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The reliability literature offers an abundance of methods for the optimal design of systems under some constraints. In most of the papers, the problem considered is: given reliabilities of each constituent component and their constraint-type data, optimize the system reliability. This amounts to the assignment of optimal redundancies to each stage of the system, with each component reliability specified. This is a partial optimization of the system reliability. At the design stage, a designer has many options, e.g., component reliability improvement and use of redundancy. A true optimal system design explores these alternatives explicitly. Our paper demonstrates the feasibility of arriving at an optimal system design using the latter concept. For simplicity, only a cost constraint is used, however, the approach is more general and can be extended to any number of constraints. A particular cost-reliability curve is used to illustrate the approach.

The problem of designing broadband multisection stepped rectangular waveguide impedance transformers, when the input and output guides have different cutoff frequencies but propagate the same mode, is formulated in general terms for direct optimization by digital computer. The formulation is sufficiently flexible to allow nonideal junction discontinuity effects and mismatched terminations to be taken into account during optimization. Constraints placed on the width, height, or length of any section need be dictated only by considerations for dominant mode propagation and the requirement of small (but not necessarily negligible) junction discontinuities. The objective of the present formulation is a minimax equal-ripple response over a predetermined frequency band satisfying the constraints selected for the particular problem. The ripple search strategy to locate the maximum reflection coefficient within the band and the razor search strategy to minimize it, as described by Bandler and Macdonald in another paper, were employed. Constrained optimum equal-ripple solutions to examples previously published by Young, Matthaei et al. and Riblet are presented. They demonstrate the considerable improvements made possible by the present formulation with regard to performance, reduction in number of sections, and physical size. The approach used in this paper should also find application in the design of broadband microwave matching or equalizing networks consisting of noncommensurate components and for which exact synthesis techniques may be unavailable.

Among the performance indices used in the processing of acoustic array signals are those that can be expressed as a ratio of quadratic forms in the beamformer weights. Popular indices in this class include directivity, array gain, efficiency factor, and supergain ratio. This paper presents a technique for optimizing a quadratic-form-ratio performance index over the beamformer weights subject to generally nonzero constraints on the value of the array power pattern in multiple directions. Previous related results include performance optimization with no constraints, with specified pattern nulls, and with constraints on the complex-valued beam pattern. When an array designer is interested just in the beam pattern magnitude, the additional specification of pattern phase required by these techniques generally penalizes the optimization. Only magnitude (power pattern) constraints are imposed in this paper. The optimum weights are shown to satisfy a characteristic equation with quadratic constraints that is related to a more familiar optimization problem for which solution methods exist. Applications include performance maximization with fixed main-lobe beamwidth and/or sidelobe values. Examples are given for a circular array.

This paper defines a mathematical model which simulates the growth of a power system and determines the least cost expansion plan for a system of distribution substations. A new approach employing linear and integer programming is used to optimize the system's substation capacities subject to the constraints of cost, load, voltage, and reserve requirements. The model has been successfully applied to a 1600 square mile urban area served by 70 distribution substations.

This paper describes two numerically stable methods for unconstrained optimization and their generalization when linear inequality constraints are added. The difference between the two methods is simply that one requires the Hessian matrix explicitly and the other does not. The methods are intimately based on the recurrence of matrix factorizations and are linked to earlier work on quasi-Newton methods and quadratic programming.

The main objective of this paper is to give a survey of recent automatic optimization methods which either have found or should find useful application in the area of computer-aided network design. Huang's family of algorithms for unconstrained optimization is reviewed. The Fletcher method and the Charalambous family of algorithms for unconstrained optimization, which abandon the "full linear search," are presented. Special emphasis is devoted to algorithms by Bandler and Charalambous on least pth and minimax optimization which can be readily programmed and used. Due to work by Bandler and Charalambous, it is shown how constrained minimax problems can be solved exactly as unconstrained minimax problems by using a new approach to nonlinear programming. The application of minirnax optimization on the design of lumped-distributed active filters, problems for future investigation, and a select list of references are also included.

Consider the problem of finding the local constrained minimum $x_0 $ of the function f on the set \[ F = \left\{ {x \in R^n |\phi _i (x) \geqq 0,\quad \psi _{j + k} (x) = 0;\quad i = 1,2, \cdots ,k;\quad j = 1,2, \cdots ,l} \right\}.\]One method of solution is to minimize the associated penalty function \[ p_0 (x) = \mu f(x) - \sum _{i = 1}^k {\min } \left( {0,\phi _i (x)} \right) + \sum _{j = 1}^l {\left| {\psi _{j + k} (x)} \right|} \quad {\text{for }} x \in R^n ,\quad \mu \geqq 0. \]Let $x(\mu )$ be the minimum of this penalty function. It is known that, provided $\mu $ is sufficiently small, $x(\mu ) = x_0 $.However, until recently, a serious drawback to this particular penalty function was that its first order derivatives are not everywhere defined. Thus, well-known gradient type methods usually applied to unconstrained optimization problems were necessarily excluded.This paper presents a method that enables a modified form of the gradient type approaches to be applied to a perturbation of the penalt...

In this paper a stochastically constrained replacement model is formulated. This model determines a sequence of replacement dates such that the total “current account” cost of all future costs and capital expenditures over an infinite time horizon for the n initial incumbent machines is minimized subject to the constraints that an expected number of machines are in a chosen utility class at any point in time. We then indicate one possible solution method for the model.

Choice among risky investments has been described using a chance constrained programming model with a finite number of states of nature. This paper presents a simple combinational algorithm for solving this model which, at worst, requires solving a number of linear programming problems.

Abstract A penalty-function technique is described that can be used to solve constrained nonlinear optimization problems. The method is based upon the strategy of transforming the original constrained optimization problem into an equivalent unconstrained problem, and then solving a sequence of unconstrained optimization problems. This strategy is particularly attractive because unconstrained optimization problems are generally much easier to solve than constrained problems. The method is applied to a cyclic steam injection process that has been studied previously using process that has been studied previously using other techniques. Detailed solutions are presented and the characteristics of the method are discussed The method works, and it exhibits several favorable characteristics, such as applicability to a wide variety of nonlinear problems and freedom from convergence difficulties or computer storage limitations. Moreover, the method can be understood and implemented on a digital computer with ease. Introduction It is difficult for the analyst to recognize the methods that are best suited to a particular problem. Recent texts on optimization theory have not aided the situation because little is presented about which techniques are most applicable to different problems. Thus, the difficulties of identifying particular optimization techniques for various specialized applications remain largely unresolved. This paper presents a penalty-function optimization algorithm is suited to a variety of nonlinear engineering optimization problems. The method is applied to a problem that has been previously optimization of a cyclic steam injection previously optimization of a cyclic steam injection process. This example has a serial structure and may, process. This example has a serial structure and may, therefore, appear to be an unlikely application for a penalty-function optimizer. There are, however, several significant advantages to the use of a penalty-function algorithm rather than dynamic penalty-function algorithm rather than dynamic programming or the maximum principle. programming or the maximum principle.

This paper describes an algorithm, based on a recently proposed method due to D. D. Morrison, for solving constrained optimization problems, and shows that it compares favorably with existing methods in several examples.

This paper deals with optimal control in multilevel systems. The decomposition of a system into N subsystems is presented as a problem of formulating the performance index P(m) as a function of N components P(P1, P2,..., PN) and of transforming the system constraint m?R into a set of constraints m1?IR1(v), m2?R2(v),..., v?Rv, where v is the coordination variable. Ways of achieving this goal as applicable to typical systems are presented. Some aspects of choosing the coordination variable and the tradeoffs involved are discussed. Lagrangian methods as used previously by Lasdon and Pearson are shown to be a particular case of parametric optimization, and the range of their applicability is specified. Simple examples of static optimization serve to illustrate the approach.

The paper develops a-method of determining the minimum amount of reactive power capacity installation required to satisfy constrained system voltage conditions. The solution gives both location and amount of reactive sources. Sensitivity parameters are the basis of the model, and linear programming is used to obtain the solution. The method is practical and static operating constraints are naturally included. Numerical examples illustrate the application of the method.

Recent developments in modern control theory and high-speed computation techniques have enabled extensive treatment of complex processes. A method is presented for building a model of an R& D process utilizing the state space approach. The specific model described in this paper contains 21 state variables and a control vector of 6 components. Its formulation leads to the Mayer's problem with inequality constraints imposed on the control vector. An algorithm based on the adjoint system technique is used for simultaneous optimization and simplification. The computation for the preceding model lasts 2.5 min and is completed within 5 iterations. It results in an improvement factor of three for the chosen index of performance. Simplification of the model reduces the 21 state variables and 6 control components into 1 state variable and 1 control component.

It is widely recognized that continuously tapered strip-line components afford certain important advantages over their discrete counterparts in practical applications. (The question of coupler directivity, for instance, must come under close scrutiny as requirements become more severe in terms of frequency and bandwidth.) It is the purpose of this paper to set forth principles by which such components may be realized as a consequence of design techniques tailored to consistency with a given set of operational requirements applicable at once to an aggregate of different microwave strip-line components. In general, some or all of the components in such a group will differ in their nominal response characteristics. Derived through judicious application of constraints in an optimization procedure, the method obviates the need for intracircuit transformers or connectors transmission line cross section geometry compromises, component overdesign, etc., while at the same time providing mutual component compatibility and preservation of the quality of quasi-TEM microwave signal processing for a given specified bandwidth. By way of example, compatible multioctave designs have been established for components having nominal, equal-ripple responses differing by 1.7 dB. Corresponding conventional designs would have required impedance ratios of as much as 3.68:1 for a dielectric spacing such that s/b=1/9. Of particular concern are couplers with symmetric and nonsymmetric local coupling coefficient distributions and dispersive all-pass phase shift network elements. The general nonsymmetric coupler is realized without abrupt discontinuities at either end and with a constant specified unbalance phase (0

The purpose of this paper is to present the theory and develop an algorithm associated with the exploration of a dual response surface system. The approach is to find conditions on a set of independent or “design” variables which maximize (or minimize) a “primary response” function subject to the condition that a “constraint response” function takes on some specified or desirable value. A method is outlined whereby a user can generate simple two dimensional plots to determine the conditions of constrained maximum primary response regardless of the number of independent variables in the system. He thus is able to reduce to simple plotting the complex task of exploring the dual response system. The procedure that is used to generate the plots depends on the nature of the individual univariate response functions. In certain situations it becomes necessary to apply the additional constraint that the located operating conditions are a certain “distance” from the origin of the independent variables (or the cent...

The development of a computer program for the optimization of dynamic systems subject to parameter and terminal state constraints is presented in this paper. The problem is handled by converting it to an equivalent algebraic optimization problem. The resulting problem is then solved by a modified version (D YS UMT) of the nonlinear programming method S UMT (Sequential Unconstrained Minimization Technique). The available program provides an efficient and convenient analysis tool to aid engineers in the modeling and designing of dynamic systems.

This paper proposes a methodology for optimizing resource allocations within a real-time information system, given the availability constraints. In essence, the methodology reformulates the availability constraints into a state transition matrix. The state probabilities, computed from the transition matrix, form a portion of the input to the optimizing algorithm. This algorithm optimizes an arbitrary objective function of system resources, subject to the reformulated availability constraints.

This paper presents a penalty function approach to the solution of inequality constrained optimal control problems. The method begins with a point interior to the constraint set and approaches the optimum from within, by solving a sequence of problems with only terminal conditions as constraints. Thus, all intermediate solutions satisfy the inequality constraints. Conditions are given which guarantee that the un "constrained" problems have solutions interior to the constraint set and that in the limit these solutions converge to the constrained optimum. For linear systems with convex objective and concave inequalities, the unconstrained problems have the property that any local minimum is global. Further, under these conditions, upper and lower bounds in the optimum are easily available. Three test problems are solved and the results presented.

Several techniques in mathematical programming involve the constrained optimization of an extremal-value function. Such functions are defined as the extremal value of a related parameterized optimization problem. This paper reviews and extends the characterization of directional derivatives for three major types of extremal-value functions. The characterization for the completely convex case is then used to construct a robust and convergent feasible direction algorithm. Such an algorithm has applications to the optimization of large-scale nonlinear decomposable systems.

In this paper, a description of a technique for improving transient performance in the velocity inertial system while minimizing the standard deviation of the system error, (i.e., standard deviation of the system velocity measurement error ?v2) is given. To accomplish this objective, a prescribed settling time constraint is formulated for the system response. The constraint equation is then imposed on the minimization process. The result shows that considerable settling time improvement can be derived without sacrificing system accuracy.

Bandlimited functions play important roles in the design of signal transmission systems; antennas; digital, analog, and optical filters; tapered transmismon lines, etc. The basic problem arising in all the above-mentioned applications consists in approximating a given nonbandlimited response with a bandlimited system function. This paper presents 1) a unified treat-merit of the mathematical properties of bandlimited system functions; 2) analytical and numerical methods for the optimization of bandlimited systems; and 3) the application of the theoretical results to practical engineering problems.

The problem of detection of an increase in runoff due to precipitation management is considered. Classical methods of detection are reviewed and their shortcomings are discussed. The concept of grouping of observations is introduced. The paper answers two fundamental questions: Given a region consisting of B basins in which changes are suspected and given that (economic, or financial, or other) constraints limit to b the number of basins where measurements can be obtained, which basins should he selected? How should the measurements in individual basins be combined? The answers are obtained by a constrained optimization procedure. When applied to the Colorado River Basin Pilot Project area the power of the test, expressed in years needed for detection, is increased by a factor of 2.

The now classical problem of optimlum dispatching miimizes the cost of production and transmission of electrical power to meet a specified demand under normal operating conditions. It appears logical and desirable to utilize and extend the methodology of optimum dispatching to problems encountered during abnormal operating conditions. This paper, as a first step in that direction, discusses a systematic approach toward minimizing the curtailment of service in a power system after a severe fault. The problem of minimizing load curtailment under a given set of emergency conditions is formulated as a problem of static optimization, subject to operational and equipment constraints. First, a feasible steady-state solution is obtained for the postfault network configuration. Starting from this initial feasible solution, the optinum (minimum curtailment) is approached by a gradient technique. An efficient computational procedure is based on the Newton-Raphson technique for solving the power flow equations, and the Kuhn-Tucker theorem for the optimization. The analytical results are verified on a 26-node example problem. Two typical emergency situations are considered: the loss of generation and the loss of an interconnection tie line. The same optimization procedure and the computed dual (Lagrangian) variables can be utilized for computer programs involving optimum service restoration, generation reserve scheduling, and system expansion studies.

In this paper, signal design in a burst-noise environment is considered. Two distinct approaches are presented, each resulting in a class of signals for which the trade-off between signal bandwidth and immunity to large-amplitude bursts is shown. In the first approach the design is treated as an optimization problem subject to energy and endpoint constraints in a fixed time interval, and a uniform probability distribution of the location of the burst within the interval. The second approach considers signal design as a two-player differential game, where design flexibliity is obtained through the second player's choice of an initial value in the resulting differential equation. Performance curves for signals designed by using both methods are compared. It is shown how the first design procedure is easily extended to M -ary alphabets, thus obtaining the full advantage of the proposed technique.

In this paper, the hourly operating cost of an interconnected power system constrained by prescribed area interchanges is minimized. The scheduled area interchanges act as constraining equations in the economic optimization procedure. The hourly cost minimization is performed with respect to active and reactive generation and is subjected to constraints on voltage and power flows. This minimization is carried out under several assumptions, notably: optimum operation of some area k of the interconnected system, optimum economic operation of the whole system with the interchange constraints satisfied, and optimum operation of the whole system with the interchange constraints disregarded for increased overall economy. These three assumptions are encountered in the operation of power pools. A numerical example involving a 30-node system, real and reactive power with adjustable transformers and three areas is solved by two optimization procedures: the Penalty Function method and the Generalized Reduced Gradient. A system equivalent that includes the effect of the area interchange controllers upon the flow is developed.

This paper presents an. application, of functional analysis to the minimal∗time control problem where two or more energy constrained systems of the same order must be taken to an initially unknown common point in state space. Two different types of problems with different forms of the energy constraint are described in detail. The development depends on the fact that the reachable region of a system with an energy constraint on the input vector is an ellipsoid.

This paper presents a quadratically convergent, rank one, variance algorithm for constrained minimization with linear constraints. The algorithm does not require a linear search for a local minimum in every iteration. Linear searches are made only when needed for stability and convergence.

This paper presents a new class of criteria for the optimum Capacity assignment in store-and-forward communication networks under a total fixed-cost constraint. Compared with conventional average-delay optimization these criteria are more sensitive to the needs of the individual user. Closed-form results are attained.

The paper gives convergence theorems for several sequential Monte-Carlo or stochastic approximation algorithms for finding a local minimum of a function $f(\bullet)$ on a set $C$ defined by $C = \{x: q^i(x) \leqq 0, i = 1, 2, \cdots, s\}. f(\bullet)$ is unknown, but "noise perturbed" values can be observed at any desired parameter $x \in C$. The algorithms generate a sequence of random variables $\{X_n\}$ such that (for a.a. $\omega$) any convergent subsequence of $\{X_n(\omega)\}$ converges to a point where a certain necessary condition for constrained optimality holds. The techniques are drawn from both stochastic approximation, and non-linear programming.

This paper presents a new function minimization algorithm for minimizing nonlinear functions of a finite number of variables subject to nonlinear equality constraints. The algorithm also provides for the explicit handling of upper and lower bounds on each of the independent variables. Although other more general inequality constraint can be transformed into an equality constraint at the expense of introducing an additional slack variable. The algorithm proposed combines the idea of a "balance function," developed independently in References 1 and 2 with a second order method for updating the balance function La grange multipliers originally developed in Reference 3. This updating technique makes use of the current estimate of the inverse Hessian of the balance function which is a byproduct of the unconstrained minimization of the balance function using the Fletcher-Powell algorithm.