QuadraticMinimizer

class QuadraticMinimizer(nGram: Int, proximal: Proximal, Aeq: DenseMatrix[Double], beq: DenseVector[Double], maxIters: Int, abstol: Double, reltol: Double, alpha: Double) extends SerializableLogging

Proximal operators and ADMM based Primal-Dual QP Solver

Reference: http://www.stanford.edu/~boyd/papers/admm/quadprog/quadprog.html

It solves problem that has the following structure

1/2 x'Hx + f'x + g(x) s.t Aeqx = b

g(x) represents the following constraints which covers ALS based matrix factorization use-cases

  1. x >= 0
  2. lb <= x <= ub
  3. L1(x)
  4. L2(x)
  5. Generic regularization on x
Value parameters:
Aeq

rhs matrix for equality constraints

abstol

ADMM absolute tolerance

alpha

over-relaxation parameter default 1.0 1.5 - 1.8 can improve convergence

beq

lhs constants for equality constraints

nGram

rank of dense gram matrix

proximal

proximal operator to be used

reltol

ADMM relative tolerance

Companion:
object
trait Serializable
class Object
trait Matchable
class Any

Type members

Classlikes

case class State

Value members

Concrete methods

Public API to get an initialState for solver hot start such that subsequent calls can reuse state memmory

Public API to get an initialState for solver hot start such that subsequent calls can reuse state memmory

Returns:

the state for the optimizer

minimize API for cases where gram matrix is updated through updateGram API. If a initialState is not provided by default it constructs it through initialize

minimize API for cases where gram matrix is updated through updateGram API. If a initialState is not provided by default it constructs it through initialize

Value parameters:
initialState

provide an optional initialState for memory optimization

q

linear term for quadratic optimization

Returns:

converged solution

minimize API for cases where gram matrix is provided by the user. If a initialState is not provided by default it constructs it through initialize

minimize API for cases where gram matrix is provided by the user. If a initialState is not provided by default it constructs it through initialize

Value parameters:
H

symmetric gram matrix of size rank x rank

initialState

provide an optional initialState for memory optimization

q

linear term for quadratic optimization

Returns:

converged solution

def minimize(upper: Array[Double], q: DenseVector[Double], initialState: State): DenseVector[Double]

minimize API for cases where upper triangular gram matrix is provided by user as primitive array. If a initialState is not provided by default it constructs it through initialize

minimize API for cases where upper triangular gram matrix is provided by user as primitive array. If a initialState is not provided by default it constructs it through initialize

Value parameters:
initialState

provide an optional initialState for memory optimization

q

linear term for quadratic optimization

upper

upper triangular gram matrix of size rank x (rank + 1)/2

Returns:

converged solution

def minimizeAndReturnState(q: DenseVector[Double], rho: Double, initialState: State, resetState: Boolean): State

minimizeAndReturnState API gives an advanced control for users who would like to use QuadraticMinimizer in 2 steps, update the gram matrix first using updateGram API and followed by doing the solve by providing a user defined initialState. It also exposes rho control to users who would like to experiment with rho parameters of the admm algorithm. Use user-defined rho only if you understand the proximal algorithm well

minimizeAndReturnState API gives an advanced control for users who would like to use QuadraticMinimizer in 2 steps, update the gram matrix first using updateGram API and followed by doing the solve by providing a user defined initialState. It also exposes rho control to users who would like to experiment with rho parameters of the admm algorithm. Use user-defined rho only if you understand the proximal algorithm well

Value parameters:
initialState

provide a initialState using initialState API

q

linear term for the quadratic optimization

resetState

use true if you want to hot start based on the provided state

rho

rho parameter for ADMM algorithm

Returns:

converged state from ADMM algorithm

minimizeAndReturnState API gives an advanced control for users who would like to use QuadraticMinimizer in 2 steps, update the gram matrix first using updateGram API and followed by doing the solve by providing a user defined initialState. rho is automatically calculated by QuadraticMinimizer from problem structure

minimizeAndReturnState API gives an advanced control for users who would like to use QuadraticMinimizer in 2 steps, update the gram matrix first using updateGram API and followed by doing the solve by providing a user defined initialState. rho is automatically calculated by QuadraticMinimizer from problem structure

Value parameters:
initialState

provide a initialState using initialState API

q

linear term for the quadratic optimization

Returns:

converged state from QuadraticMinimizer

minimizeAndReturnState API that takes a symmetric full gram matrix and the linear term for quadratic minimization

minimizeAndReturnState API that takes a symmetric full gram matrix and the linear term for quadratic minimization

Value parameters:
H

gram matrix, symmetric of size rank x rank

initialState

provide a initialState using initialState API for memory optimization

q

linear term

Returns:

converged state from QuadraticMinimizer

def minimizeAndReturnState(upper: Array[Double], q: DenseVector[Double], initialState: State): State

minimizeAndReturnState API that takes upper triangular entries of the gram matrix specified through primitive array for performance reason and the linear term for quadratic minimization

minimizeAndReturnState API that takes upper triangular entries of the gram matrix specified through primitive array for performance reason and the linear term for quadratic minimization

Value parameters:
initialState

provide a initialState using initialState API for memory optimization

q

linear term

upper

upper triangular gram matrix specified as primitive array

Returns:

converged state from QuadraticMinimizer

updateGram allows the user to seed QuadraticMinimizer with symmetric gram matrix most useful for cases where the gram matrix does not change but the linear term changes for multiple solves. It should be called iteratively from Normal Equations constructed by the user

updateGram allows the user to seed QuadraticMinimizer with symmetric gram matrix most useful for cases where the gram matrix does not change but the linear term changes for multiple solves. It should be called iteratively from Normal Equations constructed by the user

Value parameters:
H

rank * rank size full gram matrix

def updateGram(upper: Array[Double]): Unit

updateGram API allows user to seed QuadraticMinimizer with upper triangular gram matrix (memory optimization by 50%) specified through primitive arrays. It is exposed for advanced users like Spark ALS where ALS constructs normal equations as primitive arrays

updateGram API allows user to seed QuadraticMinimizer with upper triangular gram matrix (memory optimization by 50%) specified through primitive arrays. It is exposed for advanced users like Spark ALS where ALS constructs normal equations as primitive arrays

Value parameters:
upper

upper triangular gram matrix specified in primitive array

Inherited methods

protected def logger: LazyLogger
Inherited from:
SerializableLogging

Concrete fields

val full: Int
val n: Int