Interior-Point Methods as Inexact Newton Methods. Silvia Bonettini Università di Modena e Reggio Emilia Italy
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1 InteriorPoint Methods as Inexact Newton Methods Silvia Bonettini Università di Modena e Reggio Emilia Italy
2 Valeria Ruggiero Università di Ferrara Emanuele Galligani Università di Modena e Reggio Emilia Carla Durazzi Università di Ferrara
3 A Nonlinear Programming Problem Slack variables Introduce Lagrange multipliers for equality and inequality constraints Lagrangian function:
4 KarushKuhnTucker Optimality Conditions Notations: Complementarity conditions: Define a new variable
5 1. EXISTENCE Standard Assumptions 2. SMOOTHNESS The hessian matrices Lipschitz continuous in a neigborhood of 3. REGULARITY The gradients of the equality contraints and of the active inequality constraints, in the solution are linearly independent. 4. STRICT COMPLEMENTARITY exist and are 5. SECOND ORDER SUFFICIENCY The matrix is positive definite on the space
6 NONLINEAR PROGRAMMING PROBLEM NONLINEAR SYSTEM WITH BOUNDS ON SOME VARIABLES Solving the system with Newton s method If [or ], then [or ] Perturbe the complementarity conditions: perturbation parameter
7 Framework of InteriorPoint Methods Choose an initial guess Choose the perturbation parameter Solve the perturbed Newton equation Move along the direction computed at the previous step: choose the damping parameter such that the new iterate satisfies» FEASIBILITY» CENTRALITY CONDITIONS» SUFFICIENT DECREASE OF A MERIT FUNCTION (Armijo, EisenstatWalker) Update the iterate
8 Hypotheses for the convergence and is Lipschitz continuous in ; are linearly independent in ; is nonsingular ; The sequences are bounded in. [Durazzi, Ruggiero, JOTA, 2004]
9 SOLVE EXACTLY THE PERTURBED NEWTON EQUATION SOLVE THE NEWTON EQUATION WITH RESIDUAL
10 Inexact Newton Method for the Solution of a Nonlinear System Generate a sequence s.t. Residual condition forcing term Norm condition Updating rule [Eisenstat, Walker, SIAM J. Optimization, 1994]
11 CONVERGENCE: If is a limit point of the sequence s.t. is nonsingular, then and converges to IMPLEMENTATION: Residual condition: apply an iterative method to the Newton equation using the residual condition as stopping rule find a direction. Norm condition: reduce the step along the direction that satisfy the residual condition with a damping parameter until is satisfied backtracking.
12 Example contours of
13 IP Methods as Inexact Newton Methods Solve exactly the perturbed Newton equation Condition on the parameter the residual condition is satisfied with forcing term Solve approximately the perturbed Newton equation Conditions on and on the parameter the residual condition is satisfied with forcing term
14 Inexact Newton InteriorPoint Methods Choose an initial guess Choose the parameters Apply an iterative method to the perturbed Newton equation until obtaining the direction Choose the damping parameter such that the new iterate satisfies» FEASIBILITY» CENTRALITY CONDITIONS» SUFFICIENT DECREASE: backtracking until Update the iterate
15 A Nonmonotone Variant Let a fixed positive integer, and the element of the sequence such that Nonmonotone Inexact Newton Method for the solution of a nonlinear system Generate a sequence s.t. Nonmonotone residual condition forcing term Nonmonotone norm condition Updating rule [Bonettini, Optim. Methods and Software, 2004]
16 CONVERGENCE: If is a limit point of the sequence such that is nonsingular and furthermore is bounded, then and converges to. REMARK: The consequences of the nonmonotone rules are not only on the steplength, but also on the direction.
17 NONMONOTONE INEXACT NEWTON METHOD NONMONOTONE INEXACT NEWTON INTERIOR POINT METHOD Choice of the perturbation parameter Stopping rule for the inner system Nonmonotone backtracking rule
18 About the Inner Linear System
19 Solve the 2X2 System (1) Hypothesis: A is positive definite on the space The system has an unique solution, which is also the solution of the minimum problem Hestenes multiplier s scheme If is sufficiently large is positive definite [Bonettini,Galligani,Ruggiero, Rendiconti di Matematica, 2003]
20 Solve the 2X2 System (2) Preconditioned conjugate gradient method The conjugate gradient method with preconditioner M converges to a solution of the system in a finite number of steps. Implementation detail: Block solution At each step of the conjugate gradient method, a system whose matrix is the positive definite matrix has to be solved. [Durazzi, Ruggiero, Num. Linear Algebr. Appl., 2003]
21 Numerical Experience Code in Fortran90 and Matlab function files» Exact solver on the 3X3 system MA27 (HSL);» Iterative inner solver on the 2X2 system Hestenes method Preconditioned CG lipsol.f77 (NgPeyton) Numerical solution of optimal control problems Large scale nonlinear programming problems structured and sparse. [Mittelmann, Maurer, JCAM, 2000]
22 A boundary control problem [Mittelmann, Maurer, Optimization Techniques for Solving Elliptic Control Problems With Control and State Constraints, Comput. Optim and Appl, 2000] Grid Direct solver Iterative solver Monotone Non monotone Monotone Hestenes method Non monotone Monotone Preconditioned CG Non monotone (27) (27) (28) (28) (35) (33) (43) (39) (45) (44) (45) (55) 86.0
23 A distributed control problem [Mittelmann, Maurer, Optimization Techniques for Solving Elliptic Control Problems With Control and State Constraints, Comput. Optim and Appl, 2001] Grid Direct solver Iterative solver Monotone Non monotone Monotone Hestenes method Non monotone Monotone Preconditioned CG Non monotone (37) (37) (49) (44) (50) (50) (97) (90) (72) (67) (278) (237) 166.6
24 Open problems and future work Stagnation of the iterates due to:» BACKTRACKING Nonmonotone rules» FEASIBILITY REQUIREMENT (WachterBiegler example) Singularity of the matrix unbounded multipliers sequence» REGULARIZATION? Choice of the initial point AMPL interface subroutine in C with Gaetano Zanghirati (Univ. Ferrara) and Federica Tinti (Univ. Padova)
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