Functional Interface

ROL's functional interface. More...

Classes
class	ROL::Objective< Real >
	Provides the interface to evaluate objective functions. More...
class	ROL::BoundConstraint< Real >
	Provides the interface to apply upper and lower bound constraints. More...
class	ROL::EqualityConstraint< Real >
	Defines the equality constraint operator interface. More...
class	ROL::LinearOperator< Real >
	Provides the interface to apply a linear operator. More...
class	ROL::Objective_SimOpt< Real >
	Provides the interface to evaluate simulation-based objective functions. More...
class	ROL::EqualityConstraint_SimOpt< Real >
	Defines the equality constraint operator interface for simulation-based optimization. More...
class	ROL::Reduced_Objective_SimOpt< Real >
	Provides the interface to evaluate simulation-based reduced objective functions. More...
class	ROL::LogBarrierObjective< Real >
	Log barrier objective for interior point methods. More...
class	ROL::AugmentedLagrangian< Real >
	Provides the interface to evaluate the augmented Lagrangian. More...
class	ROL::InteriorPointObjective< Real >
	Adds barrier term to generic objective. More...
class	ROL::InteriorPointEqualityConstraint< Real >
	Has both inequality and equality constraints. Treat inequality constraint as equality with slack variable. More...
class	ROL::InteriorPointBoundConstraint< Real >
	Require positivity of slack variables. More...
class	ROL::MoreauYosidaPenalty< Real >
	Provides the interface to evaluate the Moreau-Yosida penalty function. More...

Detailed Description

ROL's functional interface.

ROL is used for the numerical solution of smooth optimization problems

\[ \begin{array}{rl} \displaystyle \min_{x} & f(x) \\ \mbox{subject to} & c(x) = 0 \,, \\ & a \le x \le b \,, \end{array} \]

where:

• \(f : \mathcal{X} \rightarrow \mathbb{R}\) is a Fréchet differentiable functional,
• \(c : \mathcal{X} \rightarrow \mathcal{C}\) is a Fréchet differentiable operator,
• \(\mathcal{X}\) and \(\mathcal{C}\) are Banach spaces of functions, and
• \(a \le x \le b\) defines pointwise (componentwise) bounds on \(x\).

This formulation encompasses a variety of useful problem scenarios.

First, the vector spaces \(\mathcal{X}\) and \(\mathcal{C}\), to be defined by the user through the Linear Algebra Interface, can represent real spaces, such as \(\mathcal{X} = \mathbb{R}^n\) and \(\mathcal{C} = \mathbb{R}^m\), and function spaces, such as Hilbert and Banach function spaces. ROL's vector space abstractions enable efficent implementations of general optimization algorithms, spanning traditional nonlinear programming (NLP), optimization with partial differential equation (PDE) or differential algebraic equation (DAE) constraints, and stochastic optimization.

Second, ROL's core methods can solve four types of smooth optimization problems, depending on the structure of the constraints.

• Type-U. No constraints (where \(c(x) = 0\) and \(a \le x \le b\) are absent):

  \[ \begin{array}{rl} \displaystyle \min_{x} & f(x) \end{array} \]

  These problems are known as unconstrained optimization problems. The user implements the methods of the ROL::Objective interface.
• Type-B. Bound constraints (where \(c(x) = 0\) is absent):

  \[ \begin{array}{rl} \displaystyle \min_{x} & f(x) \\ \mbox{subject to} & a \le x \le b \,. \end{array} \]

  This problem is typically handled using projections on active sets or primal-dual active-set methods. ROL provides example implementations of the projections for simple box constraints. Other projections are defined by the user. The user implements the methods of the ROL::BoundConstraint interface.
• Type-E. Equality constraints, generally nonlinear and nonconvex (where \(a \le x \le b\) is absent):

  \[ \begin{array}{rl} \displaystyle \min_{x} & f(x) \\ \mbox{subject to} & c(x) = 0 \,. \end{array} \]

  Equality constraints are handled in ROL using, for example, matrix-free composite-step methods, including sequential quadratic programming (SQP). The user implements the methods of the ROL::EqualityConstraint interface.
• Type-EB. Equality and bound constraints:

  \[ \begin{array}{rl} \displaystyle \min_{x} & f(x) \\ \mbox{subject to} & c(x) = 0 \\ & a \le x \le b \,. \end{array} \]

  This formulation includes general inequality constraints. For example, we can consider the reformulation:

  \[ \begin{array}{rlcccrl} \displaystyle \min_{x} & f(x) &&&& \displaystyle \min_{x,s} & f(x) \\ \mbox{subject to} & c(x) \le 0 & & \quad \longleftrightarrow \quad & & \mbox{subject to} & c(x) + s = 0 \,, \\ &&&&&& s \ge 0 \,. \end{array} \]

  ROL uses a combination of matrix-free composite-step methods, projection methods and primal-dual active set methods to solve these problems. The user implements the methods of the ROL::EqualityConstraint and the ROL::BoundConstraint interfaces.

Third, ROL's design enables streamlined algorithmic extensions, such as the Stochastic Optimization capability.

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