Showing papers on "Quadratically constrained quadratic program published in 1989"

Experiments in quadratic 0-1 programming

Abstract: We present computational experience with a cutting plane algorithm for 0–1 quadratic programming without constraints. Our approach is based on a reduction of this problem to a max-cut problem in a graph and on a partial linear description of the cut polytope.

A robust sequential quadratic programming method

Abstract: The sequential quadratic programming method developed by Wilson, Han and Powell may fail if the quadratic programming subproblems become infeasible, or if the associated sequence of search directions is unbounded. This paper considers techniques which circumvent these difficulties by modifying the structure of the constraint region in the quadratic programming subproblems. Furthermore, questions concerning the occurrence of an unbounded sequence of multipliers and problem feasibility are also addressed.

A Dual Method for Certain Positive Semidefinite Quadratic Programming Problems

Abstract: This paper presents a dual active set method for minimizing a sum of piecewise linear functions and a strictly convex quadratic function, subject to linear constraints. It may be used for direction finding in nondifferentiable optimization algorithms and for solving exact penalty formulations of (possibly inconsistent) strictly convex quadratic programming problems. An efficient implementation is described extending the Goldfarb and Idnani algorithm, which includes Powell's refinements. Numerical results indicate excellent accuracy of the implementation.

Quadratically constrained minimum cross-entropy analysis

Abstract: Quadratically constrained minimum cross-entropy problem has recently been studied by Zhang and Brockett through an elaborately constructed dual. In this paper, we take a geometric programming approach to analyze this problem. Unlike Zhang and Brockett, we separate the probability constraint from general quadratic constraints and use two simple geometric inequalities to derive its dual problem. Furthermore, by using the dual perturbation method, we directly prove the “strong duality theorem” and derive a “dual-to-primal” conversion formula. As a by-product, the perturbation proof gives us insights to develop a computation procedure that avoids dual non-differentiability and allows us to use a general purpose optimizer to find ane-optimal solution for the quadratically constrained minimum cross-entropy analysis.

A sequential quadratic programming algorithm for discrete optimal control problems with control inequality constraints

Abstract: The authors exploit the structure of optimal control problem and present a version of a conventional sequential quadratic programming algorithm which merely requires, at each iteration, the solution of N quadratic subproblems of dimension m (equal to that of the control) rather than the solution of one quadratic subproblem of dimension Nm as required by the mathematical programming formulation. This is achieved by extending a stagewise implementation of Newton's method for unconstrained discrete-time optimal control problems to discrete-time optimal control problems with control constraints. >

A sparse sequential quadratic programming algorithm

Abstract: Described here is the structure and theory for a sequential quadratic programming algorithm for solving sparse nonlinear optimization problems. Also provided are the details of a computer implementation of the algorithm along with test results. The algorithm maintains a sparse approximation to the Cholesky factor of the Hessian of the Lagrangian. The solution to the quadratic program generated at each step is obtained by solving a dual quadratic program using a projected conjugate gradient algorithm. An updating procedure is employed that does not destroy sparsity.

Combined control-structure optimization

Abstract: An approach for combined control-structure optimization keyed to enhancing early design trade-offs is outlined and illustrated by numerical examples. The approach employs a homotopic strategy and appears to be effective for generating families of designs that can be used in these early trade studies. Analytical results were obtained for classes of structure/control objectives with linear quadratic Gaussian (LQG) and linear quadratic regulator (LQR) costs. For these, researchers demonstrated that global optima can be computed for small values of the homotopy parameter. Conditions for local optima along the homotopy path were also given. Details of two numerical examples employing the LQR control cost were given showing variations of the optimal design variables along the homotopy path. The results of the second example suggest that introducing a second homotopy parameter relating the two parts of the control index in the LQG/LQR formulation might serve to enlarge the family of Pareto optima, but its effect on modifying the optimal structural shapes may be analogous to the original parameter lambda.

The totally singular linear quadratic problem with indefinite cost

Abstract: In this paper we study the most general version of the stationary, infinite horizon linear quadratic optimal control problem. In the literature that has appeared on this problem up to now, mostly one (or both) of the following two assumptions are made: (i) the integrand of the cost criterion is given by a positive semi-definite quadratic form, (ii) the latter quadratic form is positive definite in the control variable alone. In the present paper we propose a problem formulation for the case that neither (i) nor (ii) are imposed. Subsequently, we treat the case that the problem is completely singular.

The Linear Quadratic Tracking Problem

Abstract: In this chapter we will study a very important problem in the field of optimal control theory. In general, control theory is concerned with using the measurements in a dynamical system to “control” the state vector x. The precise meaning of the word control will be made clear as we proceed. The word “quadratic” in the title of this chapter refers to a particular class of control problems that use a quadratic form to measure the performance of a system. The reason we choose this particular performance index is that in the stochastic case it leads to a tractable solution. For reasons that will be clear later, we begin with the deterministic problem.

A decomposition-dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems

Abstract: In this paper, we shall be concerned with the solution of constrained convex minimization problems. The constrained convex minimization problems are proposed to be transformable into a convex-additively decomposed and almost separable form, e.g. by decomposition of the objective functional and the restrictions. Unconstrained dual problems are generated by using Fenchel-Rockafellar duality. This decomposition-dualization concept has the advantage that the conjugate functionals occuring in the derived dual problem are easily computable. Moreover, the minimum point of the primal constrained convex minimization problem can be obtained from any maximum point of the corresponding dual unconstrained concave problem via explicit return-formulas. In quadratic programming the decomposition-dualization approach considered here becomes applicable if the quadratic part of the objective functional is generated byH-matrices. Numerical tests for solving obstacle problems in ?1 discretized by using piecewise quadratic finite elements and in ?2 by using the five-point difference approximation are presented.

Trajectory generation for manipulators using linear quadratic optimal tracking

Abstract: The reference trajectory is normally known in advance in manipulator control, which makes it possible to apply linear quadratic optimal tracking. This gives a control system which rounds corners and generates optimal feedforward. The method may be used for references consisting of straight-line segments as an alternative to the two-step method of using splines to smooth the reference and then applying feedforward. in addition, the method can be used for more complex trajectories. The actual dynamics of the manipulator are taken into account, and this results in smooth and accurate tracking. The method has been applied in combination with the computed torque technique and excellent performance was demonstrated in a simulation study. The method has also been applied experimentally to an industrial spray-painting robot where a saw-tooth reference was tracked. The corner was rounded well, and the steady-state tracking error was eliminated by the optimal feedforward. >

Multiplier-continuation algorthms for constrained optimization

Abstract: Several path following algorithms based on the combination of three smooth penalty functions, the quadratic penalty for equality constraints and the quadratic loss and log barrier for inequality constraints, their modern counterparts, augmented Lagrangian or multiplier methods, sequential quadratic programming, and predictor-corrector continuation are described. In the first phase of this methodology, one minimizes the unconstrained or linearly constrained penalty function or augmented Lagrangian. A homotopy path generated from the functions is then followed to optimality using efficient predictor-corrector continuation methods. The continuation steps are asymptotic to those taken by sequential quadratic programming which can be used in the final steps. Numerical test results show the method to be efficient, robust, and a competitive alternative to sequential quadratic programming.