Numerical Optimization of Partial Differential Equations

Transcription

1 Numerical Optimization of Partial Differential Equations Part I: basic optimization concepts in R n Bartosz Protas Department of Mathematics & Statistics McMaster University, Hamilton, Ontario, Canada URL: Rencontres Normandes sur les aspects théoriques et numériques des EDP 5 9 November 2018, Rouen

2 Unconstrained Optimization Problems Optimality Conditions Gradient Flows Lagrange Multipliers Projected Gradients

3 A good reference for standard approaches

4 Unconstrained Optimization Problems Optimality Conditions Gradient Flows Suppose f : R N R, N 1, is a twice continuously differentiable objective function Unconstrained Optimization Problems: min f (x) x R N (for maximization problems, we can consider min[ f (x)]) A point x is a global minimizer if f ( x) f (x) for all x A point x is a local minimizer if there exists a neighborhood N of x such that f ( x) f (x) for all x N A local minimizer is strict (or strong), if it is unique in N

5 Unconstrained Optimization Problems Optimality Conditions Gradient Flows Gradient of the objective function f (x) := [ f x 1,..., f x N ] T Hessian of the objective function [ 2 f (x) ] i,j := 2 f x j x i, i, j = 1,..., N

6 Unconstrained Optimization Problems Optimality Conditions Gradient Flows Theorem (First-Order Necessary Condition) If x is a local minimizer, then f ( x) = 0. Theorem (Second-Order Sufficient Conditions) Suppose that f ( x) = 0 and 2 f ( x) is positive-definite. Then x is a strict local minimizers of f. Unfortunately, analogous characterization of global minimizers is not possible

7 How to find a local minimizer x? Unconstrained Optimization Problems Optimality Conditions Gradient Flows Consider the following initial-value problem in R N, known as the gradient flow dx(τ) = f (x(τ)), τ > 0, (GF) dτ x(0) = x 0, where τ is a pseudo-time (a parametrization) x 0 is a suitable initial guess Then, lim τ x(τ) = x In principle, the gradient flow may converge to a saddle point x s, where f (x s ) = 0 and the Hessian 2 f (x s ) is not positive-definite, but in actual computations this is very unlikely.

8 Discretize the gradient flow (GF) with Euler s explicit method (SD) { x (n+1) = x (n) τ f (x (n) ), n = 1, 2,..., x (0) = x 0, where x (n) := x(n τ), such that lim n x (n) = x τ is a fixed step size (since Euler s explicit scheme is only conditionally stable, τ must be sufficiently small) In principle, the gradient flow (GF) can be discretized with higher-order schemes, including implicit approaches, but they are not easy to apply to PDE optimization problems, hence will not be considered here.

9 Algorithm 1 (SD) 1: x (0) x 0 (initial guess) 2: n 0 3: repeat 4: compute the gradient f (x (n) ) 5: update x (n+1) = x (n) τ f (x (n) ) 6: n n + 1 7: until f (x(n) ) f (x (n 1) ) < ε f (x (n 1) ) f Input: x 0 initial guess τ fixed step size ε f tolerance in the termination condition Output: an approximation of the minimizer x

10 Computational Tests Rosenbrock s banana function f (x 1, x 2 ) = 100(x 2 x1 2 ) 2 + (1 x 1 ) 2 Global minimizer x 1 = x 2 = 1, f (1, 1) = 0 The function is known for its poor conditioning eigenvalues of the Hessian 2 f at the minimum: λ 1 0.4, λ

11 Choice of the step size τ: steepest descent is not meant to approximate the gradient flow (GF) accurately, but to minimize f (x) rapidly Sufficient decrease Armijo s condition f (x (n) + τ p (n) ) f (x (n) ) C τ f (x (n) ) T p (n) where p (n) is a search direction and C (0, 1) Figure credit: Nocedal & Wright (1999) Wolfe s condition: sufficient decrease and curvature

12 Optimize the step size at every iteration by solving the line minimization (line-search) problem τ n := argmin τ>0 f (x (n) τ f (x (n) )) Brent s method for line minimization: a combination of the golden-section search with parabolic interpolation (derivative-free) Figure credit: Numerical Recipes in C (1992) A robust implementation of Brent s method available in Numerical Recipes in C (1992), see also the function fminbnd in MATLAB

13 Algorithm 2 with Line Search (SDLS) 1: x (0) x 0 (initial guess) 2: n 0 3: repeat 4: compute the gradient f (x (n) ) 5: determine optimal step size τ n = argmin τ>0 f (x (n) τ f (x (n) )) 6: update x (n+1) = x (n) τ n f (x (n) ) 7: n n + 1 8: until f (x(n) ) f (x (n 1) ) f (x (n 1) ) < ε f Input: x 0 initial guess ε τ tolerance in line search ε f tolerance in the termination condition Output: an approximation of the minimizer x

14 Consider, for now, minimization of a quadratic form f (x) = 1 2 xt Ax b T x, where A R N N is a symmetric, positive-definite matrix and b R N Then, f (x) = Ax b =: r such that minimizing f (x) is equivalent to solving Ax = b A set of nonzero vectors [p 0, p 1,..., p k ] is said to be conjugate with respect to matrix A if p T i Ap j = 0, i, j = 0,..., k, i j (conjugacy implies linear independence)

15 Conjugate Gradient (CG) method x (n+1) = x (n) + τ n p (n), n = 1, 2,..., p (n) = r (n) + β (n) p n 1, (r (n) = f (x (n) ) = Ax (n) b), β (n) = (r(n) ) T Ap (n 1) (p (n 1) ) T, ( momentum ), Ap (n 1) τ n = (r(n) ) T p (n) (p (n) ) T, (exact formula for optimal step size), Ap (n) x 0 = x 0, p (0) = r (0) The directions p (0), p (1),..., p (n) generated by the CG method are conjugate with respect to matrix A this gives rise to a number of interesting and useful properties

16 Theorem (properties of CG iterations) The iterates generated by the CG method have the following properties { span p (0), p (1),..., p (n)} = span {r (0), r (1),..., r (n)} = span {r (0), Ar (0),..., A n r (0)} (the expanding subspace property) (r (n) ) T r (k) = (r (n) ) T p (k) = 0, i = 0,..., n 1 x (n) is the minimizer of f (x) = 1 2 xt Ax b T x over the set { x 0 + span {p (0), p (1),..., p (n)}}

17 Thus, in the method minimization of f (x) = 1 2 xt Ax b T x is performed by solving (exactly) N = dim(x) line-minimization problems along the conjugate directions { p (0), p (1),..., p (n)} As a result, convergence to x is achieved in at most N iterations What happens when f (x) is a general convex function?

18 The (linear) method admits a generalization to the nonlinear setting by: replacing the residual r (n) with the gradient f (x (n) ) computing the step size via line search τ n = argmin τ>0 f (x (n) τ f (x (n) )) using a more general expressions for the momentum term β (n) (such that the descent directions p (0), p (1),..., p (n) will only be approximately conjugate)

19 Nonlinear Conjugate Gradient (NCG) method x (n+1) = x (n) + τ n p (n), n = 1, 2,..., p (n) = f (x (n) ) + β (n) p n 1, ( T f (x )) (n) f (x (n) ) β (n) ( f (x = (n 1) )) T f (x (n 1) ) ( ) T ( ) f (x (n) ) f (x (n) ) f (x (n 1) ) ( f (x (n 1) )) T f (x (n 1) ) τ n = argmin τ>0 f (x (n) τ f (x (n) )), x 0 = x 0, p (0) = f (x (0) ) (Fletcher-Reeves), (Polak-Ribière), For quadratic functions f (x), both the Fletcher-Reeves (FR) and the Polak-Ribière (PR) variant coincide with the the linear CG In general, the descent directions p (0), p (1),..., p (n) are now only approximately conjugate

20 Algorithm 3 Polak-Ribière version of Conjugate Gradient (CG-PR) 1: x (0) x 0 (initial guess) 2: n 0 3: repeat 4: compute the gradient f (x (n) ) 5: calculate β n = ( f (x(n) )) T ( f (x (n) ) f (x (n 1) )) ( f (x (n 1) )) T f (x (n 1) ) 6: determine the descent direction p (n) = f (x (n) ) + β (n) p n 1 7: determine optimal step size τ n = argmin τ>0 f (x (n) + τ p (n) ) 8: update x (n+1) = x (n) + τ n p (n) 9: n n : until f (x(n) ) f (x (n 1) ) f (x (n 1) ) < ε f Input: x 0 initial guess, ε τ tolerance in line search tolerance in the termination condition ε f Output: an approximation of the minimizer x

21 Convergence theory quadratic case (I) Let f (x) = 1 2 xt Ax b T x, where the matrix A has eigenvalues 0 < λ 1 λ N and x A = x T Ax Theorem (Linear Convergence of ) For the Steepest-Descent approach we have the following estimate ( ) x (n+1) x 2 A λn λ 2 1 x (n) x 2 A λ N + λ 1 The rate of convergence is controlled by the spread of the eigenvalues of A

22 Convergence theory quadratic case (II) Theorem (Convergence of Linear ) For the linear approach we have the following estimate ( ) x (n+1) x 2 A λn n λ 2 1 x 0 x 2 A λ N n + λ 1 The iterates take out one eigenvalue at a time clustering of eigenvalues matters In the nonlinear setting, it is advantageous to periodically reset β n to zero (helpful in practice and simplifies some convergence proofs)

23 Lagrange Multipliers Projected Gradients What about problems with equality constraints? suppose c : R N R M, where 1 M < N then, we have an equality-constrained optimization problem min f (x) x R N subject to: c(x) = 0 If the constraint equation can be solved and we can write x = y + z, where y R N M and z = g(y) R M, then the problem is reduced to an unconstrained one with a reduced objective function min f (y + g(y)) y RN M

24 Lagrange Multipliers Projected Gradients Consider augmented objective function L : R N R L(x, λ) := f (x) λ T c(x), where λ R M is the Lagrange multiplier Differentiating the augmented objective function with respect to x x L(x, λ) := f (x) λ T c(x) Theorem (First-Order Necessary Condition) If x is a local minimizer of an equality-constrained optimization problem, then there exists λ R M such that the following equations are satisfied f ( x) λ T c( x) = 0, c( x) = 0 For inequality-constrained problems, the first-order necessary conditions become more complicated the Karush-Kuhn-Tucker (KKT) conditions

25 Lagrange Multipliers Projected Gradients How to compute equality-constrained minimizers with a gradient method? At each x R N the linearized constraint function c(x) R M N defines a (kernel) subspace with dimension rank[ c(x)] S x := {x R N, c(x)x = 0} this is the subspace tangent to the constraint manifold at x we need to project the gradient f (x) onto Sx Assuming that rank[ c(x)] = M, the projection operator P Sx : R N S x is given by [ ] 1 P Sx := I c(x) ( c(x)) T c(x) ( c(x)) T Replace f (x) with P Sx f (x) in the gradient method (SD or SDLS) nonlinear constraints satisfied with an error O(( τ) 2 ) or O(τ 2 n )

26 Lagrange Multipliers Projected Gradients Algorithm 4 Projected (PSD) 1: x (0) x 0 (initial guess) 2: n 0 3: repeat 4: compute the gradient f (x (n) ) 5: compute linearization of the constraint c(x (n) ) 6: determine the projector P Sx (n) 7: determine the projected gradient P f Sx (n) (x(n) ) 8: update x (n+1) = x (n) τ P f Sx (n) (x(n) ) 9: n n : until f (x(n) ) f (x (n 1) ) < ε f (x (n 1) ) f Input: x 0 initial guess, τ fixed step size tolerance in the termination condition ε f Output: an approximation of the minimizer x

27 Lagrange Multipliers Projected Gradients Computational Tests Rosenbrock s banana function f (x 1, x 2 ) = 100(x 2 x1 2 ) 2 + (1 x 1 ) 2 Global minimizer x 1 = x 2 = 1, f (1, 1) = 0 The function is known for its poor conditioning eigenvalues of the Hessian 2 f at the minimum: λ 1 0.4, λ Constraint c(x 1, x 2 ) = 0.05 x 4 1 x = 0