MATH529 Fundamentals of Optimization Fundamentals of Constrained - - PowerPoint PPT Presentation
MATH529 Fundamentals of Optimization Fundamentals of Constrained Optimization VIII: Algorithms Marco A. Montes de Oca Mathematical Sciences, University of Delaware, USA 1 / 24 Algorithms for Nonlinear Constrained Optimization One basic
Marco A. Montes de Oca
Mathematical Sciences, University of Delaware, USA
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One basic idea: Use Lagrangian-like functions as proxies (or analytical tools) for dealing with a constrained problem.
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In this course: Penalty Methods Interior-Point Methods
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Idea: Have a mechanism that generates solutions using information about their quality. (Favoring better quality solutions.)
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Idea: Have a mechanism that generates solutions using information about their quality. (Favoring better quality solutions.) In the process of determining the quality of those solutions, penalize those that are infeasible by reducing their quality based on the degree they violate the constraints.
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Example: Say c1(x) = x1 − x2 = 3. Given u = (2, −0.5)T, and v = (−1, 0)T, u should receive a better score than v because 2.5 is closer to 3 than −1.
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A common way to implement these ideas in order to deal with equality constraints is to define a proxy for the objective function as follows: Q(x, µ) = f (x) + µ 2
(ci(x))2 where µ > 0 is called the penalty parameter.
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Effects of the penalty parameter:
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Minimize xy subject to x2 + y2 = 1:
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Q(x, y, 1) = xy + 1
2(x2 + y2 − 1)2:
(x⋆, y⋆) = (−0.8660, 0.8660), or (x⋆, y⋆) = (0.866, −0.866).
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Q(x, y, 40) = xy + 20(x2 + y2 − 1)2: (x⋆, y⋆) = (−0.7115, 0.7115), or (x⋆, y⋆) = (0.7115, −0.7115).
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Penalty Methods: Initialize µ0 > 0, x0. for i = 1,. . . ,k
Use your favorite algorithm to find an approximate minimizer
If mk is good enough, break and return mk as solution. Else choose new µk+1 > µk, and set xx+1 = mk.
endfor
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Issues: Divergence The Hessian becomes ill-conditioned with large values of µk. It is harder to deal with inequality constraints because in this case: Q(x, µ) = f (x) + µ 2
(ci(x))2 + µ 2
(min{ci(x), 0})2 therefore, Q is no longer twice continuously differentiable.
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We saw that: ∇xQ(x, µk) = ∇f (x) + µk
ci(x)∇ci(x) Now, compare this equation with the gradient of the Lagrangian: ∇xL(x, λi) = ∇f (x) −
λi∇ci(x) At a solution point of Q, we can say that ci(xk) ≈ −λ⋆
i
µk for all i ∈ E. This means that ci(xk) → 0 as µk → ∞, but in general a solution to Q is biased.
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A way to reduce this bias, is to use what is called the augmented Lagrangian which is defined as: LA(x, λ, µ) = f (x) −
λici(x) + µ 2
(ci(x))2 The idea is then to use LA(x, λk, µk) instead of Q(x, µk) as proxy for the constrained problem.
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This works because the optimality conditions for LA(x, λk, µk) say that ∇LA(xk, λk, µk) = 0 and therefore ∇LA(xk, λk, µk) = ∇f (xk) −
(λk
i − µkci(xk))∇ci(xk) = 0
and so λ⋆
i ≈ λk i − µkci(xk)
for all i ∈ E. We can see now that ci(xk) ≈ − 1 µk (λ⋆ − λk
i ). So, ci(xk) would be
much smaller than before provided that λk
i is close to λ⋆ i .
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A method that implements the augmented Lagrangian method would use the formula λk+1
i
= λk
i − µkci(xk)
to have a better behaved algorithm that does not require µk → ∞ (at least not as fast) to have accurate solutions.
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It is possible to modify a problem with inequality constraints so that the augmented Lagrangian method can be used without modification: The idea is to transform ci(x) ≥ 0 into ci(x) − si = 0 subject to bound constraints (which are easier to deal with). Another approach is to use:
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Similar to the penalty method, but now the penalty is smooth:
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Common barrier functions:
1 x
− ln(x)
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Example: Minimize 2x2 + 9y subject to x + y ≥ 4.
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