public IEnumerable <double[]> Solve(Term objective, IEnumerable <Term> constraints, Variable[] variables, double[] startPoint)
{
var constraintSquares = constraints.Select(c => TermBuilder.Power(c, 2));
var penalizedObjective = objective + penaltyWeight * TermUtils.SafeSum(constraintSquares);
var compiledPenalizedObjective = penalizedObjective.Compile(variables);
var solution = unconstrainedOptimizer.Solve(x => compiledPenalizedObjective.Differentiate(x), startPoint, gradientNormThreshold);
yield return(solution);
}
public IEnumerable <AugmentedLagrangianIterationResult> Start(Term objective, IEnumerable <Term> constraints, Variable[] variables, double[] startPoint)
{
var compiledLagrangian = lagrangianCompiler.Compile(objective, constraints, variables);
const double CONSTRAINT_NORM_EXPONENT_INIT = 0.5;
const double CONSTRAINT_NORM_EXPONENT_REDUCE = 0.5;
const double GRADIENT_NORM_EXPONENT_INIT = 0.5;
const double GRADIENT_NORM_EXPONENT_REDUCE = 0.5;
var multipliers = new double[compiledLagrangian.ConstraintsCount];
var currentPoint = startPoint;
var constraintsPenalty = startConstraintsPenalty;
var maxConstraintsNorm = 1 / Math.Pow(constraintsPenalty, CONSTRAINT_NORM_EXPONENT_INIT);
var maxLagrangianGradientNorm = 1 / Math.Pow(constraintsPenalty, GRADIENT_NORM_EXPONENT_INIT);
while (true)
{
// we compute a new optimal point estimate
currentPoint = unconstrainedOptimizer.Solve(
objectiveWithGradient: x => compiledLagrangian.LagrangianWithGradient(x, multipliers, constraintsPenalty), // f(x) + 0.5 * mu * ||c(x)||² + lambda * c(x)
initialValue: currentPoint,
gradientNormThreshold: maxLagrangianGradientNorm);
// compute the value of each constraint function. These are the constraint violations
var constraintValues = compiledLagrangian.EvaluateConstraints(currentPoint);
var constraintsNorm = Math.Sqrt(constraintValues.Select(x => x * x).Sum());
if (constraintsNorm < maxConstraintsNorm)
{
// compute the norm of the gradient of the lagrangian
var lagrangianGradient = compiledLagrangian.LagrangianWithGradient(currentPoint, multipliers, constraintsPenalty).Item1;
var lagrangianGradientNorm = Math.Sqrt(lagrangianGradient.Sum(x => x * x));
// we now product iteration result
yield return(new AugmentedLagrangianIterationResult
{
Values = currentPoint,
ConstraintsNorm = constraintsNorm,
LagrangianGradientNorm = lagrangianGradientNorm
});
// update the current lagrange multipliers estimate according to the algorithm's update formula
// lambda <-- lambda + mu * c
for (int i = 0; i < multipliers.Length; ++i)
{
multipliers[i] = multipliers[i] + constraintValues[i] * constraintsPenalty;
}
maxConstraintsNorm = maxConstraintsNorm / Math.Pow(constraintsPenalty, CONSTRAINT_NORM_EXPONENT_REDUCE);
maxConstraintsNorm = Math.Max(maxConstraintsNorm, maxConstraintsNormLowerBound);
maxLagrangianGradientNorm = maxLagrangianGradientNorm / Math.Pow(constraintsPenalty, GRADIENT_NORM_EXPONENT_REDUCE);
maxLagrangianGradientNorm = Math.Max(maxLagrangianGradientNorm, lagrangianGradientNormLowerBound);
}
else
{
constraintsPenalty = 2 * constraintsPenalty;
constraintsPenalty = Math.Min(constraintsPenalty, constraintsPenaltyMax);
maxConstraintsNorm = 1 / Math.Pow(constraintsPenalty, CONSTRAINT_NORM_EXPONENT_INIT);
maxConstraintsNorm = Math.Max(maxConstraintsNorm, maxConstraintsNormLowerBound);
maxLagrangianGradientNorm = 1 / Math.Pow(constraintsPenalty, GRADIENT_NORM_EXPONENT_INIT);
maxLagrangianGradientNorm = Math.Max(maxLagrangianGradientNorm, lagrangianGradientNormLowerBound);
}
}
}