create_simplex.m

function [X] = create_simplex(strategy, x0)
% -- [X] = create_simplex(strategy, x0)
%
%     Create initial simplex using selected strategy. There are multiple
%     methods identified by `strategy` parameter:
%
%       '0': Default Matlab strategy using L. Pfeffer suggestion (see [1]).
%
%       '1': Regular simplex as constructed by Jacoby, Kowalik and Pizzo (see
%            pages 80-81 of [2]).
%
%       '2': Right-angled simplex generated by adding scaled unit vector along
%            all dimensions.
%
%   References:
%     [1] Fan, E. Global optimization of Lennard-Jones atomic clusters. MASTER
%         OF SCIENCE. McMaster University, COMPUTING & SOFTWARE, Hamilton,
%         Ontario, 2002.
%     [2] Jacoby, S. L., Kowalik, J. S., & Pizzo, J. T. (1972). Iterative
%         methods for nonlinear optimization problems.

    % Use column vector in computations
    x0 = x0(:);
    N = length(x0);

    switch strategy

        % Pfeffer method
        case 0
            usual_delta = 0.05;
            zero_term_delta = 0.0075;

            X(:,1) = x0;
            for i=1:N
                x = x0;
                if x(i) ~= 0
                    x(i) = (1 + usual_delta)*x0(i);
                else
                    x(i) = zero_term_delta;
                end
                X(:,i+1) = x;
            end

        % Regular simplex
        case 1
            s = max(1, norm(x0, Inf));
            alpha = ([1, 1] * (sqrt(N+1) - 1) + [N, 0]) * s / (N * sqrt(2));
            X = [x0, zeros(N)];
            X(:,2:end) = repmat(x0 + alpha(2), 1, N);
            for i=2:N+1
                X(i-1,i) = x0(i-1) + alpha(1);
            end

        % Right-angled simplex
        case 2
            s = max(1, norm(x0, Inf));
            X = repmat(x0, 1, N+1);
            X(:,2:end) = X(:,2:end) + s*eye(N);

        otherwise
            error('Unknown simplex creation strategy.');
    end

end