465 lines
11 KiB
C++
465 lines
11 KiB
C++
/**
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* @file testConstraintOptimizer.cpp
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* @brief Tests the optimization engine for SQP and BFGS Quadratic programming techniques
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* @author Alex Cunningham
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*/
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#include <iostream>
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#include <limits>
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#include <boost/tuple/tuple.hpp>
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#include <boost/optional.hpp>
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#include <CppUnitLite/TestHarness.h>
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#include <Ordering.h>
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#include <ConstraintOptimizer.h>
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#include <smallExample.h>
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#define GTSAM_MAGIC_KEY
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#include <boost/assign/std/list.hpp> // for operator +=
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using namespace boost::assign;
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using namespace std;
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using namespace gtsam;
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using namespace example;
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/* ************************************************************************* */
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// Example of a single Constrained QP problem from the matlab testCQP.m file.
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TEST( matrix, CQP_example ) {
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Matrix A = Matrix_(3, 2,
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-1.0, -1.0,
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-2.0, 1.0,
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1.0, -1.0);
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Matrix At = trans(A),
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B = 2.0 * eye(3,3);
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Vector b = Vector_(2, 4.0, -2.0),
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g = zero(3);
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Matrix G = zeros(5,5);
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insertSub(G, B, 0, 0);
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insertSub(G, A, 0, 3);
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insertSub(G, At, 3, 0);
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Vector rhs = zero(5);
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subInsert(rhs, -1.0*g, 0);
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subInsert(rhs, -1.0*b, 3);
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// solve the system with the LDL solver
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Vector actualFull = solve_ldl(G, rhs);
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Vector actual = sub(actualFull, 0, 3);
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Vector expected = Vector_(3, 2.0/7.0, 10.0/7.0, -6.0/7.0);
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CHECK(assert_equal(expected, actual));
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}
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/* ************************************************************************* */
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TEST( matrix, CQP_example_automatic ) {
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Matrix A = Matrix_(3, 2,
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-1.0, -1.0,
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-2.0, 1.0,
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1.0, -1.0);
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Matrix At = trans(A),
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B = 2.0 * eye(3,3);
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Vector g = zero(3),
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h = Vector_(2, 4.0, -2.0);
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Vector actState, actLam;
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boost::tie(actState, actLam) = solveCQP(B, A, g, h);
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Vector expected = Vector_(3, 2.0/7.0, 10.0/7.0, -6.0/7.0);
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CHECK(assert_equal(expected, actState));
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CHECK(actLam.size() == 2);
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}
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/* ************************************************************************* */
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/** SQP example from SQP tutorial */
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namespace sqp_example1 {
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/**
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* objective function with gradient and hessian
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* fx = (x2-2)^2 + x1^2;
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*/
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double objective(const Vector& x, boost::optional<Vector&> g = boost::none,
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boost::optional<Matrix&> B = boost::none) {
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double x1 = x(0), x2 = x(1);
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if (g) *g = Vector_(2, 2.0*x1, 2.0*(x2-2.0));
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if (B) *B = 2.0 * eye(2,2);
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return (x2-2)*(x2-2) + x1*x1;
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}
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/**
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* constraint function with gradient and hessian
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* cx = 4*x1^2 + x2^2 - 1;
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*/
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Vector constraint(const Vector& x, boost::optional<Matrix&> A = boost::none,
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boost::optional<Matrix&> B = boost::none) {
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double x1 = x(0), x2 = x(1);
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if (A) *A = Matrix_(2,1, 8.0*x1, 2.0*x2);
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if (B) *B = Matrix_(2,2,
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8.0, 0.0,
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0.0, 2.0);
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return Vector_(1, 4.0*x1*x1 + x2*x2 - 1.0);
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}
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/**
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* evaluates a penalty function at a given point
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*/
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double penalty(const Vector& x) {
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double constraint_gain = 1.0;
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return objective(x) + constraint_gain*sum(abs(constraint(x)));
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}
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}
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/* ************************************************************************* */
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/** SQP example from SQP tutorial (real saddle point) */
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namespace sqp_example2 {
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/**
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* objective function with gradient and hessian
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* fx = (x2-2)^2 - x1^2;
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*/
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double objective(const Vector& x, boost::optional<Vector&> g = boost::none,
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boost::optional<Matrix&> B = boost::none) {
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double x1 = x(0), x2 = x(1);
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if (g) *g = Vector_(2, -2.0*x1, 2.0*(x2-2.0));
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if (B) *B = Matrix_(2,2, -2.0, 0.0, 0.0, 2.0);
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return (x2-2)*(x2-2) - x1*x1;
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}
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/**
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* constraint function with gradient and hessian
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* cx = 4*x1^2 + x2^2 - 1;
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*/
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Vector constraint(const Vector& x, boost::optional<Matrix&> A = boost::none,
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boost::optional<Matrix&> B = boost::none) {
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double x1 = x(0), x2 = x(1);
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if (A) *A = Matrix_(2,1, 8.0*x1, 2.0*x2);
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if (B) *B = Matrix_(2,2,
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8.0, 0.0,
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0.0, 2.0);
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return Vector_(1, 4.0*x1*x1 + x2*x2 - 1.0);
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}
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/**
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* evaluates a penalty function at a given point
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*/
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double penalty(const Vector& x) {
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double constraint_gain = 1.0;
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return objective(x) + constraint_gain*sum(abs(constraint(x)));
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}
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}
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/* ************************************************************************* */
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TEST( matrix, SQP_simple_analytic ) {
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using namespace sqp_example1;
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// parameters
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double stepsize = 0.25;
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size_t maxIt = 50;
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// initial conditions
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Vector x0 = Vector_(2, 2.0, 4.0),
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lam0 = Vector_(1, -0.5);
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// current state
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Vector x = x0, lam = lam0;
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for (size_t i =0; i<maxIt; ++i) {
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// evaluate functions
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Vector dfx;
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Matrix dcx, ddfx, ddcx;
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objective(x, dfx, ddfx);
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Vector cx = constraint(x, dcx, ddcx);
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// use analytic hessian
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Matrix B = ddfx - lam(0)*ddcx;
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// solve subproblem
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Vector delta, lambda;
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boost::tie(delta, lambda) = solveCQP(B, -dcx, dfx, -cx);
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// update
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Vector step = stepsize * delta;
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x = x + step;
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lam = lambda;
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}
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// verify
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Vector expX = Vector_(2, 0.0, 1.0),
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expLambda = Vector_(1, -1.0);
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CHECK(assert_equal(expX, x, 1e-4));
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CHECK(assert_equal(expLambda, lam, 1e-4));
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}
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/* ************************************************************************* */
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TEST( matrix, SQP_simple_manual_bfgs ) {
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using namespace sqp_example1;
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// parameters
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double stepsize = 0.25;
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size_t maxIt = 50;
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// initial conditions
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Vector x0 = Vector_(2, 2.0, 4.0),
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lam0 = Vector_(1, -0.5);
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// current state
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Vector x = x0, lam = lam0;
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Matrix Bi = eye(2,2);
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Vector step, prev_dfx;
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for (size_t i=0; i<maxIt; ++i) {
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// evaluate functions
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Vector dfx; Matrix dcx;
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objective(x, dfx);
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Vector cx = constraint(x, dcx);
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// Just use dfx for the Hessian
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if (i>0) {
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Vector Bis = Bi * step,
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y = dfx - prev_dfx;
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Bi = Bi + outer_prod(y, y) / inner_prod(y, step)
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- outer_prod(Bis, Bis) / inner_prod(step, Bis);
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}
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prev_dfx = dfx;
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// solve subproblem
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Vector delta, lambda;
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boost::tie(delta, lambda) = solveCQP(Bi, -dcx, dfx, -cx);
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// update
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step = stepsize * delta;
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x = x + step;
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lam = lambda;
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}
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// verify
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Vector expX = Vector_(2, 0.0, 1.0),
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expLambda = Vector_(1, -1.0);
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CHECK(assert_equal(expX, x, 1e-4));
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CHECK(assert_equal(expLambda, lam, 1e-4));
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}
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/* ************************************************************************* */
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TEST( matrix, SQP_simple_bfgs1 ) {
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using namespace sqp_example1;
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// parameters
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size_t maxIt = 25;
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// initial conditions
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Vector x0 = Vector_(2, 2.0, 4.0),
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lam0 = Vector_(1, -0.5);
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// create a BFGSEstimator
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BFGSEstimator hessian(2);
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// current state
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Vector x = x0, lam = lam0;
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Vector step;
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for (size_t i=0; i<maxIt; ++i) {
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// evaluate functions
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Vector dfx; Matrix dcx;
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objective(x, dfx);
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Vector cx = constraint(x, dcx);
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// Just use dfx for the Hessian
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if (i>0) {
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hessian.update(dfx, step);
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} else {
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hessian.update(dfx);
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}
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// solve subproblem
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Vector delta, lambda;
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boost::tie(delta, lambda) = solveCQP(hessian.getB(), -dcx, dfx, -cx);
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// if (i == 0) print(delta, "delta1");
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// update
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step = linesearch(x,delta,penalty);
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// step = stepsize * delta;
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x = x + step;
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lam = lambda;
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}
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// verify
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Vector expX = Vector_(2, 0.0, 1.0),
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expLambda = Vector_(1, -1.0);
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CHECK(assert_equal(expX, x, 1e-4));
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CHECK(assert_equal(expLambda, lam, 1e-4));
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}
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/* ************************************************************************* */
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TEST( matrix, SQP_simple_bfgs2 ) {
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using namespace sqp_example2;
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// parameters
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double stepsize = 0.25;
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size_t maxIt = 50;
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// initial conditions
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Vector x0 = Vector_(2, 2.0, 4.0),
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lam0 = Vector_(1, -0.5);
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// create a BFGSEstimator
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BFGSEstimator hessian(2);
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// current state
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Vector x = x0, lam = lam0;
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Vector step;
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for (size_t i=0; i<maxIt; ++i) {
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// evaluate functions
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Vector dfx; Matrix dcx;
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objective(x, dfx);
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Vector cx = constraint(x, dcx);
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// Just use dfx for the Hessian
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if (i>0) {
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hessian.update(dfx, step);
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} else {
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hessian.update(dfx);
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}
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// solve subproblem
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Vector delta, lambda;
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boost::tie(delta, lambda) = solveCQP(hessian.getB(), -dcx, dfx, -cx);
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// if (i == 0) print(delta, "delta2");
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// update
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// step = linesearch(x,delta,penalty);
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step = stepsize * delta;
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x = x + step;
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lam = lambda;
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}
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// verify
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Vector expX = Vector_(2, 0.0, 1.0),
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expLambda = Vector_(1, -1.0);
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// should determine the actual values for this one
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// CHECK(assert_equal(expX, x, 1e-4));
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// CHECK(assert_equal(expLambda, lam, 1e-4));
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}
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/* ************************************************************************* */
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TEST( matrix, line_search ) {
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using namespace sqp_example2;
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// initial conditions
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Vector x0 = Vector_(2, 2.0, 4.0),
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delta = Vector_(2, 0.85, -5.575);
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Vector actual = linesearch(x0,delta,penalty);
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// check that error goes down
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double init_error = penalty(x0),
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final_error = penalty(x0 + actual);
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double actual_stepsize = dot(actual, delta)/dot(delta, delta);
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// cout << "actual_stepsize: " << actual_stepsize << endl;
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CHECK(final_error <= init_error);
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}
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/* ************************************************************************* */
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TEST( matrix, unconstrained_fg_ata ) {
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// create a graph
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GaussianFactorGraph fg = createGaussianFactorGraph();
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Matrix A; Vector b;
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Ordering ordering;
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ordering += Symbol('l', 1), Symbol('x', 1), Symbol('x', 2);
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boost::tie(A, b) = fg.matrix(ordering);
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Matrix B_ata = prod(trans(A), A);
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// solve subproblem
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Vector actual = solve_ldl(B_ata, prod(trans(A), b));
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// verify
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Vector expected = createCorrectDelta().vector();
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CHECK(assert_equal(expected,actual));
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}
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///* ************************************************************************* */
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//TEST( matrix, unconstrained_fg ) {
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// // create a graph
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// GaussianFactorGraph fg = createGaussianFactorGraph();
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//
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// Matrix A; Vector b;
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// Ordering ordering;
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// ordering += Symbol('l', 1), Symbol('x', 1), Symbol('x', 2);
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// boost::tie(A, b) = fg.matrix(ordering);
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// Matrix B_ata = prod(trans(A), A);
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//// print(B_ata, "B_ata");
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//// print(b, " b");
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//
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// // parameters
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// size_t maxIt = 50;
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// double stepsize = 0.1;
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//
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// // iterate to solve
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// VectorConfig x = createZeroDelta();
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// BFGSEstimator B(x.dim());
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//
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// Vector step;
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//
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// for (size_t i=0; i<maxIt; ++i) {
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//// cout << "Error at Iteration: " << i << " is " << fg.error(x) << endl;
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//
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// // find the gradient
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// Vector dfx = fg.gradient(x).vector();
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//// print(dfx, " dfx");
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// CHECK(assert_equal(-1.0 * prod(trans(A), b - A*x.vector()), dfx));
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//
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// // update hessian
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// if (i>0) {
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// B.update(dfx, step);
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// } else {
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// B.update(dfx);
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// }
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//
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// // solve subproblem
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//// print(B.getB(), " B_bfgs");
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// Vector delta = solve_ldl(B.getB(), -dfx);
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//// Vector delta = solve_ldl(B_ata, -dfx);
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//
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//// print(delta, " delta");
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//
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// // update
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// step = stepsize * delta;
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//// step = linesearch(x, delta, penalty); // TODO: switch here
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// x = expmap(x, step);
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//// print(step, " step");
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// }
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//
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// // verify
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// VectorConfig expected = createCorrectDelta();
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// CHECK(assert_equal(expected,x, 1e-4));
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//}
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/* ************************************************************************* */
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int main() { TestResult tr; return TestRegistry::runAllTests(tr); }
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/* ************************************************************************* */
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