Home > examples > elliptope_SDP.m

# elliptope_SDP

## PURPOSE

Solver for semidefinite programs (SDP's) with unit diagonal constraints.

## SYNOPSIS

function [Y, problem, S] = elliptope_SDP(A, p, Y0)

## DESCRIPTION

``` Solver for semidefinite programs (SDP's) with unit diagonal constraints.

function [Y, problem, S] = elliptope_SDP(A)
function [Y, problem, S] = elliptope_SDP(A, p)
function [Y, problem, S] = elliptope_SDP(A, p, Y0)

A is a real, symmetric matrix of size n.

This function uses a local optimization method in Manopt to solve the SDP

min_X  trace(A*X)  s.t.  diag(X) = 1 and X is positive semidefinite.

In practice, the symmetric matrix X of size n is parameterized
as X = Y*Y', where Y has size n x p. By default, p is taken large enough
(about sqrt(2n)) to ensure that there exists an optimal X whose rank is
smaller than p. This ensures that the SDP is equivalent to the new
problem in Y:

min_Y  trace(Y'*A*Y)  s.t.  diag(Y*Y') = 1.

The constraints on Y require each row of Y to have unit norm, which is
why Manopt is appropriate software to solve this problem. An optional
initial guess can be specified via the input Y0.

See the paper below for theory, specifically, for a proof that, for
almost all A, second-order critical points of the problem in Y are
globally optimal. In other words: there are no local traps in Y, despite
non-convexity.

Outputs:

Y: is the best point found (an nxp matrix with unit norm rows.)
To find X, form Y*Y' (or, more efficiently, study X through Y.)

problem: is the Manopt problem structure used to produce Y.

S: is a dual optimality certificate (a symmetric matrix of size n,
sparse if A is sparse). The optimality gap (in the cost
function) is at most n*min(eig(S)), for both Y and X = Y*Y'.
Hence, if min(eig(S)) is close to zero, Y is close to globally
optimal. This can be computed via eigs(S, 1, 'SR').

Paper: https://arxiv.org/abs/1606.04970

@inproceedings{boumal2016bmapproach,
author  = {Boumal, N. and Voroninski, V. and Bandeira, A.S.},
title   = {The non-convex {B}urer-{M}onteiro approach works on smooth semidefinite programs},
booktitle={Neural Information Processing Systems (NIPS 2016)},
year    = {2016}
}

## CROSS-REFERENCE INFORMATION

This function calls:
• obliquefactory Returns a manifold struct to optimize over matrices w/ unit-norm columns.
• trustregions Riemannian trust-regions solver for optimization on manifolds.
This function is called by:

## SOURCE CODE

```0001 function [Y, problem, S] = elliptope_SDP(A, p, Y0)
0002 % Solver for semidefinite programs (SDP's) with unit diagonal constraints.
0003 %
0004 % function [Y, problem, S] = elliptope_SDP(A)
0005 % function [Y, problem, S] = elliptope_SDP(A, p)
0006 % function [Y, problem, S] = elliptope_SDP(A, p, Y0)
0007 %
0008 % A is a real, symmetric matrix of size n.
0009 %
0010 % This function uses a local optimization method in Manopt to solve the SDP
0011 %
0012 %   min_X  trace(A*X)  s.t.  diag(X) = 1 and X is positive semidefinite.
0013 %
0014 % In practice, the symmetric matrix X of size n is parameterized
0015 % as X = Y*Y', where Y has size n x p. By default, p is taken large enough
0016 % (about sqrt(2n)) to ensure that there exists an optimal X whose rank is
0017 % smaller than p. This ensures that the SDP is equivalent to the new
0018 % problem in Y:
0019 %
0020 %   min_Y  trace(Y'*A*Y)  s.t.  diag(Y*Y') = 1.
0021 %
0022 % The constraints on Y require each row of Y to have unit norm, which is
0023 % why Manopt is appropriate software to solve this problem. An optional
0024 % initial guess can be specified via the input Y0.
0025 %
0026 % See the paper below for theory, specifically, for a proof that, for
0027 % almost all A, second-order critical points of the problem in Y are
0028 % globally optimal. In other words: there are no local traps in Y, despite
0029 % non-convexity.
0030 %
0031 % Outputs:
0032 %
0033 %       Y: is the best point found (an nxp matrix with unit norm rows.)
0034 %          To find X, form Y*Y' (or, more efficiently, study X through Y.)
0035 %
0036 %       problem: is the Manopt problem structure used to produce Y.
0037 %
0038 %       S: is a dual optimality certificate (a symmetric matrix of size n,
0039 %          sparse if A is sparse). The optimality gap (in the cost
0040 %          function) is at most n*min(eig(S)), for both Y and X = Y*Y'.
0041 %          Hence, if min(eig(S)) is close to zero, Y is close to globally
0042 %          optimal. This can be computed via eigs(S, 1, 'SR').
0043 %
0044 % Paper: https://arxiv.org/abs/1606.04970
0045 %
0046 % @inproceedings{boumal2016bmapproach,
0047 %   author  = {Boumal, N. and Voroninski, V. and Bandeira, A.S.},
0048 %   title   = {The non-convex {B}urer-{M}onteiro approach works on smooth semidefinite programs},
0049 %   booktitle={Neural Information Processing Systems (NIPS 2016)},
0050 %   year    = {2016}
0051 % }
0052 %
0054
0055 % This file is part of Manopt: www.manopt.org.
0056 % Original author: Nicolas Boumal, June 28, 2016
0057 % Contributors:
0058 % Change log:
0059
0060
0061     % If no inputs are provided, since this is an example file, generate
0062     % a random Erdos-Renyi graph. This is for illustration purposes only.
0063     if ~exist('A', 'var') || isempty(A)
0064         n = 100;
0065         A = triu(rand(n) <= .1, 1);
0066         A = (A+A.')/(2*n);
0067     end
0068
0069     n = size(A, 1);
0070     assert(n >= 2, 'A must be at least 2x2.');
0071     assert(isreal(A), 'A must be real.');
0072     assert(size(A, 2) == n, 'A must be square.');
0073
0074     % Force A to be symmetric
0075     A = (A+A.')/2;
0076
0077     % By default, pick a sufficiently large p (number of columns of Y).
0078     if ~exist('p', 'var') || isempty(p)
0079         p = ceil(sqrt(8*n+1)/2);
0080     end
0081
0082     assert(p >= 2 && p == round(p), 'p must be an integer >= 2.');
0083
0084     % Pick the manifold of n-by-p matrices with unit norm rows.
0085     manifold = obliquefactory(p, n, true);
0086
0087     problem.M = manifold;
0088
0089
0090     % These three, quick commented lines of code are sufficient to define
0091     % the cost function and its derivatives. This is good code to write
0092     % when prototyping. Below, a more advanced use of Manopt is shown,
0093     % where the redundant computation A*Y is avoided between the gradient
0094     % and the cost evaluation.
0095     % % problem.cost  = @(Y) .5*sum(sum((A*Y).*Y));
0096     % % problem.egrad = @(Y) A*Y;
0097     % % problem.ehess = @(Y, Ydot) A*Ydot;
0098
0099     % Products with A dominate the cost, hence we store the result.
0100     % This allows to share the results among cost, grad and hess.
0101     % This is completely optional.
0102     function store = prepare(Y, store)
0103         if ~isfield(store, 'AY')
0104             AY = A*Y;
0105             store.AY = AY;
0106             store.diagAYYt = sum(AY .* Y, 2);
0107         end
0108     end
0109
0110     % Define the cost function to be /minimized/.
0111     problem.cost = @cost;
0112     function [f, store] = cost(Y, store)
0113         store = prepare(Y, store);
0114         f = .5*sum(store.diagAYYt);
0115     end
0116
0117     % Define the Riemannian gradient.
0119     function [G, store] = grad(Y, store)
0120         store = prepare(Y, store);
0121         G = store.AY - bsxfun(@times, Y, store.diagAYYt);
0122     end
0123
0124     % If you want to, you can specify the Riemannian Hessian as well.
0125     problem.hess = @hess;
0126     function [H, store] = hess(Y, Ydot, store)
0127         store = prepare(Y, store);
0128         SYdot = A*Ydot - bsxfun(@times, Ydot, store.diagAYYt);
0129         H = manifold.proj(Y, SYdot);
0130     end
0131
0132
0133     % If no initial guess is available, tell Manopt to use a random one.
0134     if ~exist('Y0', 'var') || isempty(Y0)
0135         Y0 = [];
0136     end
0137
0138     % Call your favorite solver.
0139     opts = struct();
0140     opts.verbosity = 0;      % Set to 0 for no output, 2 for normal output
0141     opts.maxinner = 500;     % maximum Hessian calls per iteration