minima

package minima

The minima package contains classes, traits and objects for optimization to find minima.

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class Brent extends AnyRef
The Brent class is used to find roots (zeros) for a one-dimensional (scalar) function 'g'.
The Brent class is used to find roots (zeros) for a one-dimensional (scalar) function 'g'. Depending on the FunctionSelector, it can find zeros for derivatives or finite differences, which may indicate optima for function 'g'. The code is directly translated from the following:

See also
math.haifa.ac.il/ronn/NA/NAprogs/brent.java
class CheckLP extends Error
The CheckLP class checks the solution to Linear Programming (LP) problems.
The CheckLP class checks the solution to Linear Programming (LP) problems. Given a constraint matrix 'a', limit/RHS vector 'b' and cost vector 'c', determine if the values for the solution/decision vector 'x' minimizes the objective function 'f(x)', while satisfying all of the constraints, i.e.,
minimize f(x) = c x subject to a x <= b, x >= 0
Check the feasibility and optimality of the solution.
class ConjugateGradient extends Minimizer with Error
The ConjugateGradient class implements the Polak-Ribiere Conjugate Gradient (PR-CG) Algorithm for solving Non-Linear Programming (NLP) problems.
The ConjugateGradient class implements the Polak-Ribiere Conjugate Gradient (PR-CG) Algorithm for solving Non-Linear Programming (NLP) problems. PR-CG determines a search direction as a weighted combination of the steepest descent direction (-gradient) and the previous direction. The weighting is set by the beta function, which for this implementation used the Polak-Ribiere technique.
dir_k = -gradient (x) + beta * dir_k-1
minimize f(x) subject to g(x) <= 0 [ optionally g(x) == 0 ]
class CoordinateDescent extends Minimizer with Error
The CoordinateDescent class solves unconstrained Non-Linear Programming (NLP) problems using the Coordinate Descent algorithm.
The CoordinateDescent class solves unconstrained Non-Linear Programming (NLP) problems using the Coordinate Descent algorithm. Given a function 'f' and a starting point 'x0', the algorithm picks coordinate directions (cyclically) and takes steps in the those directions. The algorithm iterates until it converges.
dir_k = kth coordinate direction
minimize f(x)
class DualSimplex extends MinimizerLP
The DualSimplex class solves Linear Programming (LP) problems using a tableau based Dual Simplex Algorithm.
The DualSimplex class solves Linear Programming (LP) problems using a tableau based Dual Simplex Algorithm. It is particularly useful when re-optimizing after a constraint has been added. The algorithm starts with an infeasible super-optimal solution and moves toward (primal) feasibility and optimality.
Given a constraint matrix 'a', limit/RHS vector 'b' and cost vector 'c', find values for the solution/decision vector 'x' that minimize the objective function f(x), while satisfying all of the constraints, i.e.,
minimize f(x) = c x subject to a x <= b, x >= 0
Creates an 'MM-by-NN' simplex tableau with -- [0..M-1, 0..N-1] = a (constraint matrix) -- [0..M-1, N..M+N-1] = s (slack/surplus variable matrix) -- [0..M-1, NN-1] = b (limit/RHS vector) -- [M, 0..NN-2] = c (cost vector)
class GeneticAlgorithm extends AnyRef
The GeneticAlgorithm class performs local search to find minima of functions defined on integer vector domains (z^n).
The GeneticAlgorithm class performs local search to find minima of functions defined on integer vector domains (z^n).
minimize f(x) subject to g(x) <= 0, x in Z^n
class GoldenSectionLS extends LineSearch
The GoldenSectionLS class performs a line search on 'f(x)' to find a minimal value for 'f'.
The GoldenSectionLS class performs a line search on 'f(x)' to find a minimal value for 'f'. It requires no derivatives and only one functional evaluation per iteration. A search is conducted from 'x1' (often 0) to 'xmax'. A guess for 'xmax' must be given, but can be made larger during the expansion phase, that occurs before the recursive golden section search is called. It works on scalar functions (see GoldenSectionLSTest). If starting with a vector function 'f(x)', simply define a new function 'g(y) = x0 + direction * y' (see GoldenSectionLSTest2).
class GradientDescent extends Minimizer with Error
The GradientDescent class solves unconstrained Non-Linear Programming (NLP) problems using the Gradient Descent algorithm.
The GradientDescent class solves unconstrained Non-Linear Programming (NLP) problems using the Gradient Descent algorithm. Given a function 'f' and a starting point 'x0', the algorithm computes the gradient and takes steps in the opposite direction. The algorithm iterates until it converges. The class assumes that partial derivative functions are not available unless explicitly given via the 'setDerivatives' method.
dir_k = -gradient (x)
minimize f(x)
class GridLS extends LineSearch
The GridLS class performs a line search on 'f(x)' to find a minimal value for 'f'.
The GridLS class performs a line search on 'f(x)' to find a minimal value for 'f'. It requires no derivatives and only one functional evaluation per iteration. A search is conducted from 'x1' (often 0) to 'xmax'. A guess for 'xmax' must be given. It works on scalar functions (see GridLSTest). If starting with a vector function 'f(x)', simply define a new function 'g(y) = x0 + direction * y' (see GridLSTest2).
class IntegerGoldenSectionLS extends AnyRef
The IntegerGoldenSectionLS class performs a line search on 'f(x)' to find a minimal value for 'f'.
The IntegerGoldenSectionLS class performs a line search on 'f(x)' to find a minimal value for 'f'. It requires no derivatives and only one functional evaluation per iteration. A search is conducted from 'x1' (often 0) to 'xmax'. A guess for 'xmax' must be given, but can be made larger during the expansion phase, that occurs before the recursive golden section search is called. It works on scalar functions

See also
IntegerGoldenSectionLSTest. If starting with a vector function 'f(x)', simply define a new function 'g(y) = x0 + direction * y'.
IntegerGoldenSectionLSTest2.
class IntegerLP extends AnyRef
The IntegerLP class solves Integer Linear Programming (ILP) and Mixed Integer Linear Programming (MILP) problems recursively using the Simplex algorithm.
The IntegerLP class solves Integer Linear Programming (ILP) and Mixed Integer Linear Programming (MILP) problems recursively using the Simplex algorithm. First, an LP problem is solved. If the optimal solution vector x is entirely integer valued, the ILP is solved. If not, pick the first 'x_j' that is not integer valued. Define two new LP problems which bound 'x_j' to the integer below and above, respectively. Branch by solving each of these LP problems in turn. Prune by not exploring branches less optimal than the currently best integer solution. This technique is referred to as Branch and Bound. An exclusion set may be optionally provided for MILP problems. FIX: Use the Dual Simplex Algorithm for better performance.
Given a constraint matrix 'a', limit/RHS vector 'b' and cost vector 'c', find values for the solution/decision vector 'x' that minimize the objective function 'f(x)', while satisfying all of the constraints, i.e.,
minimize f(x) = c x subject to a x <= b, x >= 0, some x_i must be integer valued
Make 'b_i' negative to indicate a '>=' constraint.
class IntegerLocalSearch extends AnyRef
The IntegerLocalSearch class performs local search to find minima of functions defined on integer vector domains (z^n).
The IntegerLocalSearch class performs local search to find minima of functions defined on integer vector domains (z^n).
minimize f(x) subject to g(x) <= 0, x in Z^n
class IntegerNLP extends AnyRef
This IntegerNLPsolves Integer Non-Linear Programming (INLP) and Mixed Integer Linear Non-Programming (MINLP) problems recursively using the Simplex algorithm.
This IntegerNLPsolves Integer Non-Linear Programming (INLP) and Mixed Integer Linear Non-Programming (MINLP) problems recursively using the Simplex algorithm. First, an NLP problem is solved. If the optimal solution vector 'x' is entirely integer valued, the INLP is solved. If not, pick the first 'x_j' that is not integer valued. Define two new NLP problems which bound 'x_j' to the integer below and above, respectively. Branch by solving each of these NLP problems in turn. Prune by not exploring branches less optimal than the currently best integer solution. This technique is referred to as Branch and Bound. An exclusion set may be optionally provided for MINLP problems.
Given an objective function 'f(x)' and a constraint function 'g(x)', find values for the solution/decision vector 'x' that minimize the objective function 'f(x)', while satisfying the constraint function, i.e.,
minimize f(x) subject to g(x) <= 0, some x_i must integer-valued
Make b_i negative to indicate a ">=" constraint
class IntegerTabuSearch extends AnyRef
The IntegerTabuSearch class performs tabu search to find minima of functions defined on integer vector domains 'z^n'. Tabu search will not re-visit points already deemed sub-optimal.
The IntegerTabuSearch class performs tabu search to find minima of functions defined on integer vector domains 'z^n'. Tabu search will not re-visit points already deemed sub-optimal.
minimize f(x) subject to g(x) <= 0, x in Z^n
class L_BFGS_B extends Minimizer
The L_BFGS_B the class implements the Limited memory Broyden–Fletcher– Goldfarb–Shanno for Bound constrained optimization (L-BFGS-B) Quasi-Newton Algorithm for solving Non-Linear Programming (NLP) problems.
The L_BFGS_B the class implements the Limited memory Broyden–Fletcher– Goldfarb–Shanno for Bound constrained optimization (L-BFGS-B) Quasi-Newton Algorithm for solving Non-Linear Programming (NLP) problems. L-BFGS-B determines a search direction by deflecting the steepest descent direction vector (opposite the gradient) by * multiplying it by a matrix that approximates the inverse Hessian. Furthermore, only a few vectors represent the approximation of the Hessian Matrix (limited memory). The parameters estimated are also bounded within user specified lower and upper bounds.
minimize f(x) subject to g(x) <= 0 [ optionally g(x) == 0 ]
trait LineSearch extends AnyRef
The LineSearch trait specifies the basic methods for Line Search (LS) algorithms in classes extending this trait to implement.
The LineSearch trait specifies the basic methods for Line Search (LS) algorithms in classes extending this trait to implement. Line search is for one dimensional optimization problems. The algorithms perform line search to find an 'x'-value that minimizes a function 'f' that is passed into an implementing class.
x* = argmin f(x)
trait Minimizer extends AnyRef
The Minimizer trait sets the pattern for optimization algorithms for solving Non-Linear Programming (NLP) problems of the form:
The Minimizer trait sets the pattern for optimization algorithms for solving Non-Linear Programming (NLP) problems of the form:
minimize f(x) subject to g(x) <= 0 [ optionally g(x) == 0 ]
where f is the objective function to be minimized g is the constraint function to be satisfied, if any
Classes mixing in this trait must implement a function 'fg' that rolls the constraints into the objective functions as penalties for constraint violation, a one-dimensional Line Search (LS) algorithm 'lineSearch' and an iterative method (solve) that searches for improved solutions 'x'-vectors with lower objective function values (f(x)).
trait MinimizerLP extends Error
The MinimizerLP trait sets the pattern for optimization algorithms for solving Linear Programming (LP) problems of the form:
The MinimizerLP trait sets the pattern for optimization algorithms for solving Linear Programming (LP) problems of the form:
minimize c x subject to a x <= b, x >= 0
where a is the constraint matrix b is the limit/RHS vector c is the cost vector
Classes mixing in this trait must implement an objective function 'objF' an iterative method (solve) that searches for improved solutions 'x'-vectors with lower objective function values.
class NelderMeadSimplex extends Minimizer with Error
The NelderMeadSimplex solves Non-Linear Programming (NLP) problems using the Nelder-Mead Simplex algorithm.
The NelderMeadSimplex solves Non-Linear Programming (NLP) problems using the Nelder-Mead Simplex algorithm. Given a function 'f' and its dimension 'n', the algorithm moves a simplex defined by n + 1 points in order to find an optimal solution. The algorithm is derivative-free.
minimize f(x)
class NewtonRaphson extends AnyRef
The NewtonRaphson class is used to find roots (zeros) for a one-dimensional (scalar) function 'g'.
The NewtonRaphson class is used to find roots (zeros) for a one-dimensional (scalar) function 'g'. Depending on the FunctionSelector, it can find zeros for derivatives or finite differences, which may indicate optima for function 'g'. Also, for optimization, may pass the derivative of the function, since finding zeros for the derivative corresponds to finding optima for the function.
class QuadraticSimplex extends Error
The QuadraticSimplex class solves Quadratic Programming (QP) problems using the Quadratic Simplex Algorithm.
The QuadraticSimplex class solves Quadratic Programming (QP) problems using the Quadratic Simplex Algorithm. Given a constraint matrix 'a', constant vector 'b', cost matrix 'q' and cost vector 'c', find values for the solution/decision vector 'x' that minimize the objective function 'f(x)', while satisfying all of the constraints, i.e.,
minimize f(x) = 1/2 x q x + c x subject to a x <= b, x >= 0
Creates an 'MM-by-NN' simplex tableau. This implementation is restricted to linear constraints 'a x <= b' and 'q' being a positive semi-definite matrix. Pivoting must now also handle non-linear complementary slackness.

See also
www.engineering.uiowa.edu/~dbricker/lp_stacks.html
class QuasiNewton extends Minimizer with Error
The QuasiNewton the class implements the Broyden–Fletcher–Goldfarb–Shanno (BFGS) Quasi-Newton Algorithm for solving Non-Linear Programming (NLP) problems.
The QuasiNewton the class implements the Broyden–Fletcher–Goldfarb–Shanno (BFGS) Quasi-Newton Algorithm for solving Non-Linear Programming (NLP) problems. BFGS determines a search direction by deflecting the steepest descent direction vector (opposite the gradient) by * multiplying it by a matrix that approximates the inverse Hessian. Note, this implementation may be set up to work with the matrix 'b' (approximate Hessian) or directly with the 'binv' matrix (the inverse of 'b').
minimize f(x) subject to g(x) <= 0 [ optionally g(x) == 0 ]
class RevisedSimplex extends MinimizerLP
The RevisedSimplex class solves Linear Programming (LP) problems using the Revised Simplex Algorithm.
The RevisedSimplex class solves Linear Programming (LP) problems using the Revised Simplex Algorithm. Given a constraint matrix 'a', constant vector 'b' and cost vector 'c', find values for the solution/decision vector 'x' that minimize the objective function 'f(x)', while satisfying all of the constraints, i.e.,
minimize f(x) = c x subject to a x <= b, x >= 0
The Revised Simplex Algorithm operates on 'b_inv', which is the inverse of the basis-matrix ('ba' = 'B'). It has benefits over the Simplex Algorithm (less memory and reduced chance of round off errors).
class Simplex extends MinimizerLP
The Simplex class solves Linear Programming (LP) problems using a tableau based Simplex Algorithm.
The Simplex class solves Linear Programming (LP) problems using a tableau based Simplex Algorithm. Given a constraint matrix 'a', limit/RHS vector 'b' and cost vector 'c', find values for the solution/decision vector 'x' that minimize the objective function 'f(x)', while satisfying all of the constraints, i.e.,
minimize f(x) = c x subject to a x <= b, x >= 0
In case of 'a_i x >= b_i', use '-b_i' as an indicator of a '>=' constraint. The program will flip such negative b_i back to positive as well as use a surplus variable instead of the usual slack variable, i.e., a_i x <= b_i => a_i x + s_i = b_i // use slack variable s_i with coefficient 1 a_i x >= b_i => a_i x + s_i = b_i // use surplus variable s_i with coefficient -1
Creates an MM-by-NN simplex tableau with -- [0..M-1, 0..N-1] = a (constraint matrix) -- [0..M-1, N..M+N-1] = s (slack/surplus variable matrix) -- [0..M-1, NN-1] = b (limit/RHS vector) -- [M, 0..NN-2] = c (cost vector)
class Simplex2P extends MinimizerLP
The Simplex2P class solves Linear Programming (LP) problems using a tableau based Simplex Algorithm.
The Simplex2P class solves Linear Programming (LP) problems using a tableau based Simplex Algorithm. Given a constraint matrix 'a', limit/RHS vector 'b' and cost vector 'c', find values for the solution/decision vector 'x' that minimize the objective function 'f(x)', while satisfying all of the constraints, i.e.,
minimize f(x) = c x subject to a x <= b, x >= 0
In case of 'a_i x >= b_i', use -b_i as an indicator of a ">=" constraint. The program will flip such negative b_i back to positive as well as use a surplus and artificial variable instead of the usual slack variable, i.e., a_i x <= b_i => a_i x + s_i = b_i // use slack variable s_i with coefficient 1 a_i x >= b_i => a_i x + s_i = b_i // use surplus variable s_i with coefficient -1 For each '>=' constraint, an artificial variable is introduced and put into the initial basis. These artificial variables must be removed from the basis during Phase I of the Two-Phase Simplex Algorithm. After this, or if there are no artificial variables, Phase II is used to find an optimal value for 'x' and the optimum value for 'f'.
Creates an 'MM-by-nn' simplex tableau with -- [0..M-1, 0..N-1] = a (constraint matrix) -- [0..M-1, N..M+N-1] = s (slack/surplus variable matrix) -- [0..M-1, M+N..nn-2] = r (artificial variable matrix) -- [0..M-1, nn-1] = b (limit/RHS vector) -- [M, 0..nn-2] = c (cost vector)
class SimplexBG extends MinimizerLP
The SimplexBG class solves Linear Programming (LP) problems using the Bartels-Golub (BG) Simplex Algorithm.
The SimplexBG class solves Linear Programming (LP) problems using the Bartels-Golub (BG) Simplex Algorithm. Given a constraint matrix 'a', constant vector 'b' and cost vector 'c', find values for the solution/decision vector 'x' that minimize the objective function 'f(x)', while satisfying all of the constraints, i.e.,
minimize f(x) = c x subject to a x <= b, x >= 0
The BG Simplex Algorithm performs LU Factorization/Decomposition of the basis-matrix ('ba' = 'B') rather than computing inverses ('b_inv'). It has benefits over the (Revised) Simplex Algorithm (less run-time, less memory, and much reduced chance of round off errors).
class SimplexFT extends MinimizerLP
The SimplexFT class solves Linear Programming (LP) problems using the Forrest-Tomlin (FT) Simplex Algorithm.
The SimplexFT class solves Linear Programming (LP) problems using the Forrest-Tomlin (FT) Simplex Algorithm. Given a constraint matrix 'a', constant vector 'b' and cost vector 'c', find values for the solution/decision vector 'x' that minimize the objective function 'f(x)', while satisfying all of the constraints, i.e.,
minimize f(x) = c x subject to a x <= b, x >= 0
The FT Simplex Algorithm performs LU Factorization/Decomposition of the basis-matrix ('ba' = 'B') rather than computing inverses ('b_inv'). It has benefits over the (Revised) Simplex Algorithm (less run-time, less memory, and much reduced chance of round off errors).
class StochasticGradient extends Minimizer with Error
The StochasticGradient class solves unconstrained Non-Linear Programming (NLP) problems using the Stochastic Gradient Descent algorithm.
The StochasticGradient class solves unconstrained Non-Linear Programming (NLP) problems using the Stochastic Gradient Descent algorithm. Given a function 'f' and a starting point 'x0', the algorithm computes the gradient and takes steps in the opposite direction. The algorithm iterates until it converges. The algorithm is stochastic in sense that only a single batch is used in each step of the optimimation. Examples (a number of rows) are are chosen for each batch. FIX - provide option to randomly select samples in batch

See also
leon.bottou.org/publications/pdf/compstat-2010.pdf dir_k = -gradient (x) minimize f(x)
class WolfeLS extends LineSearch
The WolfeLS class performs an inexact line search on 'f' to find a point 'x' that exhibits (1) sufficient decrease ('f(x)' enough less that 'f(0)') and (2) the slope at x is less steep than the slope at 0.
The WolfeLS class performs an inexact line search on 'f' to find a point 'x' that exhibits (1) sufficient decrease ('f(x)' enough less that 'f(0)') and (2) the slope at x is less steep than the slope at 0. That is, the line search looks for a value for 'x' satisfying the two Wolfe conditions.
f(x) <= f(0) + c1 * f'(0) * x Wolfe condition 1 (Armijo condition) |f'(x)| <= |c2 * f'(0)| Wolfe condition 2 (Strong version) f'(x) >= c2 * f'(0) Wolfe condition 2 (Weak version, more robust)
It works on scalar functions (@see WolfeLSTest). If starting with a vector function f(x), simply define a new function g(y) = x0 + direction * y (@see WolfeLSTest2).

Value Members

val G_RATIO: Double
the golden ratio (1.618033988749895)
val G_SECTION: Double
the golden section number (0.6180339887498949)
object AugLagrangian
The AugLagrangian class implements the Augmented Lagrangian Method for solving equality constrained optimization problems.
The AugLagrangian class implements the Augmented Lagrangian Method for solving equality constrained optimization problems. Minimize objective function 'f' subject to constraint 'h' to find an optimal solution for 'x'.
min f(x) s.t. h(x) = 0
f = objective function h = equality contraint x = solution vector
Note: the hyper-parameters 'eta' and 'p0' will need to be tuned per problem.

See also
AugLagrangianTest for how to set up 'f', 'h' and 'grad' functions
object AugLagrangianTest extends App
The AugLagrangianTest object tests the AugLagrangian object using a simple equality constrained optimization problem defined by functions 'f' and 'h'.
The AugLagrangianTest object tests the AugLagrangian object using a simple equality constrained optimization problem defined by functions 'f' and 'h'. Caller must also supply the gradient of the Augmented Lagrangian 'grad'. > runMain scalation.minima.AugLagrangianTest
object BrentTest extends App
The BrentTest object is used to test the Brent class.
The BrentTest object is used to test the Brent class. > runMain scalation.minima.BrentTest
object ConjugateGradientTest extends App
The ConjugateGradientTest object is used to test the ConjugateGradient class.
The ConjugateGradientTest object is used to test the ConjugateGradient class. > runMain scalation.minima.ConjugateGradientTest
object CoordinateDescentTest extends App
The CoordinateDescentTest object is used to test the CoordinateDescent class.
The CoordinateDescentTest object is used to test the CoordinateDescent class. > runMain scalation.minima.CoordinateDescentTest
object DualSimplexTest extends App
The DualSimplexTest object is used to test the DualSimplex class.
object Ftran
The Ftran object ...
object FunctionSelector extends Enumeration
The FunctionSelector provides an enumeration of function types.
object GeneticAlgorithmTest extends App
The GeneticAlgorithmTest object is used to test the GeneticAlgorithm class (unconstrained).
object GoldenSectionLSTest extends App
The GoldenSectionLSTest object is used to test the GoldenSectionLS class on scalar functions.
The GoldenSectionLSTest object is used to test the GoldenSectionLS class on scalar functions. > runMain scalation.minima.GoldenSectionLSTest
object GoldenSectionLSTest2 extends App
The GoldenSectionLSTest2 object is used to test the GoldenSectionLS class on vector functions.
The GoldenSectionLSTest2 object is used to test the GoldenSectionLS class on vector functions. > runMain scalation.minima.GoldenSectionLSTest
object GradientDescentTest extends App
The GradientDescentTest object is used to test the GradientDescent class.
The GradientDescentTest object is used to test the GradientDescent class. > runMain scalation.minima.GradientDescentTest
object GridLSTest extends App
The GridLSTest object is used to test the GridLS class on scalar functions.
The GridLSTest object is used to test the GridLS class on scalar functions. > runMain scalation.minima.GridLSTest
object GridLSTest2 extends App
The GridLSTest2 object is used to test the GridLS class on vector functions.
The GridLSTest2 object is used to test the GridLS class on vector functions. > runMain scalation.minima.GridLSTest
object IntegerGoldenSectionLSTest extends App
The IntegerGoldenSectionLSTest object is used to test the IntegerGoldenSectionLS class on scalar functions.
object IntegerLPTest extends App
The IntegerLPTest object is used to test the IntegerLP class.
The IntegerLPTest object is used to test the IntegerLP class. real solution x = (.8, 1.6), f = 8.8 integer solution x = (2, 1), f = 10

See also
Linear Programming and Network Flows, Example 6.14
object IntegerLocalSearchTest extends App
The IntegerLocalSearchTest object is used to test the IntegerLocalSearch class (unconstrained).
object IntegerLocalSearchTest2 extends App
The IntegerLocalSearchTest2 object is used to test the IntegerLocalSearch class (constrained).
object IntegerNLPTest extends App
The IntegerNLPTest object is used to test the IntegerNLP class.
The IntegerNLPTest object is used to test the IntegerNLP class. real solution x = (.8, 1.6), f = 8.8 integer solution x = (2, 1), f = 10

See also
Linear Programming and Network Flows, Example 6.14
object IntegerTabuSearchTest extends App
The IntegerTabuSearchTest object is used to test the IntegerTabuSearch class (unconstrained).
object IntegerTabuSearchTest2 extends App
The IntegerTabuSearchTest2 object is used to test the IntegerTabuSearch class (constrained).
object L_BFGS_BTest extends App
The L_BFGS_BTest object is used to test the L_BFGS_B class.
The L_BFGS_BTest object is used to test the L_BFGS_B class. > runMain scalation.minima.L_BFGS_BTest
object LassoAdmm
The LassoAdmm class performs LASSO regression using Alternating Direction Method of Multipliers (ADMM).
The LassoAdmm class performs LASSO regression using Alternating Direction Method of Multipliers (ADMM). Minimize the following objective function to find an optimal solutions for 'x'.
argmin_x (1/2)||Ax − b||_2^2 + λ||x||_1
A = data matrix b = response vector λ = weighting on the l_1 penalty x = solution (coefficient vector)

See also
euler.stat.yale.edu/~tba3/stat612/lectures/lec23/lecture23.pdf
https://web.stanford.edu/~boyd/papers/admm_distr_stats.html
object LassoAdmmTest extends App
The LassoAdmmTest object tests LassoAdmm class using the following regression equation.
The LassoAdmmTest object tests LassoAdmm class using the following regression equation.
y = b dot x = b_0 + b_1*x_1 + b_2*x_2.

See also
statmaster.sdu.dk/courses/st111/module03/index.html > runMain scalation.minima.LassoAdmmTest
object LassoAdmmTest2 extends App
The LassoAdmmTest2 object tests LassoAdmm class using the following regression equation.
The LassoAdmmTest2 object tests LassoAdmm class using the following regression equation.
y = b dot x = b_0 + b_1*x_1 + b_2*x_2.

See also
www.cs.jhu.edu/~svitlana/papers/non_refereed/optimization_1.pdf > runMain scalation.minima.LassoAdmmTest2
object NLPTest1 extends App
The NLPTest1 object used to test several Non-Linear Programming (NLP) algorithms on unconstrained problems.
The NLPTest1 object used to test several Non-Linear Programming (NLP) algorithms on unconstrained problems. Algorithms: 'sdcs' - Gradient Descent with Custom Line Search 'sdgs' - Gradient Descent with Golden Section Line Search 'prcg' - Polak-Ribiere Conjugate Gradient with Golden Section Line Search 'sdws' - Gradient Descent with Wolfe Line Search 'bfgs' - Broyden–Fletcher–Goldfarb–Shanno with Wolfe Line Search
object NLPTest2 extends App
The NLPTest2 object used to test several Non-Linear Programming (NLP) algorithms on constrained problems.
object NLPTestCases1 extends App
The NLPTestCases1 object used to test several Non-Linear Programming (NLP) algorithms on unconstrained problems.
The NLPTestCases1 object used to test several Non-Linear Programming (NLP) algorithms on unconstrained problems. Algorithms: 'sdcs' - Gradient Descent with Custom Line Search 'sdgs' - Gradient Descent with Golden Section Line Search 'prcg' - Polak-Ribiere Conjugate Gradient with Golden Section Line Search 'sdws' - Gradient Descent with Wolfe Line Search 'bfgs' - Broyden–Fletcher–Goldfarb–Shanno with Wolfe Line Search
object NLPTestCases2 extends App
The NLPTestCases2 object used to test several Non-Linear Programming (NLP) algorithms on constrained problems.
object NelderMeadSimplexTest extends App
The NelderMeadSimplexTest object is used to test the NelderMeadSimplex class.
The NelderMeadSimplexTest object is used to test the NelderMeadSimplex class. > runMain scalation.minima.NelderMeadSimplexTest
object NewtonRaphsonTest extends App
The NewtonRaphsonTest object is used to test the NewtonRaphson class.
The NewtonRaphsonTest object is used to test the NewtonRaphson class. This test numerically approximates the derivative. > runMain scalation.minima.NewtonRaphsonTest
object NewtonRaphsonTest2 extends App
The NewtonRaphsonTest2 object is used to test the NewtonRaphson class.
The NewtonRaphsonTest2 object is used to test the NewtonRaphson class. This test passes in a function for the derivative. > runMain scalation.minima.NewtonRaphsonTest2
object QuadraticSimplexTest extends App
The QuadraticSimplexTest object is used to test the QuadraticSimplex class.
The QuadraticSimplexTest object is used to test the QuadraticSimplex class. > runMain scalation.minima.QuadraticSimplexTest
object QuasiNewtonTest extends App
The QuasiNewtonTest object is used to test the QuasiNewton class.
The QuasiNewtonTest object is used to test the QuasiNewton class. > runMain scalation.minima.QuasiNewtonTest
object RevisedSimplexTest extends App
The RevisedSimplexTest object is used to test the RevisedSimplex class.
The RevisedSimplexTest object is used to test the RevisedSimplex class. > runMain scalation.minima.RevisedSimplexTest
object Simplex2PTest extends App
The Simplex2PTest object is used to test the Simplex2P class.
object SimplexBGTest extends App
The SimplexBGTest object is used to test the SimplexBG class.
object SimplexFTTest extends App
The SimplexFT object is used to test the SimplexFT class.
object SimplexTest extends App
The SimplexTest object is used to test the Simplex class.
object StochasticGradientTest extends App
The StochasticGradientTest object is used to test the StochasticGradient class.
The StochasticGradientTest object is used to test the StochasticGradient class.

See also
scalation.analytics.RegressionTest3 > runMain scalation.minima.StochasticGradientTest
object WolfeLSTest extends App
The WolfeLSTest object is used to test the WolfeLS class on scalar functions.
The WolfeLSTest object is used to test the WolfeLS class on scalar functions. > runMain scalation.minima.WolfeLSTest
object WolfeLSTest2 extends App
The WolfeLSTest2 object is used to test the WolfeLS class on vector functions.
The WolfeLSTest2 object is used to test the WolfeLS class on vector functions. > runMain scalation.minima.WolfeLSTest2

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minima

package minima

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minima 

package minima

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minima