. Under a quadratic loss criterion, a calculus of variations problem, subject to constraints imposed by the system equations can be solved for the optimal release policy from the lake and aquifer and optimal feedback policy from aquifer to lake.

Through a modification of Roy's principle of safety first , the behavioral bounding constraints often encountered in descriptive applications of linear programming are interpreted as planes of support for a quadratic constraint that reflects the decision-maker's attitude toward uncertainty. In this way a linkage is established from the bounding constraint to the elements of uncertainty, the variances and covariances of relevant random variables. To illustrate the development, an application is made to interpret the “flexibility” constraints used in the recursive programming models of Henderson-Day. In contrast to the assumptions made in most studies using such constraints, we find that to be consistent with the modified safety first principle, the “flexibility” coefficients cannot be constant, but must change from period to period through a well-defined (though complicated) dependence on past levels of decision variables.

Hestenes' method of multipliers is used to approximate a quadratic optimal control problem. The global existence of a family of unconstrained problems is established. Given an initial estimate of the Lagrange multipliers, a convergent sequence of arcs is generated. They are minimizing with respect to members of the above family, and their limit is the solution to the original differentially constrained problem.

This paper deals with a general method for the optimization of multivariable systems with non-uniform sampling chosen in an optimal way. The algorithm is based on Dynamic Programming and is applied to extend the Kalman-Koepcke optimization method for a linear dynamic system, with a quadratic performance index. In the general multivariable case, the method implies the solution of a two-point boundary value problem. In the scalar case, which has only one input and one output, the sampling periods may be found separately from the control law. The computational difficulties connected with the implementation of the method are discussed and simplifications, possible for discretized continuous systems, are derived. Numerical examples are given.

A new technique for the approximate computation of optimal controls is presented, which avoids the need to solve the differential equation governing the behavior of the system. The technique is shown to be applicable to linear systems governed by partial differential equations when the cost functionals are quadratic. Both systems with distributed and boundary control are examined. For each case the quadratic cost functional is built up by taking either distributed or final observation into account.

A technique which calculates approximate solutions to optimal control problems is developed in this work. The technique, which uses an empirically found weighted quadratic Liapunov function to transform the original n-dimensional optimization problem into a scalar optimization problem, is applied to a number of optimal control problems typical of those encountered in chemical engineering practice. The problems considered include both linear and nonlinear, lumped and distributed systems with minimum time, and quadratic and final value type performance indices. The solutions of these control problems show that the Liapunov-like suboptimal method, in general, requires less computer time and storage than that required by iterative and noniterative optimal methods currently used to solve optimal control problems. The reduction in computer storage and time enables the Liapunov-like technique to handle large dimensional control problems with relatively little effort.

This paper develops a dynamic programming model for selecting an optimal combination of transportation modes over a midtiperiod planning horizon. The formulation explicitly incorporates uncertainty regarding future requirements or demands for a number of commodity classes. In addition to determining the optimal modes to employ, the model assigns individual commodity classes to various modes, determines which supply points serve which destinations, and reroutes carriers from destinations to alternative sources where they will be most effective. The model is formulated as an optimal discrete time stochastic control problem where cost is quadratic and dynamic equations linear in the state and control variables. This model may be solved in closed form by an efficient dynamic programming algorithm that permits the treatment of relatively large scale systems. Also developed is an alternative, generally suboptimal method of solution, based upon solving a sequence of convex programming problems over time. This technique may be employed for a more general class of problems. In both methods the use of “shadow prices” that arise is discussed.

The problem of scheduling tasks with deadlines and linear loss functions was discussed in some detail in two papers [McNaughton, R. Scheduling with deadlines and loss functions. Management Sci. 6(1) 1–12; Schild, A., I. J. Fredman. On scheduling tasks with associated linear loss functions. Management Sci. 7(3) 280–285] by Robert McNaughton and the authors. In this paper, the problem of scheduling with nonlinear loss functions is attacked. In particular, certain criteria will be established for quadratic loss functions and an algorithm is proposed which will yield an optimal scheduling—i.e., a minimal total loss—for nonlinear loss functions in general.