Solver4J – Newton unconstrained

The class NewtonUnconstrained implements an unconstrained Newton optimizer (see S.Boyd and L.Vandenberghe, "Convex Optimization", p. 487 for more details), and can be used to solve the following problem:

${minimize}_{x} f (x)$

where $f : R^{n} \to R$ is convex and twice differentiable.

Code example

Consider the following quadratic problem:

${minimize}_{x} \frac{1}{2} x^{T} Px + q^{T} x + r$
where $P = \frac{0.01522}{2} (\begin{matrix} 1.68 & 0.34 & 0.38 \\ 0.34 & 3.09 & - 1.59 \\ 0.38 & - 1.59 & 1.54 \end{matrix})$ , $q = (\begin{matrix} - 0.018 \\ - 0.025 \\ - 0.01 \end{matrix})$ and $r = 0$

It can be solved like this:

		
		RealMatrix PMatrix = new Array2DRowRealMatrix(new double[][] {
				{ 1.68, 0.34, 0.38 },
				{ 0.34, 3.09, -1.59 }, 
				{ 0.38, -1.59, 1.54 } });
		RealVector qVector = new ArrayRealVector(new double[] { 0.018, 0.025, 0.01 });

	    // Objective function.
		double theta = 0.01522;
		RealMatrix P = PMatrix.scalarMultiply(theta);
		RealVector q = qVector.mapMultiply(-1);
		PDQuadraticMultivariateRealFunction objectiveFunction = new PDQuadraticMultivariateRealFunction(P.getData(), q.toArray(), 0);
		
		OptimizationRequest or = new OptimizationRequest();
		or.setF0(objectiveFunction);
		or.setInitialPoint(new double[] {0.04, 0.50, 0.46});
		or.setTolerance(1.e-8);
		
	    //optimization
		NewtonUnconstrained opt = new NewtonUnconstrained();
		opt.setOptimizationRequest(or);
		opt.optimize();

This will give the solution (that we expect):

			 $sol = - P^{- 1} q$

Download test source bundle

Download the entire test source bundle from here.