Add a sparse convex solver

This is a port of the dense convex solver to a sparse one.  The syntax
is different enough it isn't worth pretending we can share code.

Change-Id: I16788db62ccc3105ed866cef0a8cefe850ac5dfb
Signed-off-by: Austin Schuh <austin.linux@gmail.com>
diff --git a/frc971/solvers/sparse_convex_test.cc b/frc971/solvers/sparse_convex_test.cc
new file mode 100644
index 0000000..e391aa4
--- /dev/null
+++ b/frc971/solvers/sparse_convex_test.cc
@@ -0,0 +1,106 @@
+#include "frc971/solvers/sparse_convex.h"
+
+#include "gtest/gtest.h"
+
+namespace frc971 {
+namespace solvers {
+namespace testing {
+
+const Eigen::IOFormat kHeavyFormat(Eigen::StreamPrecision, 0, ", ",
+                                   ",\n                        "
+                                   "                                     ",
+                                   "[", "]", "[", "]");
+
+class SimpleQP : public SparseConvexProblem {
+ public:
+  // QP of the for 0.5 * X^t Q_ X + p.T * X
+  SimpleQP(Eigen::Matrix<double, 2, 2> Q, Eigen::Matrix<double, 2, 1> p,
+           double x0_max, double x0_min, double x1_max, double x1_min)
+      : SparseConvexProblem(2, 4, 1), Q_(Q), p_(p) {
+    C_ << 1, 0, -1, 0, 0, 1, 0, -1;
+    c_ << x0_max, -x0_min, x1_max, -x1_min;
+  }
+
+  double f0(Eigen::Ref<const Eigen::VectorXd> X) const override {
+    return 0.5 * (X.transpose() * Q_ * X)(0, 0);
+  }
+
+  Eigen::SparseMatrix<double> df0(
+      Eigen::Ref<const Eigen::VectorXd> X) const override {
+    return (Q_ * X + p_).sparseView();
+  }
+
+  Eigen::SparseMatrix<double> ddf0(
+      Eigen::Ref<const Eigen::VectorXd> /*X*/) const override {
+    return Q_.sparseView();
+  }
+
+  // Returns the constraints f(X) < 0, and their derivitive.
+  Eigen::VectorXd f(
+      Eigen::Ref<const Eigen::VectorXd> X) const override {
+    return C_ * X - c_;
+  }
+  Eigen::SparseMatrix<double> df(
+      Eigen::Ref<const Eigen::VectorXd> /*X*/) const override {
+    return C_.sparseView();
+  }
+
+  // Returns the equality constraints of the form A x = b
+  Eigen::SparseMatrix<double> A() const override {
+    return Eigen::Matrix<double, 1, 2>(1, -1).sparseView();
+  }
+  Eigen::VectorXd b() const override {
+    return Eigen::Matrix<double, 1, 1>(0);
+  }
+
+ private:
+  Eigen::Matrix<double, 2, 2> Q_;
+  Eigen::Matrix<double, 2, 1> p_;
+
+  Eigen::Matrix<double, 4, 2> C_;
+  Eigen::Matrix<double, 4, 1> c_;
+};
+
+// Test a constrained quadratic problem where the constraints aren't active.
+TEST(SolverTest, SimpleQP) {
+  Eigen::Matrix<double, 2, 2> Q = Eigen::DiagonalMatrix<double, 2>(1.0, 1.0);
+  Eigen::Matrix<double, 2, 1> p(-4, -6);
+
+  SimpleQP qp(Q, p, 6, -1, 6, -1);
+  SparseSolver s;
+  Eigen::Vector2d result = s.Solve(qp, Eigen::Matrix<double, 2, 1>(0, 0));
+  LOG(INFO) << "Result is " << std::setprecision(12)
+            << result.transpose().format(kHeavyFormat);
+  EXPECT_NEAR((result - Eigen::Vector2d(5.0, 5.0)).norm(), 0.0, 1e-6);
+}
+
+// Test a constrained quadratic problem where the constraints are active.
+TEST(SolverTest, Constrained) {
+  Eigen::Matrix<double, 2, 2> Q = Eigen::DiagonalMatrix<double, 2>(1.0, 2.0);
+  Eigen::Matrix<double, 2, 1> p(-5, -10);
+
+  SimpleQP qp(Q, p, 4, -1, 5, -1);
+  SparseSolver s;
+  Eigen::Vector2d result = s.Solve(qp, Eigen::Matrix<double, 2, 1>(3, 4));
+  LOG(INFO) << "Result is " << std::setprecision(12)
+            << result.transpose().format(kHeavyFormat);
+  EXPECT_NEAR((result - Eigen::Vector2d(4.0, 4.0)).norm(), 0.0, 1e-6);
+}
+
+// Test a constrained quadratic problem where the constraints are active and the
+// initial value is the solution.
+TEST(SolverTest, ConstrainedFromSolution) {
+  Eigen::Matrix<double, 2, 2> Q = Eigen::DiagonalMatrix<double, 2>(1.0, 2.0);
+  Eigen::Matrix<double, 2, 1> p(-5, -10);
+
+  SimpleQP qp(Q, p, 4, -1, 5, -1);
+  SparseSolver s;
+  Eigen::Vector2d result = s.Solve(qp, Eigen::Matrix<double, 2, 1>(4, 4));
+  LOG(INFO) << "Result is " << std::setprecision(12)
+            << result.transpose().format(kHeavyFormat);
+  EXPECT_NEAR((result - Eigen::Vector2d(4.0, 4.0)).norm(), 0.0, 1e-6);
+}
+
+}  // namespace testing
+}  // namespace solvers
+}  // namespace frc971