L-shaped solvers

QuasiGradientAlgorithm

Functor object for the quasi-gradient algorithm.

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Algorithm parameters

• log::Bool = true: Specifices if quasi-gradient procedure should be logged on standard output or not.

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AbstractQuasiGradientAttribute

Abstract supertype for attribute objects specific to the L-shaped algorithm.

Prox

An optimizer attribute for specifying a prox policy to be used in the quasi-gradient algorithm. Options are:

• NoProx: Unconstrained.
• Polyhedron: QP projection on polyhedral space ?PolyhedronProjection for parameter descriptions. (default)
• AndersonAcceleration: Anderson acceleration of inner prox step ?AndersonAcceleratedProximal for parameter descriptions.
• Nesterov: Nesterov acceleration of inner prox step ?NesterovProximal for parameter descriptions.
• DryFriction: Dry-friction acceleration of inner prox step ?DryFrictionProximal for parameter descriptions.

ProxParameter

Abstract supertype for prox-specific attributes.

StepParameter

Abstract supertype for step-specific attributes.

StepSize

An optimizer attribute for specifying an aggregation procedure to be used in the quasi-gradient algorithm. Options are:

• Constant: Constant step size ?ConstantStep for parameter descriptions (default)
• Diminishing: Diminishing step size ?DiminishingStep for parameter descriptions.
• Polyak: Polyak step size ?PolyakStep for parameter descriptions.
• BB: Barzilai-Borwein step size ?BBStep for parameter descriptions.

SubProblems

An optimizer attribute for specifying if subproblems should be smoothed. Options are:

• Unaltered: Subproblems are solved in their original state (default)
• Smoothed: Subproblems are smoothed using Moreau envelopes ?SmoothSubProblem for parameter descriptions.

Termination

An optimizer attribute for specifying a termination criterion to be used in the quasi-gradient algorithm. Options are:

• AfterMaximumIterations: Terminate after set number of iterations ?MaximumIterations for parameter descriptions (default)
• AtObjectiveThreshold: Terminate after reaching reference objective ?ObjectiveThreshold for parameter descriptions.
• AtGradientThreshold: Terminate after reaching zero gradient ?GradientThreshold for parameter descriptions.

TerminationParameter

Abstract supertype for termination-specific attributes.

Optimizer(; <keyword arguments>)

Return a quasi-gradient optimizer. ...

Arguments

• master_optimizer::AbstractOptimizer: MathOptInterface solver capable of solving linear (and possibly quadratic) programs.

• subproblem_optimizer::AbstractOptimizer: Optionally specify a different solver for the subproblems.

• execution::Execution = Serial: Specify how algorithm should be executed (Serial, Synchronous, Asynchronous). Distributed variants requires worker cores.

• subproblems::AbstractSubProblemState = Unaltered(): Specify if a smoothing procedure should be applied.

• prox::AbstractProx = Polyhedron(): Specify proximal step.

• step::AbstractStep = Constant(): Specify step-size

• termination::AbstractTermination = AfterMaximumIterations(): Specify termination criterion

• <keyword arguments>: Algorithm specific parameters, See ?LShaped for list of possible arguments and default values.

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get_prox_attribute(stochasticprogram::StochasticProgram, name::String)

Return the value associated with the prox-specific attribute named name in stochasticprogram.

See also: set_prox_attribute, set_prox_attributes.

get_step_attribute(stochasticprogram::StochasticProgram, name::String)

Return the value associated with the step-specific attribute named name in stochasticprogram.

See also: set_step_attribute, set_step_attributes.

get_termination_attribute(stochasticprogram::StochasticProgram, name::String)

Return the value associated with the termination-specific attribute named name in stochasticprogram.

See also: set_termination_attribute, set_termination_attributes.

set_prox_attribute(stochasticprogram::StochasticProgram, name::Union{Symbol, String}, value)

Sets the prox-specific attribute identified by name to value.

set_prox_attributes(stochasticprogram::StochasticProgram, pairs::Pair...)

Given a list of attribute => value pairs or a collection of keyword arguments, calls set_prox_attribute(stochasticprogram, attribute, value) for each pair.

set_step_attribute(stochasticprogram::StochasticProgram, name::Union{Symbol, String}, value)

Sets the step-specific attribute identified by name to value.

set_step_attributes(stochasticprogram::StochasticProgram, pairs::Pair...)

Given a list of attribute => value pairs or a collection of keyword arguments, calls set_step_attribute(stochasticprogram, attribute, value) for each pair.

set_termination_attribute(stochasticprogram::StochasticProgram, name::Union{Symbol, String}, value)

Sets the termination-specific attribute identified by name to value.

set_termination_attributes(stochasticprogram::StochasticProgram, pairs::Pair...)

Given a list of attribute => value pairs or a collection of keyword arguments, calls set_termination_attribute(stochasticprogram, attribute, value) for each pair.

SerialExecution

Functor object for using serial execution in a quasi-gradient algorithm. Create by supplying a Serial object through execution in QuasiGradient.Optimizer or by setting the Execution attribute.

SynchronousExecution

Functor object for using synchronous execution in an quasi-gradient algorithm (assuming multiple Julia cores are available). Create by supplying a Synchronous object through execution in QuasiGradient.Optimizer or by setting the Execution attribute.

set_step_attribute(stochasticprogram::StochasticProgram, name::Union{Symbol, String}, value)

Sets the step-specific attribute identified by name to value.

set_step_attributes(stochasticprogram::StochasticProgram, pairs::Pair...)

Given a list of attribute => value pairs or a collection of keyword arguments, calls set_step_attribute(stochasticprogram, attribute, value) for each pair.

RawStepParameter

An optimizer attribute used for raw parameters of the step. Defers to RawParameter.

Constant

Factory object for ConstantStep. Pass to step in Quasigradient.Optimizer or set the StepSize attribute. See ?ConstantStep for parameter descriptions.

ConstantStep

Functor object for using a constant step size in a quasigradient algorithm. Create by supplying a Constant object through step to QuasiGradient.Optimizer or by setting the StepSize attribute.

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Parameters

• γ::AbstractFloat = 0.01: Step length

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Diminishing

Factory object for DiminishingStep. Pass to step in Quasigradient.Optimizer or set the StepSize attribute. See ?DiminishingStep for parameter descriptions.

DiminishingStep

Functor object for using a constant step size in a quasigradient algorithm. Create by supplying a Constant object through step to QuasiGradient.Optimizer or by setting the StepSize attribute.

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Parameters

• γ₀::AbstractFloat = 0.1: Nominal step
• η::AbstractFloat = 1.0: Diminishing factor

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Polyak

Factory object for PolyakStep. Pass to step in Quasigradient.Optimizer or set the StepSize attribute. See ?PolyakStep for parameter descriptions.

PolyakStep

Functor object for using the Polyak step size in a quasigradient algorithm. Create by supplying a Polyak object through step to QuasiGradient.Optimizer or by setting the Prox attribute.

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Parameters

• γ₀::AbstractFloat = 0.1: Nominal step
• η::AbstractFloat = 1.0: Diminishing factor

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BB

Factory object for BBStep. Pass to step in Quasigradient.Optimizer or set the StepSize attribute. See ?BBStep for parameter descriptions.

BBStep

Functor object for using the Barzilai-Borwein step size in a quasigradient algorithm. Create by supplying a BB object through step to QuasiGradient.Optimizer or by setting the StepSize attribute.

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Parameters

• γ₀::AbstractFloat = 0.1: Initial step-size and fallback if BB is numerically unstable

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set_prox_attribute(stochasticprogram::StochasticProgram, name::Union{Symbol, String}, value)

Sets the prox-specific attribute identified by name to value.

set_prox_attributes(stochasticprogram::StochasticProgram, pairs::Pair...)

Given a list of attribute => value pairs or a collection of keyword arguments, calls set_prox_attribute(stochasticprogram, attribute, value) for each pair.

NoProx

Factory object for NoProximal. Passed by default to prox in QuasiGradient.Optimizer.

NoProximal

Empty functor object for running a quasi-gradient algorithm without a prox step.

ProxPenaltyTerm

An optimizer attribute used to set the proximal term in the prox step. Options are:

• Quadratic (default)
• InfNorm
• ManhattanNorm

RawProxParameter

An optimizer attribute used for raw parameters of the proximal step. Defers to RawParameter.

Polyhedron

Factory object for PolyhedronProjection. Pass to prox in Quasigradient.Optimizer or set the Prox attribute. See ?PolyhedronProjection for parameter descriptions.

PolyhedronProjection

Functor object for using polyhedral projection in the prox step of a quasigradient algorithm. Create by supplying a Polyhedron object through prox to QuasiGradient.Optimizer or by setting the Prox attribute.

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Parameters

• penaltyterm::PenaltyTerm = Quadratic: Specify penaltyterm variant (Quadratic, InfNorm, [ManhattanNorm][@ref])

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AndersonAcceleratedProximal

Functor object for using anderson accleration in the prox step of a quasigradient algorithm. Create by supplying an AndersonAcceleration object through prox to QuasiGradient.Optimizer or by setting the Prox attribute.

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Parameters

• prox::AbstractProx = Polyhedron: Inner prox step
• m::Integer = 10: Anderson memory

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AndersonAcceleration

Factory object for AndersonAcceleratedProximal. Pass to prox in Quasigradient.Optimizer or set the Prox attribute. See ?AndersonAcceleratedProximal for parameter descriptions.

Nesterov

Factory object for NesterovProximal. Pass to prox in Quasigradient.Optimizer or set the Prox attribute. See ?NesterovProximal for parameter descriptions.

NesterovProximal

Functor object for using nesterov accleration, with FISTA updates, in the prox step of a quasigradient algorithm. Create by supplying a Nesterov object through prox to QuasiGradient.Optimizer or by setting the Prox attribute.

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Parameters

• prox::AbstractProx = Polyhedron: Inner prox step
• stabilizing_projection::Bool = false: Specify if an extra guarding projection should be performed to ensure first-stage feasibility

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DryFriction

Factory object for DryFrictionProximal. Pass to prox in Quasigradient.Optimizer or set the Prox attribute. See ?DryFrictionProximal for parameter descriptions.

DryFrictionProximal

Functor object for using dry-friction acceleration in the prox step of a quasigradient algorithm. Create by supplying a DryFriction object through prox to QuasiGradient.Optimizer or by setting the Prox attribute.

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Parameters

• prox::AbstractProx = Polyhedron: Inner prox step
• γ::AbstractFloat = 0.9: Heavy-ball parameter
• r::AbstractFloat = 1e-3: Dry-friction parameter

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set_termination_attribute(stochasticprogram::StochasticProgram, name::Union{Symbol, String}, value)

Sets the termination-specific attribute identified by name to value.

set_termination_attributes(stochasticprogram::StochasticProgram, pairs::Pair...)

Given a list of attribute => value pairs or a collection of keyword arguments, calls set_termination_attribute(stochasticprogram, attribute, value) for each pair.

RawTerminationParameter

An optimizer attribute used for raw parameters of the termination criterion. Defers to RawParameter.

AfterMaximumIterations

Factory object for MaximumIterations. Pass to terminate in Quasigradient.Optimizer or set the Termination attribute. See ?MaximumIterations for parameter descriptions.

MaximumIterations

Functor object for using maximum number of iterations as termination criterion in a quasigradient algorithm. Create by supplying a AfterMaximumIterations object through terminate to QuasiGradient.Optimizer or by setting the Termination attribute.

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Parameters

• maximum::Integer = 1000: Maximum number of iterations

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AtObjectiveThreshold

Factory object for ObjectiveThreshold. Pass to terminate in Quasigradient.Optimizer or set the Termination attribute. See ?ObjectiveThreshold for parameter descriptions.

ObjectiveThreshold

Functor object for using an objective threshold as termination criterion in a quasigradient algorithm. Create by supplying a AtObjectiveThreshold object through terminate to QuasiGradient.Optimizer or by setting the Termination attribute.

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Parameters

• reference::AbstractFloat = 0.0: Reference objective value
• τ::AbstractFloat = 1e-6: Relative tolerance

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AtGradientThreshold

Factory object for GradientThreshold. Pass to terminate in Quasigradient.Optimizer or set the Termination attribute. See ?GradientThreshold for parameter descriptions.

ObjectiveThreshold

Functor object for using a zero gradient threshold as termination criterion in a quasigradient algorithm. Create by supplying a AtGradientThreshold object through terminate to QuasiGradient.Optimizer or by setting the Termination attribute.

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Parameters

• τ::AbstractFloat = 1e-6: Numerical tolerance for zero gradient

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Unaltered

Factory object for using regular SubProblem in the quasi-gradient algorithm. Passed by default to subproblems to QuasiGradient.Optimizer.

Smoothed

Factory object for using SmoothSubProblem through Moreau envelopes in the quasi-gradient algorithm. Pass to subproblems in QuasiGradient.Optimizer or by setting the SubProblems attribute. See SmoothSubProblem for parameter descriptions.

SubProblem

Quasi-gradient subproblem.

SmoothSubProblem

Subproblem smoothed using a Moreau envelope.

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Parameters

• μ::AbstractFloat = 1.0: Moreau smoothing parameter. Controls the smoothing approximation accuracy.

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