Add a Primal-dual interior-point method solver
This sets us up to solve nonlinear, convex problems efficiently. I'm
not 100% sure this has the right interfaces, but it appears to work for
a toy problem and is worth checking in.
This is from https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
section 11.7. Parts of the code are from the multinode timestamp
solver, though that solver has enough workarounds for the problem that
is isn't easily generalized.
Change-Id: I38a727b81df3ebfc24a42a633ab1cf74cf5ac692
Signed-off-by: Austin Schuh <austin.linux@gmail.com>
diff --git a/frc971/solvers/convex.h b/frc971/solvers/convex.h
new file mode 100644
index 0000000..aa14c1a
--- /dev/null
+++ b/frc971/solvers/convex.h
@@ -0,0 +1,381 @@
+#include <sys/types.h>
+#include <unistd.h>
+
+#include <Eigen/Dense>
+#include <iomanip>
+
+#include "absl/strings/str_join.h"
+#include "glog/logging.h"
+
+namespace frc971 {
+namespace solvers {
+
+// TODO(austin): Steal JET from Ceres to generate the derivatives easily and
+// quickly?
+//
+// States is the number of inputs to the optimization problem.
+// M is the number of inequality constraints.
+// N is the number of equality constraints.
+template <size_t States, size_t M, size_t N>
+class ConvexProblem {
+ public:
+ // Returns the function to minimize and it's derivatives.
+ virtual double f0(
+ Eigen::Ref<const Eigen::Matrix<double, States, 1>> X) const = 0;
+ virtual Eigen::Matrix<double, States, 1> df0(
+ Eigen::Ref<const Eigen::Matrix<double, States, 1>> X) const = 0;
+ virtual Eigen::Matrix<double, States, States> ddf0(
+ Eigen::Ref<const Eigen::Matrix<double, States, 1>> X) const = 0;
+
+ // Returns the constraints f(X) < 0, and their derivative.
+ virtual Eigen::Matrix<double, M, 1> f(
+ Eigen::Ref<const Eigen::Matrix<double, States, 1>> X) const = 0;
+ virtual Eigen::Matrix<double, M, States> df(
+ Eigen::Ref<const Eigen::Matrix<double, States, 1>> X) const = 0;
+
+ // Returns the equality constraints of the form A x = b
+ virtual Eigen::Matrix<double, N, States> A() const = 0;
+ virtual Eigen::Matrix<double, N, 1> b() const = 0;
+};
+
+// Implements a Primal-Dual Interior point method convex solver.
+// See 11.7 of https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
+//
+// States is the number of inputs to the optimization problem.
+// M is the number of inequality constraints.
+// N is the number of equality constraints.
+template <size_t States, size_t M, size_t N>
+class Solver {
+ public:
+ // Ratio to require the cost to decrease when line searching.
+ static constexpr double kAlpha = 0.05;
+ // Line search step parameter.
+ static constexpr double kBeta = 0.5;
+ static constexpr double kMu = 2.0;
+ // Terminal condition for the primal problem (equality constraints) and dual
+ // (gradient + inequality constraints).
+ static constexpr double kEpsilonF = 1e-6;
+ // Terminal condition for nu, the surrogate duality gap.
+ static constexpr double kEpsilon = 1e-6;
+
+ // Solves the problem given a feasible initial solution.
+ Eigen::Matrix<double, States, 1> Solve(
+ const ConvexProblem<States, M, N> &problem,
+ Eigen::Ref<const Eigen::Matrix<double, States, 1>> X_initial);
+
+ private:
+ // Class to hold all the derivataves and function evaluations.
+ struct Derivatives {
+ Eigen::Matrix<double, States, 1> gradient;
+ Eigen::Matrix<double, States, States> hessian;
+
+ // Inequality function f
+ Eigen::Matrix<double, M, 1> f;
+ // df
+ Eigen::Matrix<double, M, States> df;
+
+ // ddf is assumed to be 0 because for the linear constraint distance
+ // function we are using, it is actually 0, and by assuming it is zero
+ // rather than passing it through as 0 to the solver, we can save enough CPU
+ // to make it worth it.
+
+ // A
+ Eigen::Matrix<double, N, States> A;
+ // Ax - b
+ Eigen::Matrix<double, N, 1> Axmb;
+ };
+
+ // Computes all the values for the given problem at the given state.
+ Derivatives ComputeDerivative(
+ const ConvexProblem<States, M, N> &problem,
+ const Eigen::Ref<const Eigen::Matrix<double, States + M + N, 1>> y);
+
+ // Computes Rt at the given state and with the given t_inverse. See 11.53 of
+ // cvxbook.pdf.
+ Eigen::Matrix<double, States + M + N, 1> Rt(
+ const Derivatives &derivatives,
+ Eigen::Matrix<double, States + M + N, 1> y, double t_inverse);
+
+ // Prints out all the derivatives with VLOG at the provided verbosity.
+ void PrintDerivatives(
+ const Derivatives &derivatives,
+ const Eigen::Ref<const Eigen::Matrix<double, States + M + N, 1>> y,
+ std::string_view prefix, int verbosity);
+};
+
+template <size_t States, size_t M, size_t N>
+Eigen::Matrix<double, States + M + N, 1> Solver<States, M, N>::Rt(
+ const Derivatives &derivatives, Eigen::Matrix<double, States + M + N, 1> y,
+ double t_inverse) {
+ Eigen::Matrix<double, States + M + N, 1> result;
+
+ Eigen::Ref<Eigen::Matrix<double, States, 1>> r_dual =
+ result.template block<States, 1>(0, 0);
+ Eigen::Ref<Eigen::Matrix<double, M, 1>> r_cent =
+ result.template block<M, 1>(States, 0);
+ Eigen::Ref<Eigen::Matrix<double, N, 1>> r_pri =
+ result.template block<N, 1>(States + M, 0);
+
+ Eigen::Ref<const Eigen::Matrix<double, M, 1>> lambda =
+ y.template block<M, 1>(States, 0);
+ Eigen::Ref<const Eigen::Matrix<double, N, 1>> v =
+ y.template block<N, 1>(States + M, 0);
+
+ r_dual = derivatives.gradient + derivatives.df.transpose() * lambda +
+ derivatives.A.transpose() * v;
+ r_cent = -(Eigen::DiagonalMatrix<double, M>(lambda) * derivatives.f +
+ t_inverse * Eigen::Matrix<double, M, 1>::Ones());
+ r_pri = derivatives.Axmb;
+
+ return result;
+}
+
+template <size_t States, size_t M, size_t N>
+Eigen::Matrix<double, States, 1> Solver<States, M, N>::Solve(
+ const ConvexProblem<States, M, N> &problem,
+ Eigen::Ref<const Eigen::Matrix<double, States, 1>> X_initial) {
+ const Eigen::IOFormat kHeavyFormat(Eigen::StreamPrecision, 0, ", ",
+ ",\n "
+ " ",
+ "[", "]", "[", "]");
+
+ Eigen::Matrix<double, States + M + N, 1> y =
+ Eigen::Matrix<double, States + M + N, 1>::Constant(1.0);
+ y.template block<States, 1>(0, 0) = X_initial;
+
+ Derivatives derivatives = ComputeDerivative(problem, y);
+
+ for (size_t i = 0; i < M; ++i) {
+ CHECK_LE(derivatives.f(i, 0), 0.0)
+ << ": Initial state " << X_initial.transpose().format(kHeavyFormat)
+ << " not feasible";
+ }
+
+ PrintDerivatives(derivatives, y, "", 1);
+
+ size_t iteration = 0;
+ while (true) {
+ // Solve for the primal-dual search direction by solving the newton step.
+ Eigen::Ref<const Eigen::Matrix<double, M, 1>> lambda =
+ y.template block<M, 1>(States, 0);
+
+ const double nu = -(derivatives.f.transpose() * lambda)(0, 0);
+ const double t_inverse = nu / (kMu * lambda.rows());
+ Eigen::Matrix<double, States + M + N, 1> rt_orig =
+ Rt(derivatives, y, t_inverse);
+
+ Eigen::Matrix<double, States + M + N, States + M + N> m1;
+ m1.setZero();
+ m1.template block<States, States>(0, 0) = derivatives.hessian;
+ m1.template block<States, M>(0, States) = derivatives.df.transpose();
+ m1.template block<States, N>(0, States + M) = derivatives.A.transpose();
+ m1.template block<M, States>(States, 0) =
+ -(Eigen::DiagonalMatrix<double, M>(lambda) * derivatives.df);
+ m1.template block<M, M>(States, States) -=
+ Eigen::DiagonalMatrix<double, M>(derivatives.f);
+ m1.template block<N, States>(States + M, 0) = derivatives.A;
+
+ Eigen::Matrix<double, States + M + N, 1> dy =
+ m1.colPivHouseholderQr().solve(-rt_orig);
+
+ Eigen::Ref<Eigen::Matrix<double, M, 1>> dlambda =
+ dy.template block<M, 1>(States, 0);
+
+ double s = 1.0;
+
+ // Now, time to do line search.
+ //
+ // Start by keeping lambda positive. Make sure our step doesn't let
+ // lambda cross 0.
+ for (int i = 0; i < dlambda.rows(); ++i) {
+ if (lambda(i) + s * dlambda(i) < 0.0) {
+ // Ignore tiny steps in lambda. They cause issues when we get really
+ // close to having our constraints met but haven't converged the rest
+ // of the problem and start to run into rounding issues in the matrix
+ // solve portion.
+ if (dlambda(i) < 0.0 && dlambda(i) > -1e-12) {
+ VLOG(1) << " lambda(" << i << ") " << lambda(i) << " + " << s
+ << " * " << dlambda(i) << " -> s would be now "
+ << -lambda(i) / dlambda(i);
+ dlambda(i) = 0.0;
+ VLOG(1) << " dy -> " << std::setprecision(12) << std::fixed
+ << std::setfill(' ') << dy.transpose().format(kHeavyFormat);
+ continue;
+ }
+ VLOG(1) << " lambda(" << i << ") " << lambda(i) << " + " << s << " * "
+ << dlambda(i) << " -> s now " << -lambda(i) / dlambda(i);
+ s = -lambda(i) / dlambda(i);
+ }
+ }
+
+ VLOG(1) << " After lambda line search, s is " << s;
+
+ VLOG(3) << " Initial step " << iteration << " -> " << std::setprecision(12)
+ << std::fixed << std::setfill(' ')
+ << dy.transpose().format(kHeavyFormat);
+ VLOG(3) << " rt -> "
+ << std::setprecision(12) << std::fixed << std::setfill(' ')
+ << rt_orig.transpose().format(kHeavyFormat);
+
+ const double rt_orig_squared_norm = rt_orig.squaredNorm();
+
+ Eigen::Matrix<double, States + M + N, 1> next_y;
+ Eigen::Matrix<double, States + M + N, 1> rt;
+ Derivatives next_derivatives;
+ while (true) {
+ next_y = y + s * dy;
+ next_derivatives = ComputeDerivative(problem, next_y);
+ rt = Rt(next_derivatives, next_y, t_inverse);
+
+ const Eigen::Ref<const Eigen::VectorXd> next_x =
+ next_y.block(0, 0, next_derivatives.hessian.rows(), 1);
+ const Eigen::Ref<const Eigen::VectorXd> next_lambda =
+ next_y.block(next_x.rows(), 0, next_derivatives.f.rows(), 1);
+
+ const Eigen::Ref<const Eigen::VectorXd> next_v = next_y.block(
+ next_x.rows() + next_lambda.rows(), 0, next_derivatives.A.rows(), 1);
+
+ VLOG(1) << " next_rt(" << iteration << ") is " << rt.norm() << " -> "
+ << std::setprecision(12) << std::fixed << std::setfill(' ')
+ << rt.transpose().format(kHeavyFormat);
+
+ PrintDerivatives(next_derivatives, next_y, "next_", 3);
+
+ if (next_derivatives.f.maxCoeff() > 0.0) {
+ VLOG(1) << " f_next > 0.0 -> " << next_derivatives.f.maxCoeff()
+ << ", continuing line search.";
+ s *= kBeta;
+ } else if (next_derivatives.Axmb.squaredNorm() < 0.1 &&
+ rt.squaredNorm() >
+ std::pow(1.0 - kAlpha * s, 2.0) * rt_orig_squared_norm) {
+ VLOG(1) << " |Rt| > |Rt+1| " << rt.norm() << " > " << rt_orig.norm()
+ << ", drt -> " << std::setprecision(12) << std::fixed
+ << std::setfill(' ')
+ << (rt_orig - rt).transpose().format(kHeavyFormat);
+ s *= kBeta;
+ } else {
+ break;
+ }
+ }
+
+ VLOG(1) << " Terminated line search with s " << s << ", " << rt.norm()
+ << "(|Rt+1|) < " << rt_orig.norm() << "(|Rt|)";
+ y = next_y;
+
+ const Eigen::Ref<const Eigen::VectorXd> next_lambda =
+ y.template block<M, 1>(States, 0);
+
+ // See if we hit our convergence criteria.
+ const double r_primal_squared_norm =
+ rt.template block<N, 1>(States + M, 0).squaredNorm();
+ VLOG(1) << " rt_next(" << iteration << ") is " << rt.norm() << " -> "
+ << std::setprecision(12) << std::fixed << std::setfill(' ')
+ << rt.transpose().format(kHeavyFormat);
+ if (r_primal_squared_norm < kEpsilonF * kEpsilonF) {
+ const double r_dual_squared_norm =
+ rt.template block<States, 1>(0, 0).squaredNorm();
+ if (r_dual_squared_norm < kEpsilonF * kEpsilonF) {
+ const double next_nu =
+ -(next_derivatives.f.transpose() * next_lambda)(0, 0);
+ if (next_nu < kEpsilon) {
+ VLOG(1) << " r_primal(" << iteration << ") -> "
+ << std::sqrt(r_primal_squared_norm) << " < " << kEpsilonF
+ << ", r_dual(" << iteration << ") -> "
+ << std::sqrt(r_dual_squared_norm) << " < " << kEpsilonF
+ << ", nu(" << iteration << ") -> " << next_nu << " < "
+ << kEpsilon;
+ break;
+ } else {
+ VLOG(1) << " nu(" << iteration << ") -> " << next_nu << " < "
+ << kEpsilon << ", not done yet";
+ }
+
+ } else {
+ VLOG(1) << " r_dual(" << iteration << ") -> "
+ << std::sqrt(r_dual_squared_norm) << " < " << kEpsilonF
+ << ", not done yet";
+ }
+ } else {
+ VLOG(1) << " r_primal(" << iteration << ") -> "
+ << std::sqrt(r_primal_squared_norm) << " < " << kEpsilonF
+ << ", not done yet";
+ }
+ VLOG(1) << " step(" << iteration << ") " << std::setprecision(12)
+ << (s * dy).transpose().format(kHeavyFormat);
+ VLOG(1) << " y(" << iteration << ") is now " << std::setprecision(12)
+ << y.transpose().format(kHeavyFormat);
+
+ // Very import, use the last set of derivatives we picked for our new y
+ // for the next iteration. This avoids re-computing it.
+ derivatives = std::move(next_derivatives);
+
+ ++iteration;
+ if (iteration > 100) {
+ LOG(FATAL) << "Too many iterations";
+ }
+ }
+
+ return y.template block<States, 1>(0, 0);
+}
+
+template <size_t States, size_t M, size_t N>
+typename Solver<States, M, N>::Derivatives
+Solver<States, M, N>::ComputeDerivative(
+ const ConvexProblem<States, M, N> &problem,
+ const Eigen::Ref<const Eigen::Matrix<double, States + M + N, 1>> y) {
+ const Eigen::Ref<const Eigen::Matrix<double, States, 1>> x =
+ y.template block<States, 1>(0, 0);
+
+ Derivatives derivatives;
+ derivatives.gradient = problem.df0(x);
+ derivatives.hessian = problem.ddf0(x);
+ derivatives.f = problem.f(x);
+ derivatives.df = problem.df(x);
+ derivatives.A = problem.A();
+ derivatives.Axmb =
+ derivatives.A * y.template block<States, 1>(0, 0) - problem.b();
+ return derivatives;
+}
+
+template <size_t States, size_t M, size_t N>
+void Solver<States, M, N>::PrintDerivatives(
+ const Derivatives &derivatives,
+ const Eigen::Ref<const Eigen::Matrix<double, States + M + N, 1>> y,
+ std::string_view prefix, int verbosity) {
+ const Eigen::Ref<const Eigen::VectorXd> x =
+ y.block(0, 0, derivatives.hessian.rows(), 1);
+ const Eigen::Ref<const Eigen::VectorXd> lambda =
+ y.block(x.rows(), 0, derivatives.f.rows(), 1);
+
+ if (VLOG_IS_ON(verbosity)) {
+ Eigen::IOFormat heavy(Eigen::StreamPrecision, 0, ", ",
+ ",\n "
+ " ",
+ "[", "]", "[", "]");
+ heavy.rowSeparator =
+ heavy.rowSeparator +
+ std::string(absl::StrCat(getpid()).size() + prefix.size(), ' ');
+
+ const Eigen::Ref<const Eigen::VectorXd> v =
+ y.block(x.rows() + lambda.rows(), 0, derivatives.A.rows(), 1);
+ VLOG(verbosity) << " " << prefix << "x: " << x.transpose().format(heavy);
+ VLOG(verbosity) << " " << prefix
+ << "lambda: " << lambda.transpose().format(heavy);
+ VLOG(verbosity) << " " << prefix << "v: " << v.transpose().format(heavy);
+ VLOG(verbosity) << " " << prefix
+ << "hessian: " << derivatives.hessian.format(heavy);
+ VLOG(verbosity) << " " << prefix
+ << "gradient: " << derivatives.gradient.format(heavy);
+ VLOG(verbosity) << " " << prefix
+ << "A: " << derivatives.A.format(heavy);
+ VLOG(verbosity) << " " << prefix
+ << "Ax-b: " << derivatives.Axmb.format(heavy);
+ VLOG(verbosity) << " " << prefix
+ << "f: " << derivatives.f.format(heavy);
+ VLOG(verbosity) << " " << prefix
+ << "df: " << derivatives.df.format(heavy);
+ }
+}
+
+}; // namespace solvers
+}; // namespace frc971