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function [u_norm, coeffs, u] = smallestinconvexhull(M, x, U, tol)
% Computes a minimal norm convex combination of given tangent vectors in Manopt.
%
% function [u_norm, coeffs, u] = smallestinconvexhull(M, x, U)
% function [u_norm, coeffs, u] = smallestinconvexhull(M, x, U, tol)
%
% M is a manifold as returned by a Manopt factory.
% x is a point on this manifold.
% U is a cell containing N tangent vectors U{1} to U{N} at x.
% tol (default: 1e-8): tolerance for solving the quadratic program.
%
% This function computes u, a tangent vector at x contained in the convex
% hull spanned by the N vectors U{i}, with minimal norm (according to the
% Riemannian metric on M). This is obtained by solving a convex quadratic
% program involving the Gram matrix of the given tangent vectors.
% The quadratic program is solved using Matlab's built-in quadprog,
% which requires the optimization toolbox. If this toolbox is not
% available, consider replacing with CVX for example.
%
%
% u_norm is the norm of the smallest vector u.
% coeffs is a vector of length N with entries in [0, 1] summing to 1.
% u is the sought vector: u = coeffs(1)*U{1} + ... + coeffs(N)*U{N}.
%
% Nicolas Boumal, Feb. 19, 2013
% Modified April 6, 2016 to work with Manopt.
% This file is part of Manopt: www.manopt.org.
% Original author: Nicolas Boumal, June 28, 2016.
% Contributors:
% Change log:
%
% June 28, 2016 (NB):
% Adapted for Manopt from original code by same author (Feb. 19, 2013)
% Example code: pick a manifold, a point, and a collection of tangent
% vectors at that point, then get the smallest vector in the convex hull
% of those:
%
% M = spherefactory(5);
% x = M.rand();
% N = 3;
% U = cell(N,1);
% for k = 1 : N, U{k} = M.randvec(x); end
% [u_norm, coeffs, u] = smallestinconvexhull(M, x, U)
% We simply need to solve the following quadratic program:
% minimize ||u||^2 such that u = sum_i s_i U_i, 0 <= s_i <= 1
% and sum_i s_i = 1
%
% This is equivalent to solving:
% min s'*G*s s.t. 0 <= s <= 1, s'*ones = 1, with G(i, j) = <U_i, U_j> (Gram matrix)
% Then our solution is s_1 U_1 + ... + s_N U_N.
% Compute the Gram matrix of the given tangent vectors
N = numel(U);
G = grammatrix(M, x, U);
% Solve the quadratic program.
% If the optimization toolbox is not available, consider replacing with
% CVX.
if ~exist('tol', 'var') || isempty(tol)
tol = 1e-8;
end
opts = optimset('Display', 'off', 'TolFun', tol);
[s_opt, cost_opt] ...
= quadprog(G, zeros(N, 1), ... % objective (squared norm)
[], [], ... % inequalities (none)
ones(1, N), 1, ... % equality (sum to 1)
zeros(N, 1), ... % lower bounds (s_i >= 0)
ones(N, 1), ... % upper bounds (s_i <= 1)
[], ... % we do not specify an initial guess
opts);
% Norm of the smallest tangent vector in the convex hull:
u_norm = real(sqrt(2*cost_opt));
% Keep track of optimal coefficients
coeffs = s_opt;
% If required, construct the vector explicitly.
if nargout >= 3
u = lincomb(M, x, U, coeffs);
end
end