variable.hpp

// Copyright (c) Sleipnir contributors
 
#pragma once
 
#include <algorithm>
#include <concepts>
#include <cstddef>
#include <initializer_list>
#include <source_location>
#include <type_traits>
#include <utility>
#include <vector>
 
#include <Eigen/Core>
#include <gch/small_vector.hpp>
 
#include "sleipnir/autodiff/expression.hpp"
#include "sleipnir/autodiff/expression_graph.hpp"
#include "sleipnir/autodiff/expression_type.hpp"
#include "sleipnir/autodiff/sleipnir_base.hpp"
#include "sleipnir/util/assert.hpp"
#include "sleipnir/util/concepts.hpp"
 
#ifndef SLEIPNIR_DISABLE_DIAGNOSTICS
#include "sleipnir/util/print.hpp"
#endif
 
namespace slp {
 
// Forward declarations for friend declarations in Variable
 
template <typename Scalar>
class VariableMatrix;
 
namespace detail {
 
template <typename Scalar>
VariableMatrix<Scalar> gradient_tree(
    const detail::ExpressionGraph<Scalar>& top_list,
    const VariableMatrix<Scalar>& wrt);
 
}  // namespace detail
 
template <typename Scalar, int UpLo = Eigen::Lower | Eigen::Upper>
  requires(UpLo == Eigen::Lower) || (UpLo == (Eigen::Lower | Eigen::Upper))
class Hessian;
 
template <typename Scalar>
class Jacobian;
 
template <typename Scalar_>
class Variable : public SleipnirBase {
 public:
  using Scalar = Scalar_;
 
  Variable() = default;
 
  explicit constexpr Variable(std::nullptr_t) : expr{nullptr} {}
 
  // NOLINTNEXTLINE (google-explicit-constructor)
  Variable(Scalar value)
    requires(!MatrixLike<Scalar>)
      : expr{detail::make_expression_ptr<detail::ConstantExpression<Scalar>>(
            value)} {}
 
  // NOLINTNEXTLINE (google-explicit-constructor)
  Variable(SleipnirMatrixLike<Scalar> auto value) : expr{value[0, 0].expr} {
    slp_assert(value.rows() == 1 && value.cols() == 1);
  }
 
  // NOLINTNEXTLINE (google-explicit-constructor)
  Variable(std::floating_point auto value)
      : expr{detail::make_expression_ptr<detail::ConstantExpression<Scalar>>(
            Scalar(value))} {}
 
  // NOLINTNEXTLINE (google-explicit-constructor)
  Variable(std::integral auto value)
      : expr{detail::make_expression_ptr<detail::ConstantExpression<Scalar>>(
            Scalar(value))} {}
 
  explicit Variable(const detail::ExpressionPtr<Scalar>& expr) : expr{expr} {}
 
  explicit constexpr Variable(detail::ExpressionPtr<Scalar>&& expr)
      : expr{std::move(expr)} {}
 
  Variable<Scalar>& operator=(ScalarLike auto value) {
    expr =
        detail::make_expression_ptr<detail::ConstantExpression<Scalar>>(value);
    m_graph_initialized = false;
 
    return *this;
  }
 
  void set_value(Scalar value) {
#ifndef SLEIPNIR_DISABLE_DIAGNOSTICS
    if (!expr->is_leaf) {
      auto location = std::source_location::current();
      slp::println(
          stderr,
          "WARNING: {}:{}: {}: Modified the value of a dependent variable",
          location.file_name(), location.line(), location.function_name());
    }
#endif
    expr->val = Scalar(value);
  }
 
  Scalar value() {
    if (!m_graph_initialized) {
      m_graph = detail::topological_sort(expr);
      m_graph_initialized = true;
    }
    detail::update_values(m_graph);
 
    return Scalar(expr->val);
  }
 
  ExpressionType type() const { return expr->type(); }
 
  template <ScalarLike LHS, SleipnirScalarLike<Scalar> RHS>
  friend Variable<Scalar> operator*(const LHS& lhs, const RHS& rhs) {
    return Variable{Variable<Scalar>{lhs}.expr * rhs.expr};
  }
 
  template <SleipnirScalarLike<Scalar> LHS, ScalarLike RHS>
  friend Variable<Scalar> operator*(const LHS& lhs, const RHS& rhs) {
    return Variable{lhs.expr * Variable<Scalar>{rhs}.expr};
  }
 
  friend Variable<Scalar> operator*(const Variable<Scalar>& lhs,
                                    const Variable<Scalar>& rhs) {
    return Variable{lhs.expr * rhs.expr};
  }
 
  Variable<Scalar>& operator*=(const Variable<Scalar>& rhs) {
    *this = *this * rhs;
    return *this;
  }
 
  friend Variable<Scalar> operator/(const Variable<Scalar>& lhs,
                                    const Variable<Scalar>& rhs) {
    return Variable{lhs.expr / rhs.expr};
  }
 
  Variable<Scalar>& operator/=(const Variable<Scalar>& rhs) {
    *this = *this / rhs;
    return *this;
  }
 
  friend Variable<Scalar> operator+(const Variable<Scalar>& lhs,
                                    const Variable<Scalar>& rhs) {
    return Variable{lhs.expr + rhs.expr};
  }
 
  Variable<Scalar>& operator+=(const Variable<Scalar>& rhs) {
    *this = *this + rhs;
    return *this;
  }
 
  friend Variable<Scalar> operator-(const Variable<Scalar>& lhs,
                                    const Variable<Scalar>& rhs) {
    return Variable{lhs.expr - rhs.expr};
  }
 
  Variable<Scalar>& operator-=(const Variable<Scalar>& rhs) {
    *this = *this - rhs;
    return *this;
  }
 
  friend Variable<Scalar> operator-(const Variable<Scalar>& lhs) {
    return Variable{-lhs.expr};
  }
 
  friend Variable<Scalar> operator+(const Variable<Scalar>& lhs) {
    return Variable{+lhs.expr};
  }
 
 private:
  detail::ExpressionPtr<Scalar> expr =
      detail::make_expression_ptr<detail::DecisionVariableExpression<Scalar>>();
 
  detail::ExpressionGraph<Scalar> m_graph;
 
  bool m_graph_initialized = false;
 
  template <typename Scalar>
  friend Variable<Scalar> abs(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> acos(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> asin(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> atan(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> atan2(const ScalarLike auto& y,
                                const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> atan2(const Variable<Scalar>& y,
                                const ScalarLike auto& x);
  template <typename Scalar>
  friend Variable<Scalar> atan2(const Variable<Scalar>& y,
                                const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> cbrt(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> cos(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> cosh(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> erf(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> exp(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> hypot(const ScalarLike auto& x,
                                const Variable<Scalar>& y);
  template <typename Scalar>
  friend Variable<Scalar> hypot(const Variable<Scalar>& x,
                                const ScalarLike auto& y);
  template <typename Scalar>
  friend Variable<Scalar> hypot(const Variable<Scalar>& x,
                                const Variable<Scalar>& y);
  template <typename Scalar>
  friend Variable<Scalar> if_else(
      std::type_identity_t<bool (*)(Scalar a, Scalar b)> cond,
      const Variable<Scalar>& a, const Variable<Scalar>& b,
      const Variable<Scalar>& t, const Variable<Scalar>& f);
  template <typename Scalar>
  friend Variable<Scalar> log(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> log10(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> max(const ScalarLike auto& a,
                              const Variable<Scalar>& b);
  template <typename Scalar>
  friend Variable<Scalar> max(const Variable<Scalar>& a,
                              const ScalarLike auto& b);
  template <typename Scalar>
  friend Variable<Scalar> max(const Variable<Scalar>& a,
                              const Variable<Scalar>& b);
  template <typename Scalar>
  friend Variable<Scalar> min(const ScalarLike auto& a,
                              const Variable<Scalar>& b);
  template <typename Scalar>
  friend Variable<Scalar> min(const Variable<Scalar>& a,
                              const ScalarLike auto& b);
  template <typename Scalar>
  friend Variable<Scalar> min(const Variable<Scalar>& a,
                              const Variable<Scalar>& b);
  template <typename Scalar>
  friend Variable<Scalar> pow(const ScalarLike auto& base,
                              const Variable<Scalar>& power);
  template <typename Scalar>
  friend Variable<Scalar> pow(const Variable<Scalar>& base,
                              const ScalarLike auto& power);
  template <typename Scalar>
  friend Variable<Scalar> pow(const Variable<Scalar>& base,
                              const Variable<Scalar>& power);
  template <typename Scalar>
  friend Variable<Scalar> sign(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> sin(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> sinh(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> sqrt(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> tan(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> tanh(const Variable<Scalar>& x);
  template <typename Scalar>
  friend Variable<Scalar> hypot(const Variable<Scalar>& x,
                                const Variable<Scalar>& y,
                                const Variable<Scalar>& z);
 
  template <typename Scalar>
  friend VariableMatrix<Scalar> detail::gradient_tree(
      const detail::ExpressionGraph<Scalar>& top_list,
      const VariableMatrix<Scalar>& wrt);
  template <typename Scalar, int UpLo>
    requires(UpLo == Eigen::Lower) || (UpLo == (Eigen::Lower | Eigen::Upper))
  friend class Hessian;
  template <typename Scalar>
  friend class Jacobian;
};
 
template <template <typename> typename T, typename Scalar>
  requires SleipnirMatrixLike<T<Scalar>, Scalar>
Variable(T<Scalar>) -> Variable<Scalar>;
 
template <std::floating_point T>
Variable(T) -> Variable<T>;
 
template <std::integral T>
Variable(T) -> Variable<T>;
 
template <typename Scalar>
Variable<Scalar> abs(const Variable<Scalar>& x) {
  return Variable{detail::abs(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> acos(const Variable<Scalar>& x) {
  return Variable{detail::acos(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> asin(const Variable<Scalar>& x) {
  return Variable{detail::asin(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> atan(const Variable<Scalar>& x) {
  return Variable{detail::atan(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> atan2(const ScalarLike auto& y, const Variable<Scalar>& x) {
  return Variable{detail::atan2(Variable<Scalar>(y).expr, x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> atan2(const Variable<Scalar>& y, const ScalarLike auto& x) {
  return Variable{detail::atan2(y.expr, Variable<Scalar>(x).expr)};
}
 
template <typename Scalar>
Variable<Scalar> atan2(const Variable<Scalar>& y, const Variable<Scalar>& x) {
  return Variable{detail::atan2(y.expr, x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> cbrt(const Variable<Scalar>& x) {
  return Variable{detail::cbrt(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> cos(const Variable<Scalar>& x) {
  return Variable{detail::cos(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> cosh(const Variable<Scalar>& x) {
  return Variable{detail::cosh(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> erf(const Variable<Scalar>& x) {
  return Variable{detail::erf(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> exp(const Variable<Scalar>& x) {
  return Variable{detail::exp(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> hypot(const ScalarLike auto& x, const Variable<Scalar>& y) {
  return Variable{detail::hypot(Variable<Scalar>(x).expr, y.expr)};
}
 
template <typename Scalar>
Variable<Scalar> hypot(const Variable<Scalar>& x, const ScalarLike auto& y) {
  return Variable{detail::hypot(x.expr, Variable<Scalar>(y).expr)};
}
 
template <typename Scalar>
Variable<Scalar> hypot(const Variable<Scalar>& x, const Variable<Scalar>& y) {
  return Variable{detail::hypot(x.expr, y.expr)};
}
 
template <typename Scalar>
Variable<Scalar> if_else(
    std::type_identity_t<bool (*)(Scalar a, Scalar b)> cond,
    const Variable<Scalar>& a, const Variable<Scalar>& b,
    const Variable<Scalar>& t, const Variable<Scalar>& f) {
  return Variable{detail::if_else(cond, a.expr, b.expr, t.expr, f.expr)};
}
 
template <typename Scalar>
Variable<Scalar> log(const Variable<Scalar>& x) {
  return Variable{detail::log(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> log10(const Variable<Scalar>& x) {
  return Variable{detail::log10(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> max(const ScalarLike auto& a, const Variable<Scalar>& b) {
  return Variable{detail::max(Variable<Scalar>(a).expr, b.expr)};
}
 
template <typename Scalar>
Variable<Scalar> max(const Variable<Scalar>& a, const ScalarLike auto& b) {
  return Variable{detail::max(a.expr, Variable<Scalar>(b).expr)};
}
 
template <typename Scalar>
Variable<Scalar> max(const Variable<Scalar>& a, const Variable<Scalar>& b) {
  return Variable{detail::max(a.expr, b.expr)};
}
 
template <typename Scalar>
Variable<Scalar> min(const ScalarLike auto& a, const Variable<Scalar>& b) {
  return Variable{detail::min(Variable<Scalar>(a).expr, b.expr)};
}
 
template <typename Scalar>
Variable<Scalar> min(const Variable<Scalar>& a, const ScalarLike auto& b) {
  return Variable{detail::min(a.expr, Variable<Scalar>(b).expr)};
}
 
template <typename Scalar>
Variable<Scalar> min(const Variable<Scalar>& a, const Variable<Scalar>& b) {
  return Variable{detail::min(a.expr, b.expr)};
}
 
template <typename Scalar>
Variable<Scalar> pow(const ScalarLike auto& base,
                     const Variable<Scalar>& power) {
  return Variable{detail::pow(Variable<Scalar>(base).expr, power.expr)};
}
 
template <typename Scalar>
Variable<Scalar> pow(const Variable<Scalar>& base,
                     const ScalarLike auto& power) {
  return Variable{detail::pow(base.expr, Variable<Scalar>(power).expr)};
}
 
template <typename Scalar>
Variable<Scalar> pow(const Variable<Scalar>& base,
                     const Variable<Scalar>& power) {
  return Variable{detail::pow(base.expr, power.expr)};
}
 
template <typename Scalar>
Variable<Scalar> sign(const Variable<Scalar>& x) {
  return Variable{detail::sign(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> sin(const Variable<Scalar>& x) {
  return Variable{detail::sin(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> sinh(const Variable<Scalar>& x) {
  return Variable{detail::sinh(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> sqrt(const Variable<Scalar>& x) {
  return Variable{detail::sqrt(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> tan(const Variable<Scalar>& x) {
  return Variable{detail::tan(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> tanh(const Variable<Scalar>& x) {
  return Variable{detail::tanh(x.expr)};
}
 
template <typename Scalar>
Variable<Scalar> hypot(const Variable<Scalar>& x, const Variable<Scalar>& y,
                       const Variable<Scalar>& z) {
  return Variable{detail::hypot(x.expr, y.expr, z.expr)};
}
 
// The standard form for equality constraints is c(x) = 0, and the standard form
// for inequality constraints is c(x) ≥ 0. make_constraints() takes constraints
// of the form lhs = rhs or lhs ≥ rhs and converts them to lhs - rhs = 0 or
// lhs - rhs ≥ 0.
 
template <typename Scalar, ScalarLike LHS, ScalarLike RHS>
  requires SleipnirScalarLike<LHS, Scalar> || SleipnirScalarLike<RHS, Scalar>
auto make_constraints(LHS&& lhs, RHS&& rhs) {
  gch::small_vector<Variable<Scalar>> constraints;
  constraints.emplace_back(lhs - rhs);
 
  return constraints;
}
 
template <typename Scalar, ScalarLike LHS, MatrixLike RHS>
  requires SleipnirScalarLike<LHS, Scalar> || SleipnirMatrixLike<RHS, Scalar>
auto make_constraints(LHS&& lhs, RHS&& rhs) {
  gch::small_vector<Variable<Scalar>> constraints;
  constraints.reserve(rhs.rows() * rhs.cols());
 
  for (int row = 0; row < rhs.rows(); ++row) {
    for (int col = 0; col < rhs.cols(); ++col) {
      // Make right-hand side zero
      constraints.emplace_back(lhs - rhs[row, col]);
    }
  }
 
  return constraints;
}
 
template <typename Scalar, MatrixLike LHS, ScalarLike RHS>
  requires SleipnirMatrixLike<LHS, Scalar> || SleipnirScalarLike<RHS, Scalar>
auto make_constraints(LHS&& lhs, RHS&& rhs) {
  gch::small_vector<Variable<Scalar>> constraints;
  constraints.reserve(lhs.rows() * lhs.cols());
 
  for (int row = 0; row < lhs.rows(); ++row) {
    for (int col = 0; col < lhs.cols(); ++col) {
      // Make right-hand side zero
      constraints.emplace_back(lhs[row, col] - rhs);
    }
  }
 
  return constraints;
}
 
template <typename Scalar, MatrixLike LHS, MatrixLike RHS>
  requires SleipnirMatrixLike<LHS, Scalar> || SleipnirMatrixLike<RHS, Scalar>
auto make_constraints(LHS&& lhs, RHS&& rhs) {
  slp_assert(lhs.rows() == rhs.rows() && lhs.cols() == rhs.cols());
 
  gch::small_vector<Variable<Scalar>> constraints;
  constraints.reserve(lhs.rows() * lhs.cols());
 
  for (int row = 0; row < lhs.rows(); ++row) {
    for (int col = 0; col < lhs.cols(); ++col) {
      // Make right-hand side zero
      constraints.emplace_back(lhs[row, col] - rhs[row, col]);
    }
  }
 
  return constraints;
}
 
template <typename Scalar>
struct EqualityConstraints {
  gch::small_vector<Variable<Scalar>> constraints;
 
  EqualityConstraints(
      std::initializer_list<EqualityConstraints> equality_constraints) {
    for (const auto& elem : equality_constraints) {
      constraints.insert(constraints.end(), elem.constraints.begin(),
                         elem.constraints.end());
    }
  }
 
  explicit EqualityConstraints(
      const std::vector<EqualityConstraints>& equality_constraints) {
    for (const auto& elem : equality_constraints) {
      constraints.insert(constraints.end(), elem.constraints.begin(),
                         elem.constraints.end());
    }
  }
 
  template <typename LHS, typename RHS>
    requires(ScalarLike<LHS> || MatrixLike<LHS>) &&
            (ScalarLike<RHS> || MatrixLike<RHS>) &&
            (SleipnirType<LHS> || SleipnirType<RHS>)
  EqualityConstraints(LHS&& lhs, RHS&& rhs)
      : constraints{make_constraints<Scalar>(lhs, rhs)} {}
 
  // NOLINTNEXTLINE (google-explicit-constructor)
  operator bool() {
    return std::ranges::all_of(constraints, [](auto& constraint) {
      return constraint.value() == Scalar(0);
    });
  }
};
 
template <typename Scalar>
struct InequalityConstraints {
  gch::small_vector<Variable<Scalar>> constraints;
 
  InequalityConstraints(
      std::initializer_list<InequalityConstraints> inequality_constraints) {
    for (const auto& elem : inequality_constraints) {
      constraints.insert(constraints.end(), elem.constraints.begin(),
                         elem.constraints.end());
    }
  }
 
  explicit InequalityConstraints(
      const std::vector<InequalityConstraints>& inequality_constraints) {
    for (const auto& elem : inequality_constraints) {
      constraints.insert(constraints.end(), elem.constraints.begin(),
                         elem.constraints.end());
    }
  }
 
  template <typename LHS, typename RHS>
    requires(ScalarLike<LHS> || MatrixLike<LHS>) &&
            (ScalarLike<RHS> || MatrixLike<RHS>) &&
            (SleipnirType<LHS> || SleipnirType<RHS>)
  InequalityConstraints(LHS&& lhs, RHS&& rhs)
      : constraints{make_constraints<Scalar>(lhs, rhs)} {}
 
  // NOLINTNEXTLINE (google-explicit-constructor)
  operator bool() {
    return std::ranges::all_of(constraints, [](auto& constraint) {
      return constraint.value() >= Scalar(0);
    });
  }
};
 
template <typename LHS, typename RHS>
  requires(ScalarLike<LHS> || MatrixLike<LHS>) && SleipnirType<LHS> &&
          (ScalarLike<RHS> || MatrixLike<RHS>) && (!SleipnirType<RHS>)
auto operator==(LHS&& lhs, RHS&& rhs) {
  return EqualityConstraints<typename std::decay_t<LHS>::Scalar>{lhs, rhs};
}
 
template <typename LHS, typename RHS>
  requires(ScalarLike<LHS> || MatrixLike<LHS>) && (!SleipnirType<LHS>) &&
          (ScalarLike<RHS> || MatrixLike<RHS>) && SleipnirType<RHS>
auto operator==(LHS&& lhs, RHS&& rhs) {
  return EqualityConstraints<typename std::decay_t<RHS>::Scalar>{lhs, rhs};
}
 
template <typename LHS, typename RHS>
  requires(ScalarLike<LHS> || MatrixLike<LHS>) && SleipnirType<LHS> &&
          (ScalarLike<RHS> || MatrixLike<RHS>) && SleipnirType<RHS>
auto operator==(LHS&& lhs, RHS&& rhs) {
  return EqualityConstraints<typename std::decay_t<LHS>::Scalar>{lhs, rhs};
}
 
template <typename LHS, typename RHS>
  requires(ScalarLike<LHS> || MatrixLike<LHS>) &&
          (ScalarLike<RHS> || MatrixLike<RHS>) &&
          (SleipnirType<LHS> || SleipnirType<RHS>)
auto operator<(LHS&& lhs, RHS&& rhs) {
  return rhs >= lhs;
}
 
template <typename LHS, typename RHS>
  requires(ScalarLike<LHS> || MatrixLike<LHS>) &&
          (ScalarLike<RHS> || MatrixLike<RHS>) &&
          (SleipnirType<LHS> || SleipnirType<RHS>)
auto operator<=(LHS&& lhs, RHS&& rhs) {
  return rhs >= lhs;
}
 
template <typename LHS, typename RHS>
  requires(ScalarLike<LHS> || MatrixLike<LHS>) &&
          (ScalarLike<RHS> || MatrixLike<RHS>) &&
          (SleipnirType<LHS> || SleipnirType<RHS>)
auto operator>(LHS&& lhs, RHS&& rhs) {
  return lhs >= rhs;
}
 
template <typename LHS, typename RHS>
  requires(ScalarLike<LHS> || MatrixLike<LHS>) && SleipnirType<LHS> &&
          (ScalarLike<RHS> || MatrixLike<RHS>) && (!SleipnirType<RHS>)
auto operator>=(LHS&& lhs, RHS&& rhs) {
  return InequalityConstraints<typename std::decay_t<LHS>::Scalar>{lhs, rhs};
}
 
template <typename LHS, typename RHS>
  requires(ScalarLike<LHS> || MatrixLike<LHS>) && (!SleipnirType<LHS>) &&
          (ScalarLike<RHS> || MatrixLike<RHS>) && SleipnirType<RHS>
auto operator>=(LHS&& lhs, RHS&& rhs) {
  return InequalityConstraints<typename std::decay_t<RHS>::Scalar>{lhs, rhs};
}
 
template <typename LHS, typename RHS>
  requires(ScalarLike<LHS> || MatrixLike<LHS>) && SleipnirType<LHS> &&
          (ScalarLike<RHS> || MatrixLike<RHS>) && SleipnirType<RHS>
auto operator>=(LHS&& lhs, RHS&& rhs) {
  return InequalityConstraints<typename std::decay_t<LHS>::Scalar>{lhs, rhs};
}
 
template <typename L, typename X, typename U>
  requires(ScalarLike<L> || MatrixLike<L>) && SleipnirType<X> &&
          (ScalarLike<U> || MatrixLike<U>)
auto bounds(L&& l, X&& x, U&& u) {
  return InequalityConstraints{l <= x, x <= u};
}
 
}  // namespace slp
 
namespace Eigen {
 
template <typename Scalar>
struct NumTraits<slp::Variable<Scalar>> : NumTraits<Scalar> {
  using Real = slp::Variable<Scalar>;
  using NonInteger = slp::Variable<Scalar>;
  using Nested = slp::Variable<Scalar>;
  using Literal = slp::Variable<Scalar>;
 
  static constexpr int IsComplex = 0;
  static constexpr int IsInteger = 0;
  static constexpr int IsSigned = 1;
  static constexpr int RequireInitialization = 1;
  static constexpr int ReadCost = 1;
  static constexpr int AddCost = 3;
  static constexpr int MulCost = 3;
};
 
}  // namespace Eigen