Showing papers on "Quadratically constrained quadratic program published in 1985"

On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem

Abstract: We present practical conditions under which the existence and uniqueness of a finite solution to a given equality quadratic program may be examined. Different sets of conditions allow this examination to take place when null-space, range-space or Lagrangian methods are used to find stationary points for the quadratic program.

Quadratic model for reservoir management: Application to the Central Valley Project

Abstract: A quadratic optimization model is applied to a large-scale reservoir system to obtain operation schedules. The model has the minimum possible dimensionality, treats spillage and penstock releases as decision variables and takes advantage of system-dependent features to reduce the size of the decision space. An efficient and stable quadratic programming active set algorithm is used to solve for the optimal release policies. The stability and convergence of the solution algorithm are ensured by the factorization of the reduced Hessian matrix and the accurate computation of the Lagrange multipliers. The quadratic model is compared with a simplified linear model and it is found that optimal release schedules are robust to the choice of model, both yielding an increase of nearly 27% in the total annual energy production with respect to conventional operation procedures, although the quadratic model is more flexible and of general applicability. The adequate fulfillment of other system functions such as flood control and water supply is guaranteed via constraints on storage and spillage variables.

Newton's method for constrained optimization

Abstract: We derive a quadratically convergent algorithm for minimizing a nonlinear function subject to nonlinear equality constraints. We show, following Kaufman [4], how to compute efficiently the derivative of a basis of the subspace tangent to the feasible surface. The derivation minimizes the use of Lagrange multipliers, producing multiplier estimates as a by-product of other calculations. An extension of Kantorovich's theorem shows that the algorithm maintains quadratic convergence even if the basis of the tangent space changes abruptly from iteration to iteration. The algorithm and its quadratic convergence are known but the drivation is new, simple, and suggests several new modifications of the algorithm.

Convex spline interpolants with minimal curvature

Abstract: The problem of finding convex spline interpolants with minimal mean curvature leads to a quadratic optimization problem of special structure. In the present note a corresponding dual problem without constraints is derived. Its objective function is piecewise quadratic and therefore admits an effective numerical treatment.

Quadratic geometric programming with application to machining economics

Abstract: Geometric Programming is extended to include convex quadratic functions. Generalized Geometric Programming is applied to this class of programs to obtain a convex dual program. Machining economics problems fall into this class. Such problems are studied by applying this duality to a nested set of three problems. One problem is zero degree of difficulty and the solution is obtained by solving a simple system of equations. The inclusion of a constraint restricting the force on the tool to be less than or equal to the breaking force provides a more realistic solution. This model is solved as a program with one degree of difficulty. Finally the behavior of the machining cost per part is studied parametrically as a function of axial depth.

A closed-form solution to a quadratic optimization problem in complex variables

Abstract: A special quadratic optimization problem in complex variables is investigated for a closed-form solution. Two different approaches are used. The first is a direct approach which leads to a family of solutions defined in terms of arbitrary complex constants. The second is an indirect approach based on parametrizing the objective function. It leads to a specific solution, which is a member of the above family and which is shown to be bounded.

Quadratic programming approach to reactive power optimization on the primary feeders

Abstract: In this paper we suggest the use of the quadratic programming technique to determine the optimum size and location of shunt capacitors on radial distribution feeders so as to maximize overall savings, including the cost of capacitors. The saving function which is of quadratic form is maximized for a set of linear inequality constraints by using quadratic programming. — For quadratic programming, efficient alogrithms have been developed which can easily be implemented on digital computers. — The approach is illustrated by an application to a typical distribution feeder of 23 kV.

A multi-stage quadratic programming optimization technique for optimal management of large hydro-power operations

Abstract: A multi-stage optimization technique is proposed for solving large optimization problems. The primary advantages of this technique include the utilization of little computer memory and the preservation of much of the complexities inherent in the actual problems. A demonstration of the technique is presented via its application to a complex hydro-thermal scheduling problem faced by a particular company. With this technique this complex problem is seen to be broken down into a sequence of quadratic programming sub-problems. Further simplification of each of these sub-problems is seen to be possible through the reformulation of these problems with matrices. As a result, the ease with which empirical sensitivity analyses can be performed becomes apparent and is demonstrated through the performance of such analyses on the presented hydro-thermal system.