Subgradient method
Subgradient methods are convex optimization methods which use 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 steepest 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
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.
Classical subgradient rules
Let be a convex function with domain A classical subgradient method iterates
Step size rules
Many different types of step-size rules are used by subgradient methods. This article notes five classical step-size rules for which convergence proofs are known:
- Constant step size,
- Constant step length, which gives
- Square summable but not summable step size, i.e. any step sizes satisfying
- Nonsummable diminishing, i.e. any step sizes satisfying
- Nonsummable diminishing step lengths, i.e. where
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[1] and by Bertsekas, Nedic, and Ozdaglar.[2]
Convergence results
For constant step-length and scaled subgradients having
These classical subgradient methods have poor performance and are no longer recommended for general use.[4][5] 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 Wolfe proposed "bundle methods" of descent for problems of convex minimization.[6] The meaning of the term "bundle methods" has changed significantly since that time. Modern versions and full convergence analysis were provided by Kiwiel. [7] Contemporary bundle-methods often use "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.[4][5]
Constrained optimization
Projected subgradient
One extension of the subgradient method is the projected subgradient method, which solves the constrained optimization problem
- minimize subject to
where is a convex set. The projected subgradient method uses the iteration
General constraints
The subgradient method can be extended to solve the inequality constrained problem
- minimize subject to
where are convex. The algorithm takes the same form as the unconstrained case
See also
- Stochastic gradient descent – Optimization algorithm
References
- ISBN 978-1-886529-28-1.
- ISBN 1-886529-45-0.
- ^
The approximate convergence of the constant step-size (scaled) subgradient method is stated as Exercise 6.3.14(a) in ISBN 0-387-12763-1.
- ^ a b
S2CID 9048698.
- ^ a b
Kiwiel, Krzysztof C.; Larsson, Torbjörn; Lindberg, P. O. (August 2007). "Lagrangian relaxation via ballstep subgradient methods". MR 2348241.
- ^
ISBN 1-886529-00-0.
- ^
Kiwiel, Krzysztof (1985). Methods of Descent for Nondifferentiable Optimization. Berlin: MR 0797754.
Further reading
- Bertsekas, Dimitri P. (1999). Nonlinear Programming. Belmont, MA.: Athena Scientific. ISBN 1-886529-00-0.
- Bertsekas, Dimitri P.; Nedic, Angelia; Ozdaglar, Asuman (2003). Convex Analysis and Optimization (Second ed.). Belmont, MA.: Athena Scientific. ISBN 1-886529-45-0.
- Bertsekas, Dimitri P. (2015). Convex Optimization Algorithms. Belmont, MA.: Athena Scientific. ISBN 978-1-886529-28-1.
- Shor, Naum Z. (1985). Minimization Methods for Non-differentiable Functions. ISBN 0-387-12763-1.
- MR 2199043.