A Derivative-Free Approximate Gradient Sampling Algorithm for Finite Minimax Problems

Transcription

1 1 / 33 A Derivative-Free Approximate Gradient Sampling Algorithm for Finite Minimax Problems Speaker: Julie Nutini Joint work with Warren Hare University of British Columbia (Okanagan) III Latin American Workshop on Optimization and Control January 10-13, 2012

2 2 / 33 Table of Contents 1 Introduction 2 AGS Algorithm 3 Robust AGS Algorithm 4 Numerical Results 5 Conclusion

3 3 / 33 Setting We assume that our problem is of the form min x f (x) where f (x) = max{f i : i = 1,..., N}, where each f i is continuously differentiable, i.e., f i C 1, BUT we cannot compute f i.

4 Active Set/Gradients Definition We define the active set of f at a point x to be the set of indices A( x) = {i : f ( x) = f i ( x)}. The set of active gradients of f at x is denoted by { f i ( x)} i A( x). 4 / 33

5 5 / 33 Clarke Subdifferential for Finite Max Function Proposition (Proposition , Clarke 90) Let f = max{f i : i = 1,..., N}. If f i C 1 for each i A( x), then f ( x) = conv{ f i ( x)} i A( x).

6 Method of Steepest Descent 1. Initialize: Set d 0 = Proj(0 f ). 2. Step Length: Find t k by solving min t k >0 {f (x k + t k d k )}. 3. Update: Set x k+1 = x k + t k d k. Increase k = k + 1. Loop. 6 / 33

7 7 / 33 AGS Algorithm - General Idea Replace f with an approximate subdifferential Find an approximate direction of steepest descent Minimize along nondifferentiable ridges

8 8 / 33 Approximate Gradient Sampling Algorithm (AGS Algorithm)

9 9 / 33 AGS Algorithm 1 Initialize 2 Generate Approximate Subdifferential: Sample Y = [x k, y 1,..., y m ] from around x k such that max y i x k k. i=1,...,m Calculate an approximate gradient A f i for each i A(x k ) and set G k = conv{ A f i (x k )} i A(x k ). 3 Direction: Set d k = Proj(0 G k ). 4 Check Stopping Conditions 5 (Armijo) Line Search 6 Update and Loop

10 10 / 33 Convergence AGS Algorithm

11 Remark We define the approximate subdifferential of f at x as G( x) = conv{ A f i ( x)} i A( x), where A f i ( x) is the approximate gradient of f i at x. 11 / 33

12 12 / 33 Lemma Suppose there exists an ε > 0 such that A f i ( x) f i ( x) ε. Then 1 for all w G( x), v f ( x) such that w v ε, and 2 for all v f ( x), w G( x) such that w v ε.

13 13 / 33 Proof. 1. By definition, for all w G( x) there exists a set of α i such that w = α i A f i ( x), where α i 0, α i = 1. i A( x) i A( x) Using the same α i as above, we see that v = α i f i ( x) f ( x). i A( x) Then w v = α i A f i ( x)) α i f i ( x) ε. i A( x) i A( x) Hence, for all w G( x), there exists a v f ( x) such that w v ε. (1)

14 14 / 33 Convergence Theorem Let {x k } k=0 be generated by the AGS algorithm. Suppose there exists a K such that given any set Y generated in Step 1 of the AGS algorithm, A f i (x k ) satisfies A f i (x k ) f i (x k ) K k, where k = max y i x k. y i Y Suppose t k is bounded away from 0. Then either 1 f (x k ), or 2 d k 0, k 0 and every cluster point x of the sequence {x k } k=0 satisfies 0 f ( x).

15 15 / 33 Convergence - Proof Proof - Outline. 1. The direction of steepest descent is Proj(0 f (x k )). 2. By previous lemma, G(x k ) is a good approximate of f (x k ). 3. So we can show that Proj(0 G(x k )) is still a descent direction (approximate direction of steepest descent). 4. Thus, we can show that convergence holds.

16 16 / 33 Robust Approximate Gradient Sampling Algorithm (Robust AGS Algorithm)

17 Visual Consider the function f (x) = max{f 1 (x), f 2 (x), f 3 (x)} where f 1 (x) = x x 2 2 f 2 (x) = (2 x 1 ) 2 + (2 x 2 ) 2 f 3 (x) = 2 exp(x 2 x 1 ). 17 / 33

18 18 / 33 Visual Representation Contour plot - nondifferentiable ridges

19 19 / 33 Visual Representation Regular AGS algorithm:

20 20 / 33 Visual Representation Robust AGS algorithm:

21 21 / 33 Robust Approximate Gradient Sampling Algorithm Definition Let Y = [x k, y 1, y 2,..., y m ] be a set of sampled points. The robust active set of f on Y is A(Y ) = A(y i ). y i Y

22 22 / 33 Robust Approximate Gradient Sampling Algorithm 1 Initialize 2 Generate Approximate Subdifferential: Calculate an approximate gradient A f i for each i A(Y ) and set G k = conv{ A f i (x k )} i A(x k ), and GY k = conv{ Af i (x k )} i A(Y ). 3 Direction: Set d k = Proj(0 G k ) and dy k = Proj(0 Gk Y ). 4 Check Stopping Conditions: Use d k to check stopping conditions. 5 (Armijo) Line Search: Use dy k for the search direction. 6 Update and Loop

23 23 / 33 Convergence Robust AGS Algorithm

24 24 / 33 Robust Convergence Theorem Let {x k } k=0 be generated by the robust AGS algorithm. Suppose there exists a K such that given any set Y generated in Step 1 of the robust AGS algorithm, A f i (x k ) satisfies A f i (x k ) f i (x k ) K k, where k = max y i x k. y i Y Suppose t k is bounded away from 0. Then either 1 f (x k ), or 2 d k 0, k 0 and every cluster point x of the sequence {x k } k=0 satisfies 0 f ( x).

25 25 / 33 Numerical Results

26 26 / 33 Approximate Gradients Requirement In order for convergence to be guaranteed in the AGS algorithm, A f i (x k ) must satisfy an error bound for each of the active f i : A f i (x k ) f i (x k ) K, where K > 0 and is bounded above. We used the following 3 approximate gradients: 1 Simplex Gradient (see Lemma 6.2.1, Kelley 99) 2 Centered Simplex Gradient (see Lemma 6.2.5, Kelley 99) 3 Gupal Estimate (see Theorem 3.8, Hare and Nutini 11)

27 27 / 33 Numerical Results - Overview Implementation was done in MATLAB 24 nonsmooth test problems (Lukšan-Vlček, 00) 25 trials for each problem Quality (improvement of digits of accuracy) measured by ( Fmin F ) log F 0 F

28 28 / 33 Numerical Results - Goal Determine any notable numerical differences between: 1 Version of the AGS Algorithm 1. Regular 2. Robust 2 Approximate Gradient 1. Simplex Gradient 2. Centered Simplex Gradient 3. Gupal Estimate Thus, 6 different versions of our algorithm were compared.

29 29 / 33 Performance Profile Performance Profile on AGS Algorithm (successful improvement of 1 digit) Simplex Robust Simplex Centered Simplex Robust Centered Simplex Gupal Robust Gupal ρ(τ) τ

30 30 / 33 Performance Profile Performance Profile on AGS Algorithm (successful improvement of 3 digits) Simplex Robust Simplex Centered Simplex Robust Centered Simplex Gupal Robust Gupal ρ(τ) τ

31 31 / 33 Conclusion Approximate gradient sampling algorithm for finite minimax problems Robust version Convergence (0 f ( x)) Numerical tests Notes: We have numerical results for a robust stopping condition. There is future work in developing the theoretical analysis. We are currently working on an application in seismic retrofitting.

32 32 / 33 Thank you!

33 33 / 33 References 1 J. V. Burke, A. S. Lewis, and M. L. Overton. A robust gradient sampling algorithm for nonsmooth, nonconvex optimization, SIAM J. Optim., 15(2005), pp W. Hare and J. Nutini. A derivative-free approximate gradient sampling algorithm for finite minimax problems. Submitted to SIAM J. Optim., K. C. Kiwiel. A nonderivative version of the gradient sampling algorithm for nonsmooth nonconvex optimization, SIAM J. Optim., 20(2010), pp C. T. Kelley. Iterative methods for optimization, SIAM, Philadelphia, PA, L. Lukšan and J. Vlček. Test Problems for Nonsmooth Unconstrained and Linearly Constrained Optimization. Technical Report V-798, ICS AS CR, February 2000.