fix_constraint.cpp

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Using Constraints: Example and Test

Model

\[\B{p}( y_i | \theta , u ) \sim \B{N} ( u_i + \theta_i , 1 )\]
\[\B{p}( u_i | \theta ) \sim \B{N} ( 0 , 1 )\]
\[\B{p}( \theta ) \sim \B{U} ( - \infty , + \infty )\]

where \(\B{U} ( - \infty , + \infty )\) is the improper uniform prior on \([- \infty , + \infty ]\). It follows that the Laplace approximation is exact and

\[\B{p}( y_i | \theta ) \sim \B{N} ( \theta_i , 2 )\]

The corresponding objective for the fixed effects is equivalent to:

\[\frac{1}{2} \sum_{i=0}^{N-1} ( y_i - \theta_i )^2\]

For this problem we add the explicit constraint

\[\frac{1}{2} \sum_i \theta_i^2 = 1;\]

The corresponding Lagrangian is

\[L( \theta , \lambda ) = \frac{1}{2} \sum_{i=0}^{N-1} ( y_i - \theta_i )^2 + \lambda \left( \frac{1}{2} \sum_i \theta_i^2 - 1 \right)\]
# include <cppad/cppad.hpp>
# include <cppad/mixed/cppad_mixed.hpp>

namespace {
    using CppAD::log;
    using CppAD::AD;
    //
    using CppAD::mixed::d_sparse_rcv;
    using CppAD::mixed::a1_double;
    using CppAD::mixed::d_vector;
    using CppAD::mixed::a1_vector;
    //
    class mixed_derived : public cppad_mixed {
    private:
        const d_vector&       y_;
    public:
        // constructor
        mixed_derived(
            size_t                   n_fixed        ,
            size_t                   n_random       ,
            const d_vector&          y              ) :
            cppad_mixed(n_fixed, n_random) ,
            y_(y)
        {}
        // implementation of ran_likelihood
        a1_vector ran_likelihood(
            const a1_vector&         theta  ,
            const a1_vector&         u      ) override
        {
            assert( u.size() == y_.size() );
            assert( theta.size() == y_.size() );
            a1_vector vec(1);

            // initialize part of log-density that is always smooth
            vec[0] = 0.0;

            // pi
            // sqrt_2pi = CppAD::sqrt(8.0 * CppAD::atan(1.0) );

            for(size_t i = 0; i < y_.size(); i++)
            {   a1_double mu     = u[i] + theta[i];
                a1_double sigma  = 1.0;
                a1_double res    = (y_[i] - mu) / sigma;

                // p(y_i | u, theta)
                vec[0] += res*res / 2.0;
                // following term does not depend on fixed or random effects
                // vec[0] += log(sqrt_2pi * sigma);

                // p(u_i | theta)
                vec[0] += u[i] * u[i] / 2.0;
                // following term does not depend on fixed or random effects
                // vec[0] += log(sqrt_2pi);
            }
            return vec;
        }
        // ------------------------------------------------------------------
        // fix_constraint
        a1_vector template_fix_constraint(
            const a1_vector&         fixed_vec  )
        {
            a1_vector ret_val(1);
            //
            ret_val[0] = 0.0;
            for(size_t i = 0; i < fixed_vec.size(); i++)
                ret_val[0] += fixed_vec[i] * fixed_vec[i];
            ret_val[0] /= 2.0;
            //
            return ret_val;
        }
        // a1_vector version of fix_constraint
        a1_vector fix_constraint(const a1_vector& fixed_vec) override
        {   return template_fix_constraint( fixed_vec ); }
    };
}

bool fix_constraint_xam(void)
{
    bool   ok = true;
    double inf = std::numeric_limits<double>::infinity();
    double tol = 1e-8;

    size_t n_data   = 3;
    size_t n_fixed  = n_data;
    size_t n_random = n_data;
    d_vector
        fixed_lower(n_fixed), fixed_in(n_fixed), fixed_upper(n_fixed);
    for(size_t i = 0; i < n_fixed; i++)
    {   fixed_lower[i] = - inf;
        fixed_in[i]    = 0.1;
        fixed_upper[i] = inf;
    }
    //
    // explicit constraints (in addition to l1 terms)
    d_vector fix_constraint_lower(1), fix_constraint_upper(1);
    fix_constraint_lower[0] = 1.0;
    fix_constraint_upper[0] = 1.0;
    //
    d_vector data(n_data), random_in(n_random);
    for(size_t i = 0; i < n_data; i++)
    {   data[i]       = double(i + 1);
        random_in[i] = 0.0;
    }

    // object that is derived from cppad_mixed
    mixed_derived mixed_object(n_fixed, n_random, data);
    mixed_object.initialize(fixed_in, random_in);

    // optimize the fixed effects using full Newton method
    std::string fixed_ipopt_options =
        "Integer print_level               0\n"
        "String  sb                        yes\n"
        "String  derivative_test           adaptive\n"
        "String  derivative_test_print_all yes\n"
        "Numeric tol                       1e-8\n"
        "Integer max_iter                  15\n"
    ;
    // random_ipopt_options is non-empty, so using ipopt for random effects
    std::string random_ipopt_options =
        "Integer print_level 0\n"
        "String  sb          yes\n"
        "String  derivative_test second-order\n"
    ;
    // lower and upper limits for random effects
    d_vector random_lower(n_random), random_upper(n_random);
    for(size_t i = 0; i < n_random; i++)
    {   random_lower[i] = -inf;
        random_upper[i] = +inf;
    }
    d_vector fixed_scale = fixed_in;
    CppAD::mixed::fixed_solution solution = mixed_object.optimize_fixed(
        fixed_ipopt_options,
        random_ipopt_options,
        fixed_lower,
        fixed_upper,
        fix_constraint_lower,
        fix_constraint_upper,
        fixed_scale,
        fixed_in,
        random_lower,
        random_upper,
        random_in
    );
    d_vector fixed_out = solution.fixed_opt;
    //
    // check constraint
    double sum = 0.0;
    for(size_t i = 0; i < n_fixed; i++)
        sum += fixed_out[i] * fixed_out[i];
    ok &= fabs( sum / 2.0 - 1.0 ) <= tol;

    // compute lagranges multiplier by averaging
    sum = 0.0;
    for(size_t i = 0; i < n_fixed; i++)
        sum += (fixed_out[i] - data[i]) / fixed_out[i];
    double lambda = sum / double(n_fixed);

    // check partials of Lagragian w.r.t fixed effects
    for(size_t i = 0; i < n_fixed; i++)
    {   double err  = data[i] - fixed_out[i] + lambda * fixed_out[i];
        ok         &= fabs(err) < tol;
    }
    return ok;
}