Implement rounding method using the volumetric barrier (#313)

* generalize rounding loop

* support sparse cholesky operator

* complete sparse support in max_inscribed_ball

* complete sparse support in preprocesing

* add sparse tests

* change main rounding function name

* improve explaining comments

* resolve PR comments

* changing the dates in copyrights

* use if constexpr instead of SNIFAE

* update the examples to cpp17

* update to cpp17 order polytope example

* fix templating in mat_computational_operators

* fix templating errors and change header file to mat_computational_operators

* first implementation of the volumetric barrier ellipsoid

* add criterion for step_iter

* restructure code that computes barriers' centers

* remove unused comments

* resolve PR comments

* remove NT typename from max_step()

---------

Co-authored-by: Apostolos Chalkis <apostolos.chalkis@quantagonia.com>

include/preprocess/barrier_center_ellipsoid.hpp

@@ -0,0 +1,115 @@
+// VolEsti (volume computation and sampling library)
+
+// Copyright (c) 2024 Vissarion Fisikopoulos
+// Copyright (c) 2024 Apostolos Chalkis
+// Copyright (c) 2024 Elias Tsigaridas
+
+// Licensed under GNU LGPL.3, see LICENCE file
+
+
+#ifndef BARRIER_CENTER_ELLIPSOID_HPP
+#define BARRIER_CENTER_ELLIPSOID_HPP
+
+#include <tuple>
+
+#include "preprocess/max_inscribed_ball.hpp"
+#include "preprocess/feasible_point.hpp"
+#include "preprocess/rounding_util_functions.hpp"
+
+/*
+    This implementation computes the analytic or the volumetric center of a polytope given 
+    as a set of linear inequalities P = {x | Ax <= b}. The analytic center is the tminimizer
+    of the log barrier function, i.e., the optimal solution
+    of the following optimization problem (Convex Optimization, Boyd and Vandenberghe, Section 8.5.3),
+
+    \min -\sum \log(b_i - a_i^Tx), where a_i is the i-th row of A.
+    
+    The volumetric center is the minimizer of the volumetric barrier function, i.e., the optimal
+    solution of the following optimization problem,
+
+    \min logdet \nabla^2 f(x), where f(x) the log barrier function 
+    
+    The function solves the problems by using the Newton method.
+
+    Input: (i)   Matrix A, vector b such that the polytope P = {x | Ax<=b}
+           (ii)  The number of maximum iterations, max_iters
+           (iii) Tolerance parameter grad_err_tol to bound the L2-norm of the gradient
+           (iv)  Tolerance parameter rel_pos_err_tol to check the relative progress in each iteration
+
+    Output: (i)   The Hessian of the barrier function
+            (ii)  The analytic/volumetric center of the polytope
+            (iii) A boolean variable that declares convergence
+    
+    Note: Using MT as to deal with both dense and sparse matrices, MT_dense will be the type of result matrix
+*/
+template <typename MT_dense, int BarrierType, typename NT, typename MT, typename VT>
+std::tuple<MT_dense, VT, bool>  barrier_center_ellipsoid_linear_ineq(MT const& A, VT const& b, VT const& x0,
+                                                                     unsigned int const max_iters = 500,
+                                                                     NT const grad_err_tol = 1e-08,
+                                                                     NT const rel_pos_err_tol = 1e-12)
+{
+    // Initialization
+    VT x = x0;
+    VT Ax = A * x;
+    const int n = A.cols(), m = A.rows();
+    MT H(n, n), A_trans = A.transpose();
+    VT grad(n), d(n), Ad(m), b_Ax(m), step_d(n), x_prev;
+    NT grad_err, rel_pos_err, rel_pos_err_temp, step, obj_val, obj_val_prev;
+    unsigned int iter = 0;
+    bool converged = false;
+    const NT tol_bnd = NT(0.01), tol_obj = NT(1e-06);
+
+    auto [step_iter, max_step_multiplier] = init_step<BarrierType, NT>();
+    auto llt = initialize_chol<NT>(A_trans, A);
+    get_barrier_hessian_grad<MT_dense, BarrierType>(A, A_trans, b, x, Ax, llt,
+                                                    H, grad, b_Ax, obj_val);
+    do {
+        iter++;
+        // Compute the direction
+        d.noalias() = - solve_vec<NT>(llt, H, grad);
+        Ad.noalias() = A * d;
+        // Compute the step length
+        step = std::min(max_step_multiplier * get_max_step(Ad, b_Ax), step_iter);
+        step_d.noalias() = step*d;
+        x_prev = x;
+        x += step_d;
+        Ax.noalias() += step*Ad;
+
+        // Compute the max_i\{ |step*d_i| ./ |x_i| \} 
+        rel_pos_err = std::numeric_limits<NT>::lowest();
+        for (int i = 0; i < n; i++)
+        {
+            rel_pos_err_temp = std::abs(step_d.coeff(i) / x_prev.coeff(i));
+            if (rel_pos_err_temp > rel_pos_err)
+            {
+                rel_pos_err = rel_pos_err_temp;
+            }
+        }
+
+        obj_val_prev = obj_val;
+        get_barrier_hessian_grad<MT_dense, BarrierType>(A, A_trans, b, x, Ax, llt,
+                                                        H, grad, b_Ax, obj_val);
+        grad_err = grad.norm();
+
+        if (iter >= max_iters || grad_err <= grad_err_tol || rel_pos_err <= rel_pos_err_tol)
+        {
+            converged = true;
+            break;
+        }
+        get_step_next_iteration<BarrierType>(obj_val_prev, obj_val, tol_obj, step_iter);
+    } while (true);
+
+    return std::make_tuple(MT_dense(H), x, converged);
+}
+
+template <typename MT_dense, int BarrierType, typename NT, typename MT, typename VT>
+std::tuple<MT_dense, VT, bool>  barrier_center_ellipsoid_linear_ineq(MT const& A, VT const& b,
+                                                                     unsigned int const max_iters = 500,
+                                                                     NT const grad_err_tol = 1e-08,
+                                                                     NT const rel_pos_err_tol = 1e-12) 
+{
+    VT x0 = compute_feasible_point(A, b);
+    return barrier_center_ellipsoid_linear_ineq<MT_dense, BarrierType>(A, b, x0, max_iters, grad_err_tol, rel_pos_err_tol);
+}
+
+#endif // BARRIER_CENTER_ELLIPSOID_HPP

include/preprocess/inscribed_ellipsoid_rounding.hpp

@@ -12,14 +12,9 @@
 #define INSCRIBED_ELLIPSOID_ROUNDING_HPP
 
 #include "preprocess/max_inscribed_ellipsoid.hpp"
-#include "preprocess/analytic_center_linear_ineq.h"
+#include "preprocess/barrier_center_ellipsoid.hpp"
 #include "preprocess/feasible_point.hpp"
 
-enum EllipsoidType
-{
-  MAX_ELLIPSOID = 1,
-  LOG_BARRIER = 2
-};
 
 template<typename MT, int ellipsoid_type, typename Custom_MT, typename VT, typename NT>
 inline static std::tuple<MT, VT, NT>
@@ -30,9 +25,10 @@ compute_inscribed_ellipsoid(Custom_MT A, VT b, VT const& x0,
     if constexpr (ellipsoid_type == EllipsoidType::MAX_ELLIPSOID)
     {
         return max_inscribed_ellipsoid<MT>(A, b, x0, maxiter, tol, reg);
-    } else if constexpr (ellipsoid_type == EllipsoidType::LOG_BARRIER)
+    } else if constexpr (ellipsoid_type == EllipsoidType::LOG_BARRIER ||
+                         ellipsoid_type == EllipsoidType::VOLUMETRIC_BARRIER)
     {
-        return analytic_center_linear_ineq<MT, Custom_MT, VT, NT>(A, b, x0);
+        return barrier_center_ellipsoid_linear_ineq<MT, ellipsoid_type, NT>(A, b, x0);
     } else
     {
         std::runtime_error("Unknown rounding method.");

include/preprocess/max_inscribed_ball.hpp

@@ -11,7 +11,7 @@
 #ifndef MAX_INSCRIBED_BALL_HPP
 #define MAX_INSCRIBED_BALL_HPP
 
-#include "preprocess/mat_computational_operators.h"
+#include "preprocess/rounding_util_functions.hpp"
 
 /*
     This implmentation computes the largest inscribed ball in a given convex polytope P.

include/preprocess/max_inscribed_ellipsoid.hpp

@@ -15,7 +15,7 @@
 
 #include <utility>
 #include <Eigen/Eigen>
-#include "preprocess/mat_computational_operators.h"
+#include "preprocess/rounding_util_functions.hpp"
 
 
 /*