// <copyright file="BfgsMinimizer.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://numerics.mathdotnet.com
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// http://github.com/mathnet/mathnet-numerics
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//
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// Copyright (c) 2009-2017 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
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// obtaining a copy of this software and associated documentation
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// files (the "Software"), to deal in the Software without
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// restriction, including without limitation the rights to use,
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// copies of the Software, and to permit persons to whom the
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// Software is furnished to do so, subject to the following
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// conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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using System;
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using System.Collections.Generic;
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using System.Linq;
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using IStation.Numerics.LinearAlgebra;
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using IStation.Numerics.Optimization.LineSearch;
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namespace IStation.Numerics.Optimization
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{
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/// <summary>
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/// Limited Memory version of Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm
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/// </summary>
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public class LimitedMemoryBfgsMinimizer : MinimizerBase, IUnconstrainedMinimizer
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{
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public int Memory { get; set; }
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/// <inheritdoc />
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/// <summary>
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/// Creates L-BFGS minimizer
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/// </summary>
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/// <param name="memory">Numbers of gradients and steps to store.</param>
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public LimitedMemoryBfgsMinimizer(double gradientTolerance, double parameterTolerance, double functionProgressTolerance, int memory, int maximumIterations=1000) : base(gradientTolerance, parameterTolerance, functionProgressTolerance, maximumIterations)
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{
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Memory = memory;
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}
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/// <summary>
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/// Find the minimum of the objective function given lower and upper bounds
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/// </summary>
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/// <param name="objective">The objective function, must support a gradient</param>
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/// <param name="initialGuess">The initial guess</param>
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/// <returns>The MinimizationResult which contains the minimum and the ExitCondition</returns>
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public MinimizationResult FindMinimum(IObjectiveFunction objective, Vector<double> initialGuess)
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{
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if (!objective.IsGradientSupported)
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throw new IncompatibleObjectiveException("Gradient not supported in objective function, but required for L-BFGS minimization.");
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objective.EvaluateAt(initialGuess);
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ValidateGradientAndObjective(objective);
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// Check that we're not already done
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ExitCondition currentExitCondition = ExitCriteriaSatisfied(objective, null, 0);
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if (currentExitCondition != ExitCondition.None)
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return new MinimizationResult(objective, 0, currentExitCondition);
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// Set up line search algorithm
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var lineSearcher = new WeakWolfeLineSearch(1e-4, 0.9, Math.Max(ParameterTolerance, 1e-10), 1000);
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// First step
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var lineSearchDirection = -objective.Gradient;
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var stepSize = 100 * GradientTolerance / (lineSearchDirection * lineSearchDirection);
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var previousPoint = objective;
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LineSearchResult lineSearchResult;
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try
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{
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lineSearchResult = lineSearcher.FindConformingStep(objective, lineSearchDirection, stepSize);
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}
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catch (OptimizationException e)
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{
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throw new InnerOptimizationException("Line search failed.", e);
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}
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catch (ArgumentException e)
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{
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throw new InnerOptimizationException("Line search failed.", e);
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}
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var candidate = lineSearchResult.FunctionInfoAtMinimum;
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ValidateGradientAndObjective(candidate);
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var gradient = candidate.Gradient;
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var step = candidate.Point - initialGuess;
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var yk = candidate.Gradient - previousPoint.Gradient;
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var ykhistory = new List<Vector<double>>() {yk};
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var skhistory = new List<Vector<double>>() {step};
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var rhokhistory = new List<double>() {1.0/yk.DotProduct(step)};
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// Subsequent steps
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int iterations = 1;
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int totalLineSearchSteps = lineSearchResult.Iterations;
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int iterationsWithNontrivialLineSearch = lineSearchResult.Iterations > 0 ? 0 : 1;
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previousPoint = candidate;
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while (iterations++ < MaximumIterations && previousPoint.Gradient.Norm(2) >= GradientTolerance)
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{
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lineSearchDirection = -ApplyLbfgsUpdate(previousPoint, ykhistory, skhistory, rhokhistory);
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var directionalDerivative = previousPoint.Gradient.DotProduct(lineSearchDirection);
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if (directionalDerivative > 0)
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throw new InnerOptimizationException("Direction is not a descent direction.");
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try
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{
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lineSearchResult = lineSearcher.FindConformingStep(previousPoint, lineSearchDirection, 1.0);
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}
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catch (OptimizationException e)
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{
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throw new InnerOptimizationException("Line search failed.", e);
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}
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catch (ArgumentException e)
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{
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throw new InnerOptimizationException("Line search failed.", e);
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}
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iterationsWithNontrivialLineSearch += lineSearchResult.Iterations > 0 ? 1 : 0;
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totalLineSearchSteps += lineSearchResult.Iterations;
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candidate = lineSearchResult.FunctionInfoAtMinimum;
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currentExitCondition = ExitCriteriaSatisfied(candidate, previousPoint, iterations);
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if (currentExitCondition != ExitCondition.None)
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break;
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step = candidate.Point - previousPoint.Point;
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yk = candidate.Gradient - previousPoint.Gradient;
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ykhistory.Add(yk);
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skhistory.Add(step);
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rhokhistory.Add(1.0/yk.DotProduct(step));
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previousPoint = candidate;
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if (ykhistory.Count > Memory)
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{
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ykhistory.RemoveAt(0);
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skhistory.RemoveAt(0);
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rhokhistory.RemoveAt(0);
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}
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}
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if (iterations == MaximumIterations && currentExitCondition == ExitCondition.None)
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throw new MaximumIterationsException(FormattableString.Invariant($"Maximum iterations ({MaximumIterations}) reached."));
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return new MinimizationWithLineSearchResult(candidate, iterations, ExitCondition.AbsoluteGradient, totalLineSearchSteps, iterationsWithNontrivialLineSearch);
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}
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private Vector<double> ApplyLbfgsUpdate(IObjectiveFunction previousPoint, List<Vector<double>> ykhistory, List<Vector<double>> skhistory, List<double> rhokhistory)
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{
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var q = previousPoint.Gradient.Clone();
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var alphas = new Stack<double>();
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for (int k = ykhistory.Count - 1; k >= 0; k--)
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{
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var alpha = rhokhistory[k]*q.DotProduct(skhistory[k]);
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alphas.Push(alpha);
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q -= alpha*ykhistory[k];
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}
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var yk = ykhistory.Last();
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var sk = skhistory.Last();
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q *= yk.DotProduct(sk)/yk.DotProduct(yk);
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for (int k = 0; k < ykhistory.Count; k++)
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{
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var beta = rhokhistory[k]*ykhistory[k].DotProduct(q);
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q += skhistory[k]*(alphas.Pop() - beta);
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}
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return q;
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}
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}
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}
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