Numerical solution of a nonlinear parabolic control problem by a reduced SQP method

We consider a control problem for a nonlinear diffusion equation with boundary input that occurs when heating ceramic products in a kiln. We interpret this control problem as a constrained optimization problem, and we develop a reduced SQP method that presents for this problem a new and efficient approach of its numerical solution. As opposed to Newton's method for the unconstrained problem, where at each iteration the state must be computed from a set of nonlinear equations,in the proposed algorithm only the linearized state equations need to be solved. Furthermore, by use of a secant update formula, the calculation of exact second derivatives is avoided. In this way the algorithm achieves a substantial decrease in the total cost compared to the implementation of Newton's method in [2]. Our method is practicable with regard to storage requirements, and by choosing an appropriate representation for the null space of the Jacobian of the constraints we are able to exploit the sparsity pattern of the Jacobian in the course of the iteration. We conclude with a presentation of numerical examples that demonstrate the fast two-step superlinear convergence behavior of the method.

[1]  Jean Charles Gilbert,et al.  On the local and global convergence of a reduced Quasi-Newton method1 , 1989 .

[2]  J. Burger,et al.  Functional and numerical solution of a control problem originating from heat transfer , 1991 .

[3]  Edward L. Wilson,et al.  A unified formulation for triangular and quadrilateral flat shell finite elements with six nodal degrees of freedom , 1991 .

[4]  D. Gabay.  Reduced quasi-Newton methods with feasibility improvement for nonlinearly constrained optimization , 1982 .

[5]  Jianzhon Zhang,et al.  A trust region typed dogleg method for nonlinear optimization , 1990 .

[6]  Richard H Byrd,et al.  On the convergence of constrained optimization methods with accurate Hessian information on a subspace , 1990 .

[7]  Richard H. Byrd,et al.  An example of irregular convergence in some constrained optimization methods that use the projected hessian , 1985, Math. Program..

[8]  Jean Charles Gilbert.  Une Methode de Quasi-Newton Reduite en Optimisation Sous Contraintes Avec Priorite a la Restauration , 1986 .

[9]  E. Sachs,et al.  A prospective look at SQP methods for semilinear parabolic control problems , 1991 .

[10]  Ya-Xiang Yuan,et al.  An only 2-step Q-superlinear convergence example for some algorithms that use reduced hessian approximations , 1985, Math. Program..

[11]  David Q. Mayne,et al.  Differential dynamic programming , 1972, The Mathematical Gazette.

[12]  Jorge Nocedal,et al.  An analysis of reduced Hessian methods for constrained optimization , 1991, Math. Program..

[13]  Karl Kunisch,et al.  Reduced SQP methods for parameter identification problems , 1992 .

[14]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[15]  T. Coleman,et al.  On the Local Convergence of a Quasi-Newton Method for the Nonlinear Programming Problem , 1984 .