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{{Short description|Concept in convex optimization mathematics}}
'''Subgradient methods''' are [[convex optimization]] methods which use [[Subderivative|subderivatives]]. Originally developed by [[Naum Z. Shor]] and others in the 1960s and 1970s, subgradient methods are convergent when applied even to a non-differentiable objective function. When the objective function is differentiable, sub-gradient methods for unconstrained problems use the same search direction as the method of [[gradient descent|gradient descent]].
Subgradient methods are slower than Newton's method when applied to minimize twice continuously differentiable convex functions. However, Newton's method fails to converge on problems that have non-differentiable kinks.
In recent years, some [[interior-point methods]] have been suggested for convex minimization problems, but subgradient projection methods and related bundle methods of descent remain competitive. For convex minimization problems with
Subgradient projection methods are often applied to large-scale problems with decomposition techniques. Such decomposition methods often allow a simple distributed method for a problem.
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==Classical subgradient rules==
Let <math>f : \
A where <math>g^{(k)}</math> denotes
If <math>f \ </math> is differentiable, then its only subgradient is the gradient vector <math>\nabla f</math> itself. It may happen that <math>-g^{(k)}</math> is not a descent direction for <math>f \ </math> at <math>x^{(k)}.</math>
===Step size rules===
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*Constant step size, <math>\alpha_k = \alpha.</math>
*Constant step length, <math>\alpha_k = \gamma/\lVert g^{(k)} \rVert_2,</math>
*Square summable but not summable step size, i.e. any step sizes satisfying <math display="block">\alpha_k\geq0,\qquad\sum_{k=1}^\infty \alpha_k^2 < \infty,\qquad \sum_{k=1}^\infty \alpha_k = \infty.</math>
*Nonsummable diminishing step lengths, i.e. <math>\alpha_k = \gamma_k/\lVert g^{(k)} \rVert_2,</math> where <math display="block">\gamma_k \geq 0,\qquad \lim_{k\to\infty} \gamma_k = 0,\qquad \sum_{k=1}^\infty \gamma_k = \infty.</math>
For all five rules, the step-sizes are determined "off-line", before the method is iterated; the step-sizes do not depend on preceding iterations. This "off-line" property of subgradient methods differs from the "on-line" step-size rules used for descent methods for differentiable functions: Many methods for minimizing differentiable functions satisfy Wolfe's sufficient conditions for convergence, where step-sizes typically depend on the current point and the current search-direction. An extensive discussion of stepsize rules for subgradient methods, including incremental versions, is given in the books by Bertsekas<ref>{{cite book
| last = Bertsekas
| first = Dimitri P.
| author-link = Dimitri P. Bertsekas
| title = Convex Optimization Algorithms
| edition = Second
| publisher = Athena Scientific
| year = 2015
| ___location = Belmont, MA.
| isbn = 978-1-886529-28-1
}}</ref> and by Bertsekas, Nedic, and Ozdaglar.<ref>{{cite book
|last1=Bertsekas
|first1=Dimitri P.
|last2=Nedic
|first2=Angelia
|last3 = Ozdaglar
|first3 = Asuman
| title = Convex Analysis and Optimization
| edition = Second
| publisher = Athena Scientific
| year = 2003
| ___location = Belmont, MA.
| isbn = 1-886529-45-0 }}</ref>
===Convergence results===
For constant step-length and scaled subgradients having [[Euclidean norm]] equal to one, the subgradient method converges to an arbitrarily close approximation to the minimum value, that is
:<math>\lim_{k\to\infty} f_{\rm{best}}^{(k)} - f^* <\epsilon</math> by a result of [[Naum Z. Shor|Shor]].<ref>
The approximate convergence of the constant step-size (scaled) subgradient method is stated as Exercise 6.3.14(a) in [[Dimitri P. Bertsekas|Bertsekas]] (page 636): {{cite book
| last = Bertsekas
| first = Dimitri P.
|
| title = Nonlinear Programming
| edition = Second
| publisher = Athena Scientific
|
| ___location = Cambridge, MA.
| isbn = 1-886529-00-0
}} On page 636, Bertsekas attributes this result to
| last = Shor
| first = Naum Z.
|
| title = Minimization Methods for Non-differentiable Functions
| publisher = [[Springer-Verlag]]
| isbn = 0-387-12763-1
|
}}
</ref>
These classical subgradient methods have poor performance and are no longer recommended for general use.<ref name="Lem"/><ref name="KLL"/> However, they are still used widely in specialized applications because they are simple and they can be easily adapted to take advantage of the special structure of the problem at hand.
==Subgradient-projection and bundle methods==
During the 1970s, [[Claude Lemaréchal]] and Phil {{cite book
| last = Bertsekas
| first = Dimitri P.
|
| title = Nonlinear Programming
| edition = Second
| publisher = Athena Scientific
|
| ___location = Cambridge, MA.
| isbn = 1-886529-00-0
}}
</ref> The meaning of the term "bundle methods" has changed significantly since that time. Modern versions and full convergence analysis were provided by Kiwiel.
<ref>
{{cite book|last=Kiwiel|first=Krzysztof|title=Methods of Descent for Nondifferentiable Optimization|publisher=[[Springer Verlag]]|___location=Berlin|year=1985|pages=362|isbn=978-3540156420 |mr=0797754}}
</ref> Contemporary bundle-methods often use "[[level set|level]] control" rules for choosing step-sizes, developing techniques from the "subgradient-projection" method of Boris T. Polyak (1969). However, there are problems on which bundle methods offer little advantage over subgradient-projection methods.<ref name="Lem">
{{cite book| last=Lemaréchal|first=Claude|author-link=Claude Lemaréchal|chapter=Lagrangian relaxation|pages=112–156|title=Computational combinatorial optimization: Papers from the Spring School held in Schloß Dagstuhl, May 15–19, 2000|editor=Michael Jünger and Denis Naddef|series=Lecture Notes in Computer Science|volume=2241|publisher=Springer-Verlag| ___location=Berlin|year=2001|isbn=3-540-42877-1|mr=1900016|doi=10.1007/3-540-45586-8_4|s2cid=9048698 }}</ref><ref name="KLL">
{{cite journal|last1=Kiwiel|first1=Krzysztof C.|last2=Larsson |first2=Torbjörn|last3=Lindberg|first3=P. O.|title=Lagrangian relaxation via ballstep subgradient methods|url=http://rcin.org.pl/Content/139438/PDF/RB-2002-76.pdf |journal=[[Mathematics of Operations Research]]|volume=32|date=August 2007|number=3|pages=669–686|mr=2348241|doi=10.1287/moor.1070.0261}}
</ref>
==Constrained optimization==
===Projected subgradient===
One extension of the subgradient method is the '''projected subgradient method''', which solves the constrained [[Mathematical optimization|optimization]] problem
:minimize <math>f(x) \ </math> subject to <math display=block>x \in \mathcal{C}</math>
where <math>\mathcal{C}</math> is a [[convex set]].
The projected subgradient method uses the iteration
<math display=block>x^{(k+1)} = P \left(x^{(k)} - \alpha_k g^{(k)}\right)</math>
where <math>P</math> is projection on <math>\mathcal{C}</math> and <math>g^{(k)}</math> is any subgradient of <math>f \ </math> at <math>x^{(k)}.</math>
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The subgradient method can be extended to solve the inequality constrained problem
:minimize <math>f_0(x) \ </math> subject to <math display=block>f_i (x) \leq 0,\quad i = 1,\ldots,m</math>
where <math>f_i</math> are convex. The algorithm takes the same form as the unconstrained case
<math display=block>x^{(k+1)} = x^{(k)} - \alpha_k g^{(k)} \ </math>
where <math>\alpha_k>0</math> is a step size, and <math>g^{(k)}</math> is a subgradient of the objective or one of the constraint functions at <math>x. \ </math> Take
<math display=block>g^{(k)} =
\begin{cases}
\partial f_0 (x) & \text{ if } f_i(x) \leq 0 \; \forall i = 1 \dots m \\
\partial f_j (x) & \text{ for some } j \text{ such that } f_j(x) > 0
\end{cases}</math>
where <math>\partial f</math> denotes the [[subdifferential]] of <math>f. \ </math> If the current point is feasible, the algorithm uses an objective subgradient; if the current point is infeasible, the algorithm chooses a subgradient of any violated constraint.
==See also==
* {{annotated link|Stochastic gradient descent}}
==References==
{{reflist}}
==Further reading==
* {{cite book
| last = Bertsekas
| first = Dimitri P.
| title = Nonlinear Programming
| publisher = Athena Scientific
|
| ___location =
| isbn = 1-886529-00-0
}}
* {{cite book
|last1=Bertsekas
|first1=Dimitri P.
|last2=Nedic
|first2=Angelia
|last3 = Ozdaglar
|first3 = Asuman
| title = Convex Analysis and Optimization
| edition = Second
| publisher = Athena Scientific
| year = 2003
| ___location = Belmont, MA.
| isbn = 1-886529-45-0
}}
* {{cite book
| last = Bertsekas
| first = Dimitri P.
| title = Convex Optimization Algorithms
| publisher = Athena Scientific
| year = 2015
| ___location = Belmont, MA.
| isbn = 978-1-886529-28-1
}}
* {{cite book
| last = Shor
| first = Naum Z.
| title = Minimization Methods for Non-differentiable Functions
| publisher = [[Springer-Verlag]]
| isbn = 0-387-12763-1
|
}}
* {{cite book|last=Ruszczyński|author-link=Andrzej Piotr Ruszczyński|first=Andrzej|title=Nonlinear Optimization|publisher=[[Princeton University Press]]|___location=Princeton, NJ|year=2006|pages=xii+454|isbn=978-0691119151 |mr=2199043}}
==External links==
* [http://www.stanford.edu/class/ee364a/ EE364A] and [http://www.stanford.edu/class/ee364b/ EE364B], Stanford's convex optimization course sequence.
{{Convex analysis and variational analysis}}
{{optimization algorithms|convex}}
[[Category:Convex analysis]]
[[Category:Convex optimization]]
[[Category:Mathematical optimization]]
[[Category:Optimization algorithms and methods]]
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