MATLAB数值分析：第十章示例程序解析

%% 10-1 列出所有优化参数列表

optimset

%% 实验结果

% ex10_1

% MaxFunEvals: [ positive scalar ]

% MaxIter: [ positive scalar ]

% TolFun: [ positive scalar ]

% TolX: [ positive scalar ]

% FunValCheck: [ on | {off} ]

% OutputFcn: [ function | {[]} ]

% PlotFcns: [ function | {[]} ]

% Algorithm: [ active-set | interior-point | interior-point-convex | levenberg-marquardt | ...

% sqp | trust-region-dogleg | trust-region-reflective ]

% AlwaysHonorConstraints: [ none | {bounds} ]

% DerivativeCheck: [ on | {off} ]

% Diagnostics: [ on | {off} ]

% DiffMaxChange: [ positive scalar | {Inf} ]

% DiffMinChange: [ positive scalar | {0} ]

% FinDiffRelStep: [ positive vector | positive scalar | {[]} ]

% FinDiffType: [ {forward} | central ]

% GoalsExactAchieve: [ positive scalar | {0} ]

% GradConstr: [ on | {off} ]

% GradObj: [ on | {off} ]

% HessFcn: [ function | {[]} ]

% HessMult: [ function | {[]} ]

% HessPattern: [ sparse matrix | {sparse(ones(numberOfVariables))} ]

% HessUpdate: [ dfp | steepdesc | {bfgs} ]

% InitBarrierParam: [ positive scalar | {0.1} ]

% InitTrustRegionRadius: [ positive scalar | {sqrt(numberOfVariables)} ]

% Jacobian: [ on | {off} ]

% JacobMult: [ function | {[]} ]

% JacobPattern: [ sparse matrix | {sparse(ones(Jrows,Jcols))} ]

% LargeScale: [ on | off ]

% MaxNodes: [ positive scalar | {1000*numberOfVariables} ]

% MaxPCGIter: [ positive scalar | {max(1,floor(numberOfVariables/2))} ]

% MaxProjCGIter: [ positive scalar | {2*(numberOfVariables-numberOfEqualities)} ]

% MaxSQPIter: [ positive scalar | {10*max(numberOfVariables,numberOfInequalities+numberOfBounds)} ]

% MaxTime: [ positive scalar | {7200} ]

% MeritFunction: [ singleobj | {multiobj} ]

% MinAbsMax: [ positive scalar | {0} ]

% ObjectiveLimit: [ scalar | {-1e20} ]

% PrecondBandWidth: [ positive scalar | 0 | Inf ]

% RelLineSrchBnd: [ positive scalar | {[]} ]

% RelLineSrchBndDuration: [ positive scalar | {1} ]

% ScaleProblem: [ none | obj-and-constr | jacobian ]

% SubproblemAlgorithm: [ cg | {ldl-factorization} ]

% TolCon: [ positive scalar ]

% TolConSQP: [ positive scalar | {1e-6} ]

% TolPCG: [ positive scalar | {0.1} ]

% TolProjCG: [ positive scalar | {1e-2} ]

% TolProjCGAbs: [ positive scalar | {1e-10} ]

% TypicalX: [ vector | {ones(numberOfVariables,1)} ]

% UseParallel: [ logical scalar | true | {false} ]

cut-off

%% 10-2 对下列非线性方程组按线性规划问题进行求解：

% min f(x) = -5x1 - 4x2 - 6x3

% x1 - x2 + x3 ≤ 20

% 3x1 + 2x2 + 4x3 ≤ 42

% 3x1 + 2x2 ≤ 20

% x1, x2, x3 ≥ 0

f = [-5;-4;-6] % 输入目标函数系数矩阵

A = [1 -1 1;3 2 4;3 2 0] % 输入不等式约束系数矩阵

b = [20;42;30] % 输入右端点

lb = zeros(3,1);

[x,fval,exitflag,output,lambda] = linprog(f,A,b,[],[],lb,[],[],optimset('Display','iter'))

%% 实验结果

% ex10_2

% f =

% -5

% -4

% -6

% A =

% 1 -1 1

% 3 2 4

% 3 2 0

% b =

% 20

% 42

% 30

% LP preprocessing removed 0 inequalities, 0 equalities,

% 0 variables, and added 0 non-zero elements.

% Iter Time Fval Primal Infeas Dual Infeas

% 0 0.013 0.000000e+00 0.000000e+00 3.880779e+00

% 3 0.014 -7.800000e+01 0.000000e+00 0.000000e+00

% Optimal solution found.

% x =

% 0

% 15.0000

% 3.0000

% fval =

% -78

% exitflag =

% 1

% output =

% 包含以下字段的 struct:

% iterations: 3

% constrviolation: 0

% message: 'Optimal solution found.'

% algorithm: 'dual-simplex'

% firstorderopt: 1.7764e-15

% lambda =

% 包含以下字段的 struct:

% lower: [3×1 double]

% upper: [3×1 double]

% eqlin: []

% ineqlin: [3×1 double]

cut-off

%% 10-3 对下列非线性方程组按线性规划问题进行求解：

f = [-80,-125] % 输入目标函数系数矩阵

A = [8 5;6 4;4 5] % 输入不等式约束系数矩阵

b = [3500;1800;2800] % 输入右端点

lb = [0;0];

[x,fval,exitflag,output,lambda] = linprog(f,A,b,[],[],lb,[],[],optimset('Display','iter'))

%% 实验结果

% ex10_3

% f =

% -80 -125

% A =

% 8 5

% 6 4

% 4 5

% b =

% 3500

% 1800

% 2800

% LP preprocessing removed 0 inequalities, 0 equalities,

% 0 variables, and added 0 non-zero elements.

% Iter Time Fval Primal Infeas Dual Infeas

% 0 0 0.000000e+00 0.000000e+00 2.482324e+01

% 1 0.001 -7.000000e+04 4.800041e+02 0.000000e+00

% 2 0.001 -5.625000e+04 0.000000e+00 0.000000e+00

% Optimal solution found.

% x =

% 0

% 450.0000

% fval =

% -5.6250e+04

% exitflag =

% 1

% output =

% 包含以下字段的 struct:

% iterations: 2

% constrviolation: 0

% message: 'Optimal solution found.'

% algorithm: 'dual-simplex'

% firstorderopt: 7.1054e-12

% lambda =

% 包含以下字段的 struct:

% lower: [2×1 double]

% upper: [2×1 double]

% eqlin: []

% ineqlin: [3×1 double]

cut-off

%% 10-4 对下列非线性方程组按线性规划问题进行求解：

f = [-0.1 -0.122 -0.15 -0.12 -0.08] % 输入目标函数系数矩阵

A = [1 -1 -1 -1 -1;0 -1 1 -1 1] % 输入不等式约束系数矩阵

b = [0;0] % 输入右端点

Aeq = [1 1 1 1 1]

beq = [1]

lb = [0;0;0;0;0]

[x,fval,exitflag,output,lambda] = linprog(f,A,b,Aeq,beq,lb,[],[],optimset('Display','iter'))

%% 实验结果

% ex10_4

% f =

% -0.1000 -0.1220 -0.1500 -0.1200 -0.0800

% A =

% 1 -1 -1 -1 -1

% 0 -1 1 -1 1

% b =

% 0

% Aeq =

% 1 1 1 1 1

% beq =

% 1

% lb =

% 0

% LP preprocessing removed 1 inequalities, 0 equalities,

% 2 variables, and 8 non-zero elements.

% Iter Time Fval Primal Infeas Dual Infeas

% 0 0.001 0.000000e+00 1.000000e+00 2.275610e-01

% 1 0.001 -1.500000e-01 1.000000e+00 0.000000e+00

% 2 0.001 -1.360000e-01 0.000000e+00 0.000000e+00

% Optimal solution found.

% x =

% 0

% 0.5000

% 0

% fval =

% -0.1360

% exitflag =

% 1

% output =

% 包含以下字段的 struct:

% iterations: 2

% constrviolation: 0

% message: 'Optimal solution found.'

% algorithm: 'dual-simplex'

% firstorderopt: 1.3878e-17

% lambda =

% 包含以下字段的 struct:

% lower: [5×1 double]

% upper: [5×1 double]

% eqlin: 0.1360

% ineqlin: [2×1 double]

cut-off

%% 10-5 求下面函数在区间[-1.2,1]内的最小值

% f(x) = 100*(x2-x1^2)^2 + (a-x1)^2

a = sqrt(2) % 给定未知系数a的值

banana = @(x)100*(x(2)-x(1)^2)^2+(a-x(1))^2 % 使用内联形式定义函数

[x,fval,exitflag,output] = fminsearch(banana,[-1.2,1],optimset('TolX',1e-8)) % 调用fminsearch函数求解最小值

%% 实验结果

% ex10_5

% a =

% 1.4142

% banana =

% 包含以下值的 function_handle:

% @(x)100*(x(2)-x(1)^2)^2+(a-x(1))^2

% x =

% 1.4142 2.0000

% fval =

% 4.2065e-18

% exitflag =

% 1

% output =

% 包含以下字段的 struct:

% iterations: 131

% funcCount: 249

% algorithm: 'Nelder-Mead simplex direct search'

% message: '优化已终止:↵ 当前的 x 满足使用 1.000000e-08 的 OPTIONS.TolX 的终止条件，↵F(X) 满足使用 1.000000e-04 的 OPTIONS.TolFun 的收敛条件↵'

cut-off

%% 10-6 分别用fminunc函数与fminsearch函数求解无约束优化问题

% f(x) = 3*x1^2 + 2*x1*x2 + x2^2

x0 = [1,1];

[x,fval,exitflag,output,grad,hessian] = fminunc(@ex10_6a,x0) % fminunc函数

[x,fval,exitflag,output] = fminsearch(@ex10_6a,x0) % fminsearch函数

function f = ex10_6a(x)

f = 3*x(1)^2 + 2*x(1)*x(2) + x(2)^2;

end

%% 实验结果

% ex10_6

% Computing finite-difference Hessian using objective function.

% Local minimum found.

% Optimization completed because the size of the gradient is less than

% the value of the optimality tolerance.

% <stopping criteria details>

% x =

% 1.0e-06 *

% 0.2541 -0.2029

% fval =

% 1.3173e-13

% exitflag =

% 1

% output =

% 包含以下字段的 struct:

% iterations: 8

% funcCount: 27

% stepsize: 3.9161e-05

% lssteplength: 1

% firstorderopt: 1.1633e-06

% algorithm: 'quasi-newton'

% message: '↵Local minimum found.↵↵Optimization completed because the size of the gradient is less than↵the value of the optimality tolerance.↵↵<stopping criteria details>↵↵Optimization completed: The first-order optimality measure, 1.292597e-07, is less ↵than options.OptimalityTolerance = 1.000000e-06.↵↵'

% grad =

% 1.0e-05 *

% 0.1163

% 0.0087

% hessian =

% 6.0000 2.0000

% 2.0000 2.0000

% x =

% 1.0e-04 *

% -0.0675 0.1715

% fval =

% 1.9920e-10

% exitflag =

% 1

% output =

% 包含以下字段的 struct:

% iterations: 46

% funcCount: 89

% algorithm: 'Nelder-Mead simplex direct search'

% message: '优化已终止:↵ 当前的 x 满足使用 1.000000e-04 的 OPTIONS.TolX 的终止条件，↵F(X) 满足使用 1.000000e-04 的 OPTIONS.TolFun 的收敛条件↵'

cut-off

%% 10-7 计算下面函数在(-2,2)内的最小值

% f(x) = (x^3 + x)/(x^4 - x^2 + 1)

% help fminbnd

% fminbnd - 查找单变量函数在定区间上的最小值

% fminbnd 是一个一维最小值，用于求由以下条件指定的问题的最小值：

% x = fminbnd(fun,x1,x2)

% x = fminbnd(fun,x1,x2,options)

% x = fminbnd(problem)

% [x,fval] = fminbnd(___)

% [x,fval,exitflag] = fminbnd(___)

% [x,fval,exitflag,output] = fminbnd(___)

[x,fval,exitflag,output] = fminbnd(@ex10_7a,-2,2) % fminbnd函数

function f = ex10_7a(x)

f = (x^3 + x)/(x^4 - x^2 + 1);

end

%% 实验结果

% ex10_7

% x =

% -1.0000

% fval =

% -2.0000

% exitflag =

% 1

% output =

% 包含以下字段的 struct:

% iterations: 11

% funcCount: 12

% algorithm: 'golden section search, parabolic interpolation'

% message: '优化已终止:↵ 当前的 x 满足使用 1.000000e-04 的 OPTIONS.TolX 的终止条件↵'

cut-off

%% 10-8 求解在约束条件0≤x1+2*x2+3*x3≤72下，函数f(x)=-x1*x2*x3的最小值的最优解和最优解的数值

A = [-1 -2 -2;1 2 2] % 不等式左边系数

b = [0;72] % 不等式右边系数

x0 = [10;10;10] % 给定初始值

[x,fval,exitflag,output,lambda] = fmincon(@ex10_8a,x0,A,b) % fminunc函数

function f = ex10_8a(x)

f = -x(1) * x(2) * x(3);

end

%% 实验结果

% ex10_8

% A =

% -1 -2 -2

% 1 2 2

% b =

% 0

% 72

% x0 =

% 10

% Local minimum found that satisfies the constraints.

% Optimization completed because the objective function is non-decreasing in

% feasible directions, to within the value of the optimality tolerance,

% and constraints are satisfied to within the value of the constraint tolerance.

% <stopping criteria details>

% x =

% 24.0000

% 12.0000

% fval =

% -3.4560e+03

% exitflag =

% 1

% output =

% 包含以下字段的 struct:

% iterations: 9

% funcCount: 43

% constrviolation: 0

% stepsize: 6.5312e-05

% algorithm: 'interior-point'

% firstorderopt: 1.2552e-05

% cgiterations: 0

% message: '↵Local minimum found that satisfies the constraints.↵↵Optimization completed because the objective function is non-decreasing in ↵feasible directions, to within the value of the optimality tolerance,↵and constraints are satisfied to within the value of the constraint tolerance.↵↵<stopping criteria details>↵↵Optimization completed: The relative first-order optimality measure, 4.358349e-08,↵is less than options.OptimalityTolerance = 1.000000e-06, and the relative maximum constraint↵violation, 0.000000e+00, is less than options.ConstraintTolerance = 1.000000e-06.↵↵'

% bestfeasible: [1×1 struct]

% lambda =

% 包含以下字段的 struct:

% eqlin: [0×1 double]

% eqnonlin: [0×1 double]

% ineqlin: [2×1 double]

% lower: [3×1 double]

% upper: [3×1 double]

% ineqnonlin: [0×1 double]

cut-off

%% 10-9 求积下面函数的最小值

% f(x,y,z) = x*y + y*z + z*x

% 约束条件：2*x + y + 2*z = 2; x^2 + y^2 + z^2 = 4

% 求初始值 x0 = [0 1 1]

% help fmincon

% fmincon - 寻找约束非线性多变量函数的最小值

% 非线性规划求解器。

% x = fmincon(fun,x0,A,b)

% x = fmincon(fun,x0,A,b,Aeq,beq)

% x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)

% x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)

% x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

% x = fmincon(problem)

% [x,fval] = fmincon(___)

% [x,fval,exitflag,output] = fmincon(___)

% [x,fval,exitflag,output,lambda,grad,hessian] = fmincon(___)

x0 = [0 1 1]

Aeq = [2 1 2]

Beq = 2

[x,fval,exitflag] = fmincon(@ex10_9a,x0,[],[],Aeq,Beq,[],[],@ex10_9b)

function y= ex10_9a(x)

y = x(1)*x(2) + x(2)*x(3) + x(3)*x(1);

end

function [c,ceq] = ex10_9b(x)

c = [];

ceq = x(1)^2 + x(2)^2 + x(3)^2 - 4;

end

%% 实验结果

% ex10_9

% x0 =

% 0 1 1

% Aeq =

% 2 1 2

% Beq =

% 2

% Local minimum found that satisfies the constraints.

% Optimization completed because the objective function is non-decreasing in

% feasible directions, to within the value of the optimality tolerance,

% and constraints are satisfied to within the value of the constraint tolerance.

% <stopping criteria details>

% x =

% 0.8889 -1.5556 0.8889

% fval =

% -1.9753

% exitflag =

% 1

cut-off

%% 10-10 求解下面二次规划问题

% min f(x) = 1/2*x1^2 + x2^2 - x1*x2 - 2*x1 - 6*x2

% 约束条件：

% x1 + x2 ≤ 2

% -x1 + 2x2 ≤ 2

% 2x1 + 2x ≤ 3

% x1 ≥ 0, x2 ≥ 0

H = [1 -1;-1 2]

f = [-2;-6]

A = [1 1;-1 2;2 1]

b = [2;2;3]

lb = zeros(2,1)

opts = optimset('Algorithm','interior-point-convex','Display','iter');

[x fval eflag output lambda] = quadprog(H,f,A,b,[],[],[],[],[],opts)

%% 实验结果

% ex10_10

% H =

% 1 -1

% -1 2

% f =

% -2

% -6

% A =

% 1 1

% -1 2

% 2 1

% b =

% 2

% 3

% lb =

% 0

% Iter Fval Primal Infeas Dual Infeas Complementarity

% 0 -8.884885e+00 3.214286e+00 1.071429e-01 1.000000e+00

% 1 -8.331868e+00 1.321041e-01 4.403472e-03 1.910489e-01

% 2 -8.212804e+00 1.676295e-03 5.587652e-05 1.009601e-02

% 3 -8.222204e+00 8.381476e-07 2.793826e-08 1.809485e-05

% 4 -8.222222e+00 3.019807e-14 1.352696e-12 7.525735e-13

% Minimum found that satisfies the constraints.

% Optimization completed because the objective function is non-decreasing in

% feasible directions, to within the value of the optimality tolerance,

% and constraints are satisfied to within the value of the constraint tolerance.

% <stopping criteria details>

% x =

% 0.6667

% 1.3333

% fval =

% -8.2222

% eflag =

% 1

% output =

% 包含以下字段的 struct:

% message: '↵Minimum found that satisfies the constraints.↵↵Optimization completed because the objective function is non-decreasing in ↵feasible directions, to within the value of the optimality tolerance,↵and constraints are satisfied to within the value of the constraint tolerance.↵↵<stopping criteria details>↵↵Optimization completed: The relative dual feasibility, 2.254493e-13,↵is less than options.OptimalityTolerance = 1.000000e-08, the complementarity measure,↵7.525735e-13, is less than options.OptimalityTolerance, and the relative maximum constraint↵violation, 5.033011e-15, is less than options.ConstraintTolerance = 1.000000e-08.↵↵'

% algorithm: 'interior-point-convex'

% firstorderopt: 1.3527e-12

% constrviolation: 1.7764e-15

% iterations: 4

% linearsolver: 'dense'

% cgiterations: []

% lambda =

% 包含以下字段的 struct:

% ineqlin: [3×1 double]

% eqlin: [0×1 double]

% lower: [2×1 double]

% upper: [2×1 double]

cut-off

%% 10-11 求解下面的二次规划问题

% max f(x) = -x1^2 -x2^2 + 3*x1 + 5*x2

% 约束条件

% 3*x1 + 2*x2 ≤ 5

% x1 ≥ 0, x2 ≥ 0

% help quadprog

% quadprog - 二次规划

% 具有线性约束的二次目标函数的求解器。

% x = quadprog(H,f)

% x = quadprog(H,f,A,b)

% x = quadprog(H,f,A,b,Aeq,beq)

% x = quadprog(H,f,A,b,Aeq,beq,lb,ub)

% x = quadprog(H,f,A,b,Aeq,beq,lb,ub,x0)

% x = quadprog(H,f,A,b,Aeq,beq,lb,ub,x0,options)

% x = quadprog(problem)

% [x,fval] = quadprog(___)

% [x,fval,exitflag,output] = quadprog(___)

% [x,fval,exitflag,output,lambda] = quadprog(___)

H = [2 0;0 2]

f = [-3 -5]

A = [3 2]

b = [5]

lb = [0 0]

[x,fval,exitflag] = quadprog(H,f,A,b,[],[],lb)

%% 实验结果

% ex10_11

% H =

% 2 0

% 0 2

% f =

% -3 -5

% A =

% 3 2

% b =

% 5

% lb =

% 0 0

% Minimum found that satisfies the constraints.

% Optimization completed because the objective function is non-decreasing in

% feasible directions, to within the value of the optimality tolerance,

% and constraints are satisfied to within the value of the constraint tolerance.

% <stopping criteria details>

% x =

% 0.4615

% 1.8077

% fval =

% -6.9423

% exitflag =

% 1

cut-off

%% 10-12 求解下面的函数最小最大值问题：

% min(x) max(Fi) {f1(x),f2(x),f3(x),f4(x),f5(x)}

% 约束条件

% x1^2 + x2^2 ≤ 8

% x1 + x2 ≤ 3

% -3 ≤ x1 ≤ 3

% -2 ≤ x2 ≤ 2

% 其中

% f1(x) = 2*x1^2 + x2^2 - 48*x1 - 40*x2 + 304

% f2(x) = -x1^2 - 3*x2^2

% f3(x) = x1 + 3*x2 - 18

% f4(x) = -x1 - x2

% f5(x) = x1 + x2 - 8

% help fminimax

% fminimax - 求解 minimax 约束问题

% fminimax 寻找能够最小化一组目标函数最大值的点。

% x = fminimax(fun,x0)

% x = fminimax(fun,x0,A,b)

% x = fminimax(fun,x0,A,b,Aeq,beq)

% x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub)

% x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)

% x = fminimax(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

% x = fminimax(problem)

% [x,fval] = fminimax(___)

% [x,fval,maxfval,exitflag,output] = fminimax(___)

% [x,fval,maxfval,exitflag,output,lambda] = fminimax(___)

x0 = [0.1;0.1]

[x,fval,maxfval] = fminimax(@ex10_12a,x0)

options = optimset('MinAbsMax',5); % 最小绝对值

[x,fval,maxfval,exitflag,output,lambda] = fminimax(@ex10_12a,x0,[],[],[],[],[],[],[],options)

function f = ex10_12a(x)

f(1) = 2*x(1)^2 + x(2)^2 - 48*x(1) - 40*x(2) + 304;

f(2) = -x(1)^2 - 3*x(2)^2;

f(3) = x(1) + 3*x(2) - 18;

f(4) = -x(1) - x(2);

f(5) = x(1) + x(2) - 8;

end

%% 实验结果

% ex10_12

% x0 =

% 0.1000

% Local minimum possible. Constraints satisfied.

% fminimax stopped because the size of the current search direction is less than

% twice the value of the step size tolerance and constraints are

% satisfied to within the value of the constraint tolerance.

% <stopping criteria details>

% x =

% 4.0000

% fval =

% 0.0000 -64.0000 -2.0000 -8.0000 -0.0000

% maxfval =

% 4.2405e-11

% Local minimum possible. Constraints satisfied.

% fminimax stopped because the predicted change in the objective function

% is less than the value of the function tolerance and constraints

% are satisfied to within the value of the constraint tolerance.

% <stopping criteria details>

% x =

% 4.9256

% 2.0796

% fval =

% 37.2356 -37.2356 -6.8357 -7.0052 -0.9948

% maxfval =

% 37.2356

% exitflag =

% 5

% output =

% 包含以下字段的 struct:

% iterations: 7

% funcCount: 35

% lssteplength: 1

% stepsize: 2.7799e-06

% algorithm: 'active-set'

% firstorderopt: []

% constrviolation: 2.8906e-10

% message: '↵Local minimum possible. Constraints satisfied.↵↵fminimax stopped because the predicted change in the objective function↵is less than the value of the function tolerance and constraints ↵are satisfied to within the value of the constraint tolerance.↵↵<stopping criteria details>↵↵Optimization stopped because the predicted change in the objective function,↵2.524100e-10, is less than options.FunctionTolerance = 1.000000e-06, and the maximum constraint↵violation, 2.890630e-10, is less than options.ConstraintTolerance = 1.000000e-06.↵↵'

% lambda =

% 包含以下字段的 struct:

% lower: [2×1 double]

% upper: [2×1 double]

% eqlin: [0×1 double]

% eqnonlin: [0×1 double]

% ineqlin: [0×1 double]

% ineqnonlin: [0×1 double]

cut-off

%% 10-13 求解如下“半无限”多元约束优化问题

% max f(x) = 8*x1 + 4.5*x2^2 + 3*x3

% 约束

% 1.5*x3^2 - x2^2 ≥ 3

% x1 + 4*x2 - 5*x3 ≤ 32

% x1 + 3*x2 + 2*x3 ≤ 29

% x1,x2,x3 ≥ 0

% 半无限约束K_i(w,x)为：

% K1(x,w1) = sin(w1*x1).*cos(w1*x1)-1/1000*(w1-45).^2-sin(w1*x3)-x2-1;

% K2(x,w2) = sin(w2*x2).*cos(w2*x2)-1/1000*(w2-45).^2-sin(w2*x3)-x3-1;

% help fseminf

% fseminf - 求解半无限约束多变量非线性函数的最小值

% 此 MATLAB 函数从 x0 开始，求解满足 seminfcon 中定义的 ntheta 半无限约束条件的函

% 数 fun 的最小值。

% x = fseminf(fun,x0,ntheta,seminfcon)

% x = fseminf(fun,x0,ntheta,seminfcon,A,b)

% x = fseminf(fun,x0,ntheta,seminfcon,A,b,Aeq,beq)

% x = fseminf(fun,x0,ntheta,seminfcon,A,b,Aeq,beq,lb,ub)

% x = fseminf(fun,x0,ntheta,seminfcon,A,b,Aeq,beq,lb,ub,options)

% x = fseminf(problem)

% [x,fval] = fseminf(...)

% [x,fval,exitflag] = fseminf(...)

% [x,fval,exitflag,output] = fseminf(...)

% [x,fval,exitflag,output,lambda] = fseminf(...)

A = [1 4 5;1 3 2];

b = [32 29];

Aeq = [];

beq = [];

lb = zeros(1,size(A,2));

x0 = ones(size(A,2),1);

[x,feval] = fseminf(@ex10_13a,x0,2,@ex10_13b,A,b,Aeq,beq,lb)

function f = ex10_13a(x)

f = 8*x(1) + 4.5*x(2) + 3*x(3);

f = -f;

end

function [c,ceq,K1,K2,s] = ex10_13b(x,s)

if isnan(s(1,1))

s = [0.2 0;0.2 0];

end

% 采样的数据点

w1 = 1:s(1,1):100;

w2 = 1:s(2,1):100;

% 半无限约束

K1 = sin(w1*x(1)).*cos(w1*x(1))-1/1000*(w1-45).^2-sin(w1*x(3))-x(2)-1;

K2 = sin(w2*x(2)).*cos(w2*x(2))-1/1000*(w2-45).^2-sin(w2*x(3))-x(3)-1;

% 非线性约束

c = 3-1.5*x(3)^2-x(2)^2;

ceq = [];

end

%% 实验结果

% ex10_13

% Solver stopped prematurely.

% fseminf stopped because it exceeded the function evaluation limit,

% options.MaxFunctionEvaluations = 3.000000e+02.

% x =

% 9.4493

% 2.0942

% 2.8347

% feval =

% -93.5225

cut-off

%% 10-14 求解以下的多目标最小化问题

% x = (A+BKC)x+Bu

% y = Cx

% 其中

% A = [-0.5 0 0;0 -2 10;0 1 -2]

% B = [1 0;-2 2;0 1]

% C = [1 0 0;0 0 1]

% K0 = [-1 -1;-1 -1]

% goal = [-5 -3 -1]

% help fgoalattain

% fgoalattain - 求解涉及多目标的目标达到问题

% fgoalattain 求解目标达到问题，这是多目标优化问题最小化的一种表示。

% x = fgoalattain(fun,x0,goal,weight)

% x = fgoalattain(fun,x0,goal,weight,A,b)

% x = fgoalattain(fun,x0,goal,weight,A,b,Aeq,beq)

% x = fgoalattain(fun,x0,goal,weight,A,b,Aeq,beq,lb,ub)

% x = fgoalattain(fun,x0,goal,weight,A,b,Aeq,beq,lb,ub,nonlcon)

% x = fgoalattain(fun,x0,goal,weight,A,b,Aeq,beq,lb,ub,nonlcon,options)

% x = fgoalattain(problem)

% [x,fval] = fgoalattain(___)

% [x,fval,attainfactor,exitflag,output] = fgoalattain(___)

% [x,fval,attainfactor,exitflag,output,lambda] = fgoalattain(___)

A = [-0.5 0 0;0 -2 10;0 1 -2]

B = [1 0;-2 2;0 1]

C = [1 0 0;0 0 1]

K0 = [-1 -1;-1 -1] % 初始化控制矩阵

goal = [-5 -3 -1] % 优化目标特征值

weight = abs(goal) % 目标函数权重

lb = -4 * ones(size(K0)) % 决策变量下限

ub = 4 * ones(size(K0)) % 决策变量上限

options = optimset('Display', 'iter'); % 属性参数设置

[K, fval, attainfactor] = fgoalattain(@(K)ex10_14a(K,A,B,C),K0,goal,weight,[],[],[],[],lb,ub,[],options)

function F = ex10_14a(K,A,B,C)

F = sort(eig(A+B*K*C));

end

%% 实验结果

% ex10_14

% A =

% -0.5000 0 0

% 0 -2.0000 10.0000

% 0 1.0000 -2.0000

% B =

% 1 0

% -2 2

% 0 1

% C =

% 1 0 0

% 0 0 1

% K0 =

% -1 -1

% goal =

% -5 -3 -1

% weight =

% 5 3 1

% lb =

% -4 -4

% ub =

% 4 4

% Attainment Max Line search Directional

% Iter F-count factor constraint steplength derivative Procedure

% 0 6 0 1.88521

% 1 13 1.031 0.02998 1 0.745

% 2 20 0.3525 0.06863 1 -0.613

% 3 27 -0.1706 0.1071 1 -0.223 Hessian modified

% 4 34 -0.2236 0.06654 1 -0.234 Hessian modified twice

% 5 41 -0.3568 0.007894 1 -0.0812

% 6 48 -0.3645 0.000145 1 -0.164 Hessian modified

% 7 55 -0.3645 0 1 -0.00515 Hessian modified

% 8 62 -0.3675 0.0001546 1 -0.00812 Hessian modified twice

% 9 69 -0.3889 0.008328 1 -0.00751 Hessian modified

% 10 76 -0.3862 0 1 0.00568

% 11 83 -0.3863 2.955e-13 1 -0.998 Hessian modified twice

% Local minimum possible. Constraints satisfied.

% fgoalattain stopped because the size of the current search direction is less than

% twice the value of the step size tolerance and constraints are

% satisfied to within the value of the constraint tolerance.

% <stopping criteria details>

% K =

% -4.0000 -0.2564

% -4.0000 -4.0000

% fval =

% -6.9313

% -4.1588

% -1.4099

% attainfactor =

% -0.3863

cut-off

%% 10-15 在工程实验中，得到下面一组数据

% ti = [0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0]

% yi = [0 3.2 4.0 4.5 5.8 6.9 8.1 9.9 10.8]

% 求系数a,b,cd，使得函数f(t) = a + b*cos(t) + c*sin(t) + d*t^3

% help lsqcurvefit

% lsqcurvefit - 用最小二乘求解非线性曲线拟合（数据拟合）问题

% 非线性最小二乘求解器

% x = lsqcurvefit(fun,x0,xdata,ydata)

% x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub)

% x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,options)

% x = lsqcurvefit(problem)

% [x,resnorm] = lsqcurvefit(___)

% [x,resnorm,residual,exitflag,output] = lsqcurvefit(___)

% [x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqcurvefit(___)

ti = [0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0]

yi = [0 3.2 4.0 4.5 5.8 6.9 8.1 9.9 10.8]

x0 = [1 1 1 1]'

[x,resnorm,residual,exitflag,output,lambda,J] = lsqcurvefit(ex10_15a,x0,ti,yi)

function f = ex10_15a(x,ti)

n = length(ti);

for i = 1:n

f(i) = x(1) + x(2) * cos(ti(i)) + x(3) * sin(ti(i)) + x(4) * ti(i)^2;

end

%% 实验结果

% 输入参数的数目不足。

% 出错 ex10_15>ex10_15a (第 19 行)

% n = length(ti);

% 出错 ex10_15 (第 16 行)

% [x,resnorm,residual,exitflag,output,lambda,J] = lsqcurvefit(ex10_15a,x0,ti,yi)

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