目录
- MATLAB代码
-
- linear_inequalities_constraints_correction函数
- linear_programming_correction函数
- [Problem 1](#Problem 1)
- [Problem 2](#Problem 2)
- [Problem 3](#Problem 3)
- [Problem 4](#Problem 4)
- 研究目标
MATLAB代码
linear_inequalities_constraints_correction函数
matlab
function [X_cor, correction_info] = linear_inequalities_constraints_correction(X_gen, lb, ub, A_ieq, b_ieq, ...
W_multiplicator)
% LINEAR_INEQUALITIES_CONSTRAINTS_CORRECTION Correction onto generated random vector.
% A methodology of correction onto generated random vector,
% facilitating the corrected vector to range within variable
% boundaries and to meet linear constraints
if ~exist('W_multiplicator', 'var')
W_multiplicator = 1000;
end
max_runtime = ceil(W_multiplicator / size(X_gen, 2));
M = size(A_ieq, 1);
[N_D, N_P] = size(X_gen);
b_ieq_lb = diag(A_ieq * ((A_ieq > 0)' .* lb + (A_ieq < 0)' .* ub));
b_ieq_ub = min([diag(A_ieq * ((A_ieq > 0)' .* ub + (A_ieq < 0)' .* lb)), b_ieq], [], 2);
if any(b_ieq < b_ieq_lb)
X_cor = [];
correction_info.is_feasible = false(1, N_P);
correction_info.does_X_cor_exceed.by_bit = true(N_D, N_P);
correction_info.does_X_cor_exceed.globally = true(1, N_P);
correction_info.is_inequality_less_than_b.by_bit = false(M, N_P);
correction_info.is_inequality_less_than_b.globally = false;
else
X_cor = X_gen;
run = 0;
while 1
X_cor = X_gen;
iEq_gen = A_ieq * X_gen;
does_A_ieq_X_gen_exceed = (iEq_gen > b_ieq);
for indi = 1:N_P
if any(does_A_ieq_X_gen_exceed(:, indi))
flag_A_ieq_X_gen_exceed = find(does_A_ieq_X_gen_exceed(:, indi));
row_A_ieq_candi = flag_A_ieq_X_gen_exceed(randi(length(flag_A_ieq_X_gen_exceed)));
A_ieq_candi = A_ieq(row_A_ieq_candi, :);
Eq_candi_gen = A_ieq_candi * X_gen(:, indi);
b_ieq_k_lb = b_ieq_lb(row_A_ieq_candi, :);
b_ieq_k_ub = b_ieq_ub(row_A_ieq_candi, :);
delta_plus = A_ieq_candi * ((A_ieq_candi > 0)' .* (X_gen(:, indi) - lb) ...
+ (A_ieq_candi < 0)' .* (X_gen(:, indi) - ub));
x_cor_base = zeros(N_D, 1);
if Eq_candi_gen > b_ieq_k_ub
b_eq_k_best = [b_ieq_k_lb + (delta_plus * (b_ieq_lb - A_ieq * ((A_ieq_candi > 0)' ...
.* lb + (A_ieq_candi < 0)' .* ub + (A_ieq_candi == 0)' .* X_gen(:, indi)))) ...
./ (A_ieq * ((A_ieq_candi > 0)' .* (X_gen(:, indi) - lb) ...
+ (A_ieq_candi < 0)' .* (X_gen(:, indi) - ub))), ...
b_ieq_k_lb + (delta_plus * (b_ieq_ub - A_ieq * ((A_ieq_candi > 0)' ...
.* lb + (A_ieq_candi < 0)' .* ub + (A_ieq_candi == 0)' .* X_gen(:, indi)))) ...
./ (A_ieq * ((A_ieq_candi > 0)' .* (X_gen(:, indi) - lb) ...
+ (A_ieq_candi < 0)' .* (X_gen(:, indi) - ub)))];
b_eq_k_min = max(min(b_eq_k_best, [], 2), [], 1);
b_eq_k_max = min(max(b_eq_k_best, [], 2), [], 1);
if b_eq_k_min <= b_eq_k_max
b_eq_k = b_eq_k_min + rand * (b_eq_k_max - b_eq_k_min);
else
b_eq_k = b_eq_k_best(row_A_ieq_candi, 1) + rand * (b_eq_k_best(row_A_ieq_candi, 2) ...
- b_eq_k_best(row_A_ieq_candi, 1));
end
x_cor_base(A_ieq_candi > 0) = (((b_eq_k - b_ieq_k_lb) / delta_plus) ...
* (X_gen(A_ieq_candi > 0, indi) - lb(A_ieq_candi > 0)) ...
+ lb(A_ieq_candi > 0))';
x_cor_base(A_ieq_candi == 0) = (X_gen(A_ieq_candi == 0))';
x_cor_base(A_ieq_candi < 0) = (((b_eq_k - b_ieq_k_lb) / delta_plus) ...
* (X_gen(A_ieq_candi < 0, indi) - ub(A_ieq_candi < 0)) ...
+ ub(A_ieq_candi < 0))';
end
X_cor(:, indi) = x_cor_base;
end
end
for indi = 1:N_P
X_cor(abs(X_cor(:, indi) - lb) <= 1e-12, indi) = lb(abs(X_cor(:, indi) - lb) <= 1e-12);
X_cor(abs(X_cor(:, indi) - ub) <= 1e-12, indi) = ub(abs(X_cor(:, indi) - ub) <= 1e-12);
end
correction_info.is_feasible = all((X_cor - lb >= -1e-12) & (X_cor - ub <= 1e-12)) ...
& all((A_ieq * X_cor - b_ieq) <= 1e-12);
correction_info.does_X_cor_exceed.by_bit = (X_cor - lb < -1e-12) | (X_cor - ub > 1e-12);
correction_info.does_X_cor_exceed.globally = any((X_cor - lb < -1e-12) | (X_cor - ub > 1e-12));
correction_info.is_inequality_less_than_b.by_bit = (A_ieq * X_cor - b_ieq) <= 1e-12;
correction_info.is_inequality_less_than_b.globally = all((A_ieq * X_cor - b_ieq) <= 1e-12);
run = run + 1;
% Feasible, Theorem 5
if all(correction_info.is_feasible)
break;
end
% Infeasible, Theorems 4 & 5
if ~any(correction_info.is_feasible) && run >= max_runtime
X_cor = [];
correction_info.is_feasible = false(1, N_P);
correction_info.does_X_cor_exceed.by_bit = true(N_D, N_P);
correction_info.does_X_cor_exceed.globally = true(1, N_P);
correction_info.is_equation_equal_to_b.by_bit = false(M, N_P);
correction_info.is_equation_equal_to_b.globally = false;
break;
end
X_gen = X_cor;
end
end
endlinear_programming_correction函数
matlab
function [X_cor, correction_info] = linear_programming_correction(X_gen, lb, ub, A_eq, b_eq, A_ieq, b_ieq, ...
W_multiplicator)
% LINEAR_PROGRAMMING_CORRECTION Correction onto generated random vector.
% A methodology of correction onto generated random vector,
% facilitating the corrected vector to range within variable
% boundaries and to meet linear constraints
if ~exist('W_multiplicator', 'var')
W_multiplicator = 1000;
end
Ab_eq = [A_eq, b_eq];
M_eq = size(A_eq, 1);
M_ieq = size(A_ieq, 1);
[N_D, N_P] = size(X_gen);
% Invoke Python for an accurate computation of the reduced row echelon form of Ab_eq
Ab_eq_py = py.numpy.array(Ab_eq);
Ab_eq_RREF_sym = py.sympy.Matrix(Ab_eq_py).rref();
Ab_eq_RREF_raw = double(py.numpy.asarray(Ab_eq_RREF_sym(1), dtype = 'float'));
if size(Ab_eq, 1) == 1
Ab_eq_RREF = Ab_eq_RREF_raw;
else
Ab_eq_RREF = reshape(Ab_eq_RREF_raw, size(Ab_eq_RREF_raw, 2), size(Ab_eq_RREF_raw, 3));
end
R_A_eq = sum(any(Ab_eq_RREF(:, 1:end - 1), 2), 1);
R_Ab_eq = sum(any(Ab_eq_RREF, 2), 1);
A_eq_tilde = Ab_eq_RREF(:, R_A_eq + 1:end - 1);
b_eq_tilde = Ab_eq_RREF(:, end);
if R_A_eq < R_Ab_eq
X_cor = [];
correction_info.is_feasible = false(1, N_P);
correction_info.does_X_cor_exceed.by_bit = true(N_D, N_P);
correction_info.does_X_cor_exceed.globally = true(1, N_P);
correction_info.is_equation_equal_to_b.by_bit = false(M_eq, N_P);
correction_info.is_equation_equal_to_b.globally = false;
correction_info.is_inequality_less_than_b.by_bit = false(M_ieq, N_P);
correction_info.is_inequality_less_than_b.globally = false;
else
lb_base = lb(R_A_eq + 1:end, :);
ub_base = ub(R_A_eq + 1:end, :);
b_eq_tilde_plus = b_eq_tilde - lb(1:R_A_eq, end);
b_eq_tilde_minus = b_eq_tilde - ub(1:R_A_eq, end);
if isempty(A_eq)
A_new = A_ieq;
b_new = b_ieq;
[X_cor, correction_info] = linear_inequalities_constraints_correction(X_gen, lb, ub, A_new, ...
b_new, W_multiplicator);
elseif isempty(A_ieq)
if R_A_eq == 1
A_new = [-A_eq; A_eq];
b_new = [-b_eq; b_eq];
[X_cor, correction_info] = linear_inequalities_constraints_correction(X_gen, lb, ub, A_new, ...
b_new, W_multiplicator);
else
A_new = [-A_eq_tilde; A_eq_tilde];
b_new = [-b_eq_tilde_minus; b_eq_tilde_plus];
X_gen_base = X_gen(R_A_eq + 1:end, :);
[X_cor_base, correction_info] = linear_inequalities_constraints_correction(X_gen_base, lb_base, ...
ub_base, A_new, b_new, W_multiplicator);
X_cor = [b_eq_tilde - A_eq_tilde * X_cor_base; X_cor_base];
end
else
if R_A_eq == 1
A_new = [-A_eq; A_eq; A_ieq];
b_new = [-b_eq; b_eq; b_ieq];
[X_cor, correction_info] = linear_inequalities_constraints_correction(X_gen, lb, ub, A_new, ...
b_new, W_multiplicator);
else
A_new = [-A_eq_tilde; A_eq_tilde; A_ieq(:, R_A_eq + 1:end) - A_ieq(:, 1:R_A_eq) * A_eq_tilde];
b_new = [-b_eq_tilde_minus; b_eq_tilde_plus; b_ieq - A_ieq(:, 1:R_A_eq) * b_eq_tilde];
X_gen_base = X_gen(R_A_eq + 1:end, :);
[X_cor_base, correction_info] = linear_inequalities_constraints_correction(X_gen_base, lb_base, ...
ub_base, A_new, b_new, W_multiplicator);
X_cor = [b_eq_tilde - A_eq_tilde * X_cor_base; X_cor_base];
end
end
if isempty(A_eq)
correction_info.is_feasible = all((X_cor - lb >= -1e-12) & (X_cor - ub <= 1e-12)) ...
& all((A_ieq * X_cor - b_ieq) <= 1e-12);
correction_info.does_X_cor_exceed.by_bit = (X_cor - lb < -1e-12) | (X_cor - ub > 1e-12);
correction_info.does_X_cor_exceed.globally = any((X_cor - lb < -1e-12) | (X_cor - ub > 1e-12));
correction_info.is_equation_equal_to_b.by_bit = [];
correction_info.is_equation_equal_to_b.globally = [];
correction_info.is_inequality_less_than_b.by_bit = (A_ieq * X_cor - b_ieq) <= 1e-12;
correction_info.is_inequality_less_than_b.globally = all((A_ieq * X_cor - b_ieq) <= 1e-12);
elseif isempty(A_ieq)
correction_info.is_feasible = all((X_cor - lb >= -1e-12) & (X_cor - ub <= 1e-12)) ...
& all(abs(A_eq * X_cor - b_eq) <= 1e-12);
correction_info.does_X_cor_exceed.by_bit = (X_cor - lb < -1e-12) | (X_cor - ub > 1e-12);
correction_info.does_X_cor_exceed.globally = any((X_cor - lb < -1e-12) | (X_cor - ub > 1e-12));
correction_info.is_equation_equal_to_b.by_bit = abs(A_eq * X_cor - b_eq) <= 1e-12;
correction_info.is_equation_equal_to_b.globally = all(abs(A_eq * X_cor - b_eq) <= 1e-12);
correction_info.is_inequality_less_than_b.by_bit = [];
correction_info.is_inequality_less_than_b.globally = [];
else
correction_info.is_feasible = all((X_cor - lb >= -1e-12) & (X_cor - ub <= 1e-12)) ...
& all(abs(A_eq * X_cor - b_eq) <= 1e-12) & ...
all((A_ieq * X_cor - b_ieq) <= 1e-12);
correction_info.does_X_cor_exceed.by_bit = (X_cor - lb < -1e-12) | (X_cor - ub > 1e-12);
correction_info.does_X_cor_exceed.globally = any((X_cor - lb < -1e-12) | (X_cor - ub > 1e-12));
correction_info.is_equation_equal_to_b.by_bit = abs(A_eq * X_cor - b_eq) <= 1e-12;
correction_info.is_equation_equal_to_b.globally = all(abs(A_eq * X_cor - b_eq) <= 1e-12);
correction_info.is_inequality_less_than_b.by_bit = (A_ieq * X_cor - b_ieq) <= 1e-12;
correction_info.is_inequality_less_than_b.globally = all((A_ieq * X_cor - b_ieq) <= 1e-12);
end
end
endProblem 1
matlab
clear;
close all;
clc;
lb = [-3, -1, 2, -4, 0, 1, -6, -5]';
ub = [6, 4, 10, 8, 11, 3, 4, 4]';
X_gen = lb + rand(1, 100) .* (ub - lb);
A_eq = [6, -3, 1, 7, -5, 0, -3, 8];
b_eq = 12;
[X_cor, correction_info] = linear_programming_correction(X_gen, lb, ub, [], [], A_eq, b_eq);Problem 2
matlab
clear;
close all;
clc;
lb = [-5, -3, -6, 0]';
ub = [7, 1, -2, 6]';
X_gen = lb + rand(1, 100) .* (ub - lb);
A_eq = [1, 1, 1, 1; 0, 2, 1, 1];
b_eq = [-1, 1]';
[X_cor, correction_info] = linear_programming_correction(X_gen, lb, ub, A_eq, b_eq, [], []);Problem 3
matlab
clear;
close all;
clc;
lb = [-8, -15, -2, 0, -3, -10]';
ub = [9, 7, 4, 5, 8, 2]';
X_gen = lb + rand(1, 100) .* (ub - lb);
A_eq = [4, 3, 5, -7, 6, -8];
b_eq = -2;
A_ieq = [-7, -4, 8, -5, 0, 1];
b_ieq = 14;
[X_cor, correction_info] = linear_programming_correction(X_gen, lb, ub, A_eq, b_eq, A_ieq, b_ieq);Problem 4
matlab
clear;
close all;
clc;
lb = [-8, -15, -2, 1, -3, -10]';
ub = [9, 7, 11, 7, 8, 2]';
X_gen = lb + rand(1, 100) .* (ub - lb);
A_eq = [4, 3, 5, -7, 6, -8; -7, -4, 8, -5, 0, 8];
b_eq = [-2; 14];
A_ieq = [10, 3, -3, 6, -7, 2; 2, -3, -7, 5, -6, 3];
b_ieq = [9; -8];
[X_cor, correction_info] = linear_programming_correction(X_gen, lb, ub, A_eq, b_eq, A_ieq, b_ieq);研究目标
(1) 探究满足某些非线性约束的生成随机数修正方法；
(2) 探究随机向量为离散向量和连续-离散混合向量时的生成随机数修正方法；
(3) 将生成随机数修正方法应用到基于启发式算法的优化问题求解中，使得启发式算法能够始终在优化问题的可行域中搜寻问题的解，从而加快启发式优化算法的收敛速度。