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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
A Derivative-Free Approximate GradientSampling Algorithm for Finite Minimax
Problems
Speaker: Julie NutiniJoint work with Warren Hare
University of British Columbia (Okanagan)
III Latin American Workshop on Optimization and ControlJanuary 10-13, 2012
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Table of Contents
1 Introduction
2 AGS Algorithm
3 Robust AGS Algorithm
4 Numerical Results
5 Conclusion
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Setting
We assume that our problem is of the form
minx
f (x) where f (x) = max{fi : i = 1, . . . ,N},
where each fi is continuously differentiable, i.e., fi ∈ C1,BUT we cannot compute ∇fi .
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Active Set/Gradients
DefinitionWe define the active set of f at a point x to be the set of indices
A(x) = {i : f (x) = fi(x)}.
The set of active gradients of f at x is denoted by
{∇fi(x)}i∈A(x).
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Clarke Subdifferential for Finite Max Function
Proposition (Proposition 2.3.12, Clarke ‘90)
Let f = max{fi : i = 1, . . . ,N}.If fi ∈ C1 for each i ∈ A(x), then
∂f (x) = conv{∇fi(x)}i∈A(x).
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Method of Steepest Descent
1. Initialize: Set d0 = −Proj(0|∂f ).
2. Step Length: Find tk by solving mintk>0{f (xk + tkdk )}.
3. Update: Set xk+1 = xk + tkdk . Increase k = k + 1. Loop.
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
AGS Algorithm - General Idea
Replace ∂f with an approximate subdifferential
Find an approximate direction of steepest descent
Minimize along nondifferentiable ridges
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Approximate Gradient Sampling Algorithm(AGS Algorithm)
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
AGS Algorithm
1 Initialize
2 Generate Approximate Subdifferential:Sample Y = [xk , y1, . . . , ym] from around xk such that
maxi=1,...,m
|y i − xk | ≤ ∆k .
Calculate an approximate gradient ∇Afi for each i ∈ A(xk ) and set
Gk = conv{∇Afi(xk )}i∈A(xk ).
3 Direction: Set dk = −Proj(0|Gk ).
4 Check Stopping Conditions
5 (Armijo) Line Search
6 Update and Loop
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
ConvergenceAGS Algorithm
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
RemarkWe define the approximate subdifferential of f at x as
G(x) = conv{∇Afi(x)}i∈A(x),
where ∇Afi(x) is the approximate gradient of fi at x .
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Lemma
Suppose there exists an ε > 0 such that |∇Afi(x)−∇fi(x)| ≤ ε.
Then1 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.
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Proof.1. By definition, for all w ∈ G(x) there exists a set of αi suchthat
w =∑
i∈A(x)
αi∇Afi(x), where αi ≥ 0,∑
i∈A(x)
αi = 1.
Using the same αi as above, we see that
v =∑
i∈A(x)
αi∇fi(x) ∈ ∂f (x).
Then |w − v | = |∑
i∈A(x)
αi∇Afi(x))−∑
i∈A(x)
αi∇fi(x)| ≤ ε.
Hence, for all w ∈ G(x), there exists a v ∈ ∂f (x) such that
|w − v | ≤ ε. (1)
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Convergence
Theorem
Let {xk}∞k=0 be generated by the AGS algorithm.Suppose there exists a K such that given any set Y generated inStep 1 of the AGS algorithm, ∇Afi(xk ) satisfies
|∇Afi(xk )−∇fi(xk )| ≤ K ∆k , where ∆k = maxy i∈Y|y i − xk |.
Suppose tk is bounded away from 0.Then either
1 f (xk ) ↓ −∞, or2 |dk | → 0, ∆k ↓ 0 and every cluster point x of the sequence{xk}∞k=0 satisfies 0 ∈ ∂f (x).
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Convergence - Proof
Proof - Outline.
1. The direction of steepest descent is −Proj(0|∂f (xk )).
2. By previous lemma, G(xk ) is a good approximate of ∂f (xk ).
3. So we can show that −Proj(0|G(xk )) is still a descentdirection (approximate direction of steepest descent).
4. Thus, we can show that convergence holds.
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Robust Approximate Gradient SamplingAlgorithm
(Robust AGS Algorithm)
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Visual
Consider the function
f (x) = max{f1(x), f2(x), f3(x)}where f1(x) = x2
1 + x22
f2(x) = (2− x1)2 + (2− x2)2
f3(x) = 2 ∗ exp(x2 − x1).
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Visual Representation
Contour plot - ‘nondifferentiable ridges’
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Visual Representation
Regular AGS algorithm:
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Visual Representation
Robust AGS algorithm:
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Robust Approximate Gradient Sampling Algorithm
Definition
Let Y = [xk , y1, y2, . . . , ym] be a set of sampled points.The robust active set of f on Y is
A(Y ) =⋃
y i∈Y
A(y i).
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Robust Approximate Gradient Sampling Algorithm
1 Initialize
2 Generate Approximate Subdifferential:Calculate an approximate gradient ∇Afi for each i ∈ A(Y ) and set
Gk = conv{∇Afi(xk )}i∈A(xk ),
and GkY = conv{∇Afi(xk )}i∈A(Y ).
3 Direction: Set dk = −Proj(0|Gk ) and dkY = −Proj(0|Gk
Y ).
4 Check Stopping Conditions: Use dk to check stopping conditions.
5 (Armijo) Line Search: Use dkY for the search direction.
6 Update and Loop
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
ConvergenceRobust AGS Algorithm
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Robust Convergence
Theorem
Let {xk}∞k=0 be generated by the robust AGS algorithm.Suppose there exists a K such that given any set Y generated inStep 1 of the robust AGS algorithm, ∇Afi(xk ) satisfies
|∇Afi(xk )−∇fi(xk )| ≤ K ∆k , where ∆k = maxy i∈Y|y i − xk |.
Suppose tk is bounded away from 0.Then either
1 f (xk ) ↓ −∞, or2 |dk | → 0, ∆k ↓ 0 and every cluster point x of the sequence{xk}∞k=0 satisfies 0 ∈ ∂f (x).
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Numerical Results
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Approximate Gradients
RequirementIn order for convergence to be guaranteed in the AGS algorithm,∇Afi(xk ) must satisfy an error bound for each of the active fi :
|∇Afi(xk )−∇fi(xk )| ≤ 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)
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Numerical Results - Overview
Implementation was done in MATLAB
24 nonsmooth test problems (Luksan-Vlcek, ‘00)25 trials for each problemQuality (improvement of digits of accuracy) measured by
− log(|Fmin − F ∗||F0 − F ∗|
)
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Numerical Results - Goal
Determine any notable numerical differences between:
1 Version of the AGS Algorithm1. Regular2. Robust
2 Approximate Gradient1. Simplex Gradient2. Centered Simplex Gradient3. Gupal Estimate
Thus, 6 different versions of our algorithm were compared.
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Performance Profile
100
101
102
103
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Performance Profile on AGS Algorithm (successful improvement of 1 digit)
τ
ρ(τ
)
Simplex
Robust Simplex
Centered Simplex
Robust Centered Simplex
Gupal
Robust Gupal
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Performance Profile
100
101
102
103
0
0.1
0.2
0.3
0.4
0.5
0.6
Performance Profile on AGS Algorithm (successful improvement of 3 digits)
τ
ρ(τ
)
Simplex
Robust Simplex
Centered Simplex
Robust Centered Simplex
Gupal
Robust Gupal
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Conclusion
Approximate gradient sampling algorithm for finite minimaxproblems
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.
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
Thank you!
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Outline Introduction AGS Algorithm Robust AGS Algorithm Numerical Results Conclusion
References
1 J. V. Burke, A. S. Lewis, and M. L. Overton. A robust gradientsampling algorithm for nonsmooth, nonconvex optimization,SIAM J. Optim., 15(2005), pp. 751-779.
2 W. Hare and J. Nutini. A derivative-free approximate gradientsampling algorithm for finite minimax problems. Submitted toSIAM J. Optim., 2011.
3 K. C. Kiwiel. A nonderivative version of the gradient samplingalgorithm for nonsmooth nonconvex optimization, SIAM J.Optim., 20(2010), pp. 1983-1994.
4 C. T. Kelley. Iterative methods for optimization, SIAM,Philadelphia, PA, 1999.
5 L. Luksan and J. Vlcek. Test Problems for NonsmoothUnconstrained and Linearly Constrained Optimization.Technical Report V-798, ICS AS CR, February 2000.
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