% Frobenious norm diagonal scaling example % Section 4.5.4 example in Boyd & Vandenberghe "Convex Optimization" % (see page 163 for more details) % % Given a square matrix M, the goal is to find a positive vector d % such that ||DMD^{-1}||_F is minimized, where D = diag(d). % The problem can be cast into an unconstrained GP: % % minimize sum_{i,j=1}^{n} M_ij^2*d_i^2/d_j^2 % % with variable d (vector in R^n) and data matrix M in R^{n-by-n}. % % Almir Mutapcic 10/15/05 randn('state',0); % matrix size (M is an n-by-n matrix) n = 4; M = randn(n,n); % GP variables gpvar d(n); % objective expressed using a for loop obj = posynomial; % initialize an empty posynomial for i = 1:n for j = 1:n obj = obj + M(i,j)^2*d(i)^2*d(j)^-2; end end % objective can also be expressed using matrices % (can also use it below in the problem solving routine) % obj_vect = sum( sum( diag(d.^2)*(M.^2)*diag(d.^-2) ) ); % solve the unconstrained GP problem [opt_frob_norm, solution, status] = gpsolve(obj,[]); % assign GP variables to their values (d now becomes an array of doubles) assign(solution); % construct matrix D and display results disp(' ') disp('Optimal diagonal scaling d is: '), d D = diag(d); frob_norm_M = norm(M,'fro'); frob_norm_A = norm(D*M*inv(D),'fro'); fprintf(1,['The Frobenius norm for M before the scaling is %3.4f\n'... 'and after the optimal scaling (DMD^{-1}) it is %3.4f.\n'],... frob_norm_M,frob_norm_A);

Problem succesfully solved. Optimal diagonal scaling d is: d = 0.9706 0.8148 0.8697 1.4529 The Frobenius norm for M before the scaling is 3.6126 and after the optimal scaling (DMD^{-1}) it is 3.2523.