{"id":108562,"date":"2026-09-22T07:08:46","date_gmt":"2026-09-21T23:08:46","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/108562.html"},"modified":"2026-09-22T07:08:46","modified_gmt":"2026-09-21T23:08:46","slug":"c%e4%bb%a3%e7%a0%81%e5%ae%9e%e7%8e%b0matlab%e4%b8%ad%e7%9a%84fgoalattain%e5%87%bd%e6%95%b0%e5%8a%9f%e8%83%bd","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/108562.html","title":{"rendered":"C++\u4ee3\u7801\u5b9e\u73b0MATLAB\u4e2d\u7684fgoalattain\u51fd\u6570\u529f\u80fd"},"content":{"rendered":"

\/\/ fgoalattain.cpp<\/span>
\n\/\/ \u96f6\u7b2c\u4e09\u65b9\u4f9d\u8d56：\u4ec5\u9700 C++17 \u6807\u51c6\u5e93<\/span>
\n\/\/ \u7f16\u8bd1: g++ -std=c++17 -O2 fgoalattain_standalone.cpp -o fgoalattain<\/span>
\n#<\/span>include<\/span> <iostream><\/span><\/span>
\n#<\/span>include<\/span> <vector><\/span><\/span>
\n#<\/span>include<\/span> <cmath><\/span><\/span>
\n#<\/span>include<\/span> <functional><\/span><\/span>
\n#<\/span>include<\/span> <algorithm><\/span><\/span>
\n#<\/span>include<\/span> <stdexcept><\/span><\/span>
\n#<\/span>include<\/span> <string><\/span><\/span>
\n#<\/span>include<\/span> <iomanip><\/span><\/span>
\nusing<\/span> Vec =<\/span> std::<\/span>vector<<\/span>double<\/span>><\/span>;<\/span>
\nusing<\/span> Mat =<\/span> std::<\/span>vector<<\/span>Vec><\/span>;<\/span>
\n\/\/ ================= \u57fa\u7840\u5411\u91cf\u8fd0\u7b97 =================<\/span>
\nVec operator<\/span>+<\/span>(<\/span>const<\/span> Vec&<\/span> a,<\/span> const<\/span> Vec&<\/span> b)<\/span> {<\/span>
\n Vec r<\/span>(<\/span>a.<\/span>size<\/span>(<\/span>)<\/span>)<\/span>;<\/span>
\n for<\/span> (<\/span>size_t i =<\/span> 0<\/span>;<\/span> i <<\/span> a.<\/span>size<\/span>(<\/span>)<\/span>;<\/span> ++<\/span>i)<\/span> r[<\/span>i]<\/span> =<\/span> a[<\/span>i]<\/span> +<\/span> b[<\/span>i]<\/span>;<\/span>
\n return<\/span> r;<\/span>
\n}<\/span>
\nVec operator<\/span>–<\/span>(<\/span>const<\/span> Vec&<\/span> a,<\/span> const<\/span> Vec&<\/span> b)<\/span> {<\/span>
\n Vec r<\/span>(<\/span>a.<\/span>size<\/span>(<\/span>)<\/span>)<\/span>;<\/span>
\n for<\/span> (<\/span>size_t i =<\/span> 0<\/span>;<\/span> i <<\/span> a.<\/span>size<\/span>(<\/span>)<\/span>;<\/span> ++<\/span>i)<\/span> r[<\/span>i]<\/span> =<\/span> a[<\/span>i]<\/span> –<\/span> b[<\/span>i]<\/span>;<\/span>
\n return<\/span> r;<\/span>
\n}<\/span>
\nVec operator<\/span>*<\/span>(<\/span>double<\/span> s,<\/span> const<\/span> Vec&<\/span> a)<\/span> {<\/span>
\n Vec r<\/span>(<\/span>a.<\/span>size<\/span>(<\/span>)<\/span>)<\/span>;<\/span>
\n for<\/span> (<\/span>size_t i =<\/span> 0<\/span>;<\/span> i <<\/span> a.<\/span>size<\/span>(<\/span>)<\/span>;<\/span> ++<\/span>i)<\/span> r[<\/span>i]<\/span> =<\/span> s *<\/span> a[<\/span>i]<\/span>;<\/span>
\n return<\/span> r;<\/span>
\n}<\/span>
\ndouble<\/span> dot<\/span>(<\/span>const<\/span> Vec&<\/span> a,<\/span> const<\/span> Vec&<\/span> b)<\/span> {<\/span>
\n double<\/span> s =<\/span> 0<\/span>;<\/span>
\n for<\/span> (<\/span>size_t i =<\/span> 0<\/span>;<\/span> i <<\/span> a.<\/span>size<\/span>(<\/span>)<\/span>;<\/span> ++<\/span>i)<\/span> s +=<\/span> a[<\/span>i]<\/span> *<\/span> b[<\/span>i]<\/span>;<\/span>
\n return<\/span> s;<\/span>
\n}<\/span>
\ndouble<\/span> norm2<\/span>(<\/span>const<\/span> Vec&<\/span> a)<\/span> {<\/span> return<\/span> std::<\/span>sqrt<\/span>(<\/span>dot<\/span>(<\/span>a,<\/span> a)<\/span>)<\/span>;<\/span> }<\/span>
\n\/\/ ================= \u9ad8\u65af\u6d88\u5143\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4 A x = b =================<\/span>
\nVec solve_lin<\/span>(<\/span>const<\/span> Mat&<\/span> A_,<\/span> const<\/span> Vec&<\/span> b_)<\/span> {<\/span>
\n int<\/span> n =<\/span> (<\/span>int<\/span>)<\/span>b_.<\/span>size<\/span>(<\/span>)<\/span>;<\/span>
\n Mat A =<\/span> A_;<\/span>
\n Vec b =<\/span> b_;<\/span>
\n for<\/span> (<\/span>int<\/span> k =<\/span> 0<\/span>;<\/span> k <<\/span> n;<\/span> ++<\/span>k)<\/span> {<\/span>
\n int<\/span> piv =<\/span> k;<\/span>
\n double<\/span> mx =<\/span> std::<\/span>abs<\/span>(<\/span>A[<\/span>k]<\/span>[<\/span>k]<\/span>)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> k +<\/span> 1<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span>
\n if<\/span> (<\/span>std::<\/span>abs<\/span>(<\/span>A[<\/span>i]<\/span>[<\/span>k]<\/span>)<\/span> ><\/span> mx)<\/span> {<\/span> mx =<\/span> std::<\/span>abs<\/span>(<\/span>A[<\/span>i]<\/span>[<\/span>k]<\/span>)<\/span>;<\/span> piv =<\/span> i;<\/span> }<\/span>
\n if<\/span> (<\/span>mx <<\/span> 1e-14<\/span>)<\/span> throw<\/span> std::<\/span>runtime_error<\/span>(<\/span>"Singular matrix"<\/span>)<\/span>;<\/span>
\n std::<\/span>swap<\/span>(<\/span>A[<\/span>k]<\/span>,<\/span> A[<\/span>piv]<\/span>)<\/span>;<\/span>
\n std::<\/span>swap<\/span>(<\/span>b[<\/span>k]<\/span>,<\/span> b[<\/span>piv]<\/span>)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> k +<\/span> 1<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> {<\/span>
\n double<\/span> m =<\/span> A[<\/span>i]<\/span>[<\/span>k]<\/span> \/<\/span> A[<\/span>k]<\/span>[<\/span>k]<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> j =<\/span> k;<\/span> j <<\/span> n;<\/span> ++<\/span>j)<\/span> A[<\/span>i]<\/span>[<\/span>j]<\/span> -=<\/span> m *<\/span> A[<\/span>k]<\/span>[<\/span>j]<\/span>;<\/span>
\n b[<\/span>i]<\/span> -=<\/span> m *<\/span> b[<\/span>k]<\/span>;<\/span>
\n }<\/span>
\n }<\/span>
\n Vec x<\/span>(<\/span>n)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> n –<\/span> 1<\/span>;<\/span> i >=<\/span> 0<\/span>;<\/span> —<\/span>i)<\/span> {<\/span>
\n double<\/span> s =<\/span> b[<\/span>i]<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> j =<\/span> i +<\/span> 1<\/span>;<\/span> j <<\/span> n;<\/span> ++<\/span>j)<\/span> s -=<\/span> A[<\/span>i]<\/span>[<\/span>j]<\/span> *<\/span> x[<\/span>j]<\/span>;<\/span>
\n x[<\/span>i]<\/span> =<\/span> s \/<\/span> A[<\/span>i]<\/span>[<\/span>i]<\/span>;<\/span>
\n }<\/span>
\n return<\/span> x;<\/span>
\n}<\/span>
\n\/\/ ================= \u95ee\u9898\u5b9a\u4e49 =================<\/span>
\nstruct<\/span> GoalAttainProblem<\/span> {<\/span>
\n int<\/span> n_vars =<\/span> 0<\/span>;<\/span> \/\/ \u8bbe\u8ba1\u53d8\u91cf x \u7684\u7ef4\u5ea6<\/span>
\n int<\/span> n_objectives =<\/span> 0<\/span>;<\/span> \/\/ \u76ee\u6807\u51fd\u6570 F \u7684\u7ef4\u5ea6<\/span>
\n Vec goal;<\/span> \/\/ \u76ee\u6807\u5411\u91cf<\/span>
\n Vec weight;<\/span> \/\/ \u6743\u91cd\u5411\u91cf<\/span>
\n std::<\/span>function<<\/span>Vec<\/span>(<\/span>const<\/span> Vec&<\/span>)<\/span>><\/span> F;<\/span> \/\/ \u76ee\u6807\u51fd\u6570 F(x)<\/span>
\n std::<\/span>function<<\/span>Vec<\/span>(<\/span>const<\/span> Vec&<\/span>)<\/span>><\/span> nonlin_ineq;<\/span> \/\/ \u975e\u7ebf\u6027\u4e0d\u7b49\u5f0f c(x) <= 0<\/span>
\n std::<\/span>function<<\/span>Vec<\/span>(<\/span>const<\/span> Vec&<\/span>)<\/span>><\/span> nonlin_eq;<\/span> \/\/ \u975e\u7ebf\u6027\u7b49\u5f0f ceq(x) = 0<\/span>
\n Vec lb,<\/span> ub;<\/span> \/\/ \u53d8\u91cf\u4e0a\u4e0b\u754c<\/span>
\n}<\/span>;<\/span>
\n\/\/ ================= \u6c42\u89e3\u7ed3\u679c =================<\/span>
\nstruct<\/span> SolveResult<\/span> {<\/span>
\n Vec x;<\/span> \/\/ \u6700\u4f18\u89e3<\/span>
\n Vec fval;<\/span> \/\/ F(x*)<\/span>
\n double<\/span> gamma;<\/span> \/\/ \u8fbe\u5230\u56e0\u5b50<\/span>
\n bool<\/span> success =<\/span> false<\/span>;<\/span>
\n std::<\/span>string msg;<\/span>
\n}<\/span>;<\/span>
\n\/\/ ================= \u4e3b\u6c42\u89e3\u51fd\u6570：fgoalattain =================<\/span>
\nSolveResult fgoalattain<\/span>(<\/span>const<\/span> GoalAttainProblem&<\/span> P,<\/span> const<\/span> Vec&<\/span> x0,<\/span>
\n bool<\/span> verbose =<\/span> false<\/span>)<\/span>
\n{<\/span>
\n const<\/span> int<\/span> nv =<\/span> P.<\/span>n_vars;<\/span>
\n const<\/span> int<\/span> no =<\/span> P.<\/span>n_objectives;<\/span>
\n const<\/span> int<\/span> n =<\/span> nv +<\/span> 1<\/span>;<\/span> \/\/ \u589e\u5e7f\u53d8\u91cf z = [x; gamma]<\/span>
\n \/\/ \u63a2\u6d4b\u7528\u6237\u7ea6\u675f\u7684\u7ef4\u5ea6<\/span>
\n Vec probe<\/span>(<\/span>nv,<\/span> 0.0<\/span>)<\/span>;<\/span>
\n const<\/span> int<\/span> n_user_ineq =<\/span> (<\/span>int<\/span>)<\/span>P.<\/span>nonlin_ineq<\/span>(<\/span>probe)<\/span>.<\/span>size<\/span>(<\/span>)<\/span>;<\/span>
\n const<\/span> int<\/span> n_user_eq =<\/span> (<\/span>int<\/span>)<\/span>P.<\/span>nonlin_eq<\/span>(<\/span>probe)<\/span>.<\/span>size<\/span>(<\/span>)<\/span>;<\/span>
\n \/\/ \u4e0d\u7b49\u5f0f\u7ea6\u675f\u51fd\u6570（\u8fd4\u56de c(z) <= 0）<\/span>
\n auto<\/span> eval_ineq =<\/span> [<\/span>&<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span> z)<\/span> -><\/span> Vec {<\/span>
\n Vec x<\/span>(<\/span>z.<\/span>begin<\/span>(<\/span>)<\/span>,<\/span> z.<\/span>begin<\/span>(<\/span>)<\/span> +<\/span> nv)<\/span>;<\/span>
\n double<\/span> g =<\/span> z[<\/span>nv]<\/span>;<\/span>
\n Vec Fx =<\/span> P.<\/span>F<\/span>(<\/span>x)<\/span>;<\/span>
\n Vec c<\/span>(<\/span>no +<\/span> n_user_ineq)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> no;<\/span> ++<\/span>i)<\/span>
\n c[<\/span>i]<\/span> =<\/span> Fx[<\/span>i]<\/span> –<\/span> P.<\/span>weight[<\/span>i]<\/span> *<\/span> g –<\/span> P.<\/span>goal[<\/span>i]<\/span>;<\/span>
\n Vec u =<\/span> P.<\/span>nonlin_ineq<\/span>(<\/span>x)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n_user_ineq;<\/span> ++<\/span>i)<\/span> c[<\/span>no +<\/span> i]<\/span> =<\/span> u[<\/span>i]<\/span>;<\/span>
\n return<\/span> c;<\/span>
\n }<\/span>;<\/span>
\n \/\/ \u7b49\u5f0f\u7ea6\u675f\u51fd\u6570<\/span>
\n auto<\/span> eval_eq =<\/span> [<\/span>&<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span> z)<\/span> -><\/span> Vec {<\/span>
\n Vec x<\/span>(<\/span>z.<\/span>begin<\/span>(<\/span>)<\/span>,<\/span> z.<\/span>begin<\/span>(<\/span>)<\/span> +<\/span> nv)<\/span>;<\/span>
\n return<\/span> P.<\/span>nonlin_eq<\/span>(<\/span>x)<\/span>;<\/span>
\n }<\/span>;<\/span>
\n \/\/ ———- \u6784\u9020\u521d\u59cb\u70b9：\u4fdd\u8bc1\u4e25\u683c\u53ef\u884c ———-<\/span>
\n Vec z<\/span>(<\/span>n)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> nv;<\/span> ++<\/span>i)<\/span> z[<\/span>i]<\/span> =<\/span> x0[<\/span>i]<\/span>;<\/span>
\n Vec F0 =<\/span> P.<\/span>F<\/span>(<\/span>x0)<\/span>;<\/span>
\n double<\/span> gmin =<\/span> –<\/span>1e300<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> no;<\/span> ++<\/span>i)<\/span> {<\/span>
\n if<\/span> (<\/span>std::<\/span>abs<\/span>(<\/span>P.<\/span>weight[<\/span>i]<\/span>)<\/span> ><\/span> 1e-15<\/span>)<\/span> {<\/span>
\n double<\/span> gv =<\/span> (<\/span>F0[<\/span>i]<\/span> –<\/span> P.<\/span>goal[<\/span>i]<\/span>)<\/span> \/<\/span> P.<\/span>weight[<\/span>i]<\/span>;<\/span>
\n if<\/span> (<\/span>gv ><\/span> gmin)<\/span> gmin =<\/span> gv;<\/span>
\n }<\/span>
\n }<\/span>
\n if<\/span> (<\/span>gmin <<\/span> –<\/span>1e299<\/span>)<\/span> gmin =<\/span> 0.0<\/span>;<\/span>
\n z[<\/span>nv]<\/span> =<\/span> gmin +<\/span> 1.0<\/span>;<\/span>
\n \/\/ \u589e\u5927 gamma \u76f4\u5230\u76ee\u6807\u8fbe\u5230\u7ea6\u675f\u5168\u90e8\u6ee1\u8db3<\/span>
\n for<\/span> (<\/span>int<\/span> t =<\/span> 0<\/span>;<\/span> t <<\/span> 200<\/span>;<\/span> ++<\/span>t)<\/span> {<\/span>
\n Vec c =<\/span> eval_ineq<\/span>(<\/span>z)<\/span>;<\/span>
\n double<\/span> mx =<\/span> –<\/span>1e300<\/span>;<\/span>
\n for<\/span> (<\/span>double<\/span> v :<\/span> c)<\/span> mx =<\/span> std::<\/span>max<\/span>(<\/span>mx,<\/span> v)<\/span>;<\/span>
\n if<\/span> (<\/span>mx <<\/span> –<\/span>1e-3<\/span>)<\/span> break<\/span>;<\/span>
\n z[<\/span>nv]<\/span> +=<\/span> std::<\/span>max<\/span>(<\/span>1.0<\/span>,<\/span> 2.0<\/span> *<\/span> mx)<\/span>;<\/span>
\n }<\/span>
\n \/\/ \u6700\u7ec8\u53ef\u884c\u6027\u68c0\u67e5<\/span>
\n {<\/span>
\n Vec c =<\/span> eval_ineq<\/span>(<\/span>z)<\/span>;<\/span>
\n bool<\/span> feas =<\/span> true<\/span>;<\/span>
\n for<\/span> (<\/span>double<\/span> v :<\/span> c)<\/span> if<\/span> (<\/span>v >=<\/span> –<\/span>1e-10<\/span>)<\/span> {<\/span> feas =<\/span> false<\/span>;<\/span> break<\/span>;<\/span> }<\/span>
\n if<\/span> (<\/span>!<\/span>feas)<\/span> {<\/span>
\n SolveResult r;<\/span>
\n r.<\/span>x =<\/span> x0;<\/span> r.<\/span>fval =<\/span> F0;<\/span> r.<\/span>gamma =<\/span> z[<\/span>nv]<\/span>;<\/span>
\n r.<\/span>success =<\/span> false<\/span>;<\/span>
\n r.<\/span>msg =<\/span> "Initial point not strictly feasible for user constraints"<\/span>;<\/span>
\n return<\/span> r;<\/span>
\n }<\/span>
\n }<\/span>
\n \/\/ ———- \u5185\u70b9\u6cd5\u4e3b\u5faa\u73af ———-<\/span>
\n double<\/span> mu =<\/span> 1.0<\/span>;<\/span>
\n const<\/span> double<\/span> mu_floor =<\/span> 1e-9<\/span>;<\/span>
\n const<\/span> int<\/span> outer_max =<\/span> 50<\/span>;<\/span>
\n const<\/span> double<\/span> inner_tol =<\/span> 1e-8<\/span>;<\/span>
\n \/\/ \u969c\u788d\u76ee\u6807\u51fd\u6570 phi_mu(z) = gamma – mu*sum log(-c_i(z)) + \u7b49\u5f0f\u60e9\u7f5a<\/span>
\n auto<\/span> phi =<\/span> [<\/span>&<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span> zz)<\/span> -><\/span> double<\/span> {<\/span>
\n Vec c =<\/span> eval_ineq<\/span>(<\/span>zz)<\/span>;<\/span>
\n double<\/span> v =<\/span> zz[<\/span>nv]<\/span>;<\/span>
\n double<\/span> pen =<\/span> 0.0<\/span>,<\/span> infeas =<\/span> 0.0<\/span>;<\/span>
\n for<\/span> (<\/span>double<\/span> ci :<\/span> c)<\/span> {<\/span>
\n if<\/span> (<\/span>ci >=<\/span> 0<\/span>)<\/span> infeas +=<\/span> ci *<\/span> ci;<\/span>
\n else<\/span> pen -=<\/span> mu *<\/span> std::<\/span>log<\/span>(<\/span>–<\/span>ci)<\/span>;<\/span>
\n }<\/span>
\n Vec eq =<\/span> eval_eq<\/span>(<\/span>zz)<\/span>;<\/span>
\n double<\/span> eqpen =<\/span> 0.0<\/span>;<\/span>
\n for<\/span> (<\/span>double<\/span> h :<\/span> eq)<\/span> eqpen +=<\/span> h *<\/span> h;<\/span>
\n \/\/ \u4e0d\u53ef\u884c\u65f6\u7ed9\u4e00\u4e2a\u5927\u7684\u5149\u6ed1\u60e9\u7f5a，\u628a\u8fed\u4ee3\u62c9\u56de\u53ef\u884c\u57df<\/span>
\n return<\/span> v +<\/span> pen +<\/span> (<\/span>1.0<\/span> \/<\/span> mu)<\/span> *<\/span> eqpen +<\/span> 1e6<\/span> *<\/span> infeas;<\/span>
\n }<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> outer =<\/span> 0<\/span>;<\/span> outer <<\/span> outer_max;<\/span> ++<\/span>outer)<\/span> {<\/span>
\n \/\/ — \u5185\u5c42 Newton \u8fed\u4ee3 —<\/span>
\n for<\/span> (<\/span>int<\/span> it =<\/span> 0<\/span>;<\/span> it <<\/span> 200<\/span>;<\/span> ++<\/span>it)<\/span> {<\/span>
\n double<\/span> f0 =<\/span> phi<\/span>(<\/span>z)<\/span>;<\/span>
\n if<\/span> (<\/span>f0 ><\/span> 1e100<\/span>)<\/span> break<\/span>;<\/span>
\n \/\/ \u6570\u503c\u68af\u5ea6（\u4e2d\u5fc3\u5dee\u5206）<\/span>
\n Vec g<\/span>(<\/span>n,<\/span> 0.0<\/span>)<\/span>;<\/span>
\n const<\/span> double<\/span> eps =<\/span> 1e-7<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> {<\/span>
\n double<\/span> h =<\/span> eps *<\/span> std::<\/span>max<\/span>(<\/span>1.0<\/span>,<\/span> std::<\/span>abs<\/span>(<\/span>z[<\/span>i]<\/span>)<\/span>)<\/span>;<\/span>
\n Vec zp =<\/span> z,<\/span> zm =<\/span> z;<\/span>
\n zp[<\/span>i]<\/span> +=<\/span> h;<\/span>
\n zm[<\/span>i]<\/span> -=<\/span> h;<\/span>
\n g[<\/span>i]<\/span> =<\/span> (<\/span>phi<\/span>(<\/span>zp)<\/span> –<\/span> phi<\/span>(<\/span>zm)<\/span>)<\/span> \/<\/span> (<\/span>2.0<\/span> *<\/span> h)<\/span>;<\/span>
\n }<\/span>
\n if<\/span> (<\/span>norm2<\/span>(<\/span>g)<\/span> <<\/span> inner_tol *<\/span> (<\/span>1.0<\/span> +<\/span> std::<\/span>abs<\/span>(<\/span>f0)<\/span>)<\/span>)<\/span> break<\/span>;<\/span>
\n \/\/ \u6570\u503c Hessian（\u4e2d\u5fc3\u5dee\u5206）<\/span>
\n Mat H<\/span>(<\/span>n,<\/span> Vec<\/span>(<\/span>n,<\/span> 0.0<\/span>)<\/span>)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> {<\/span>
\n for<\/span> (<\/span>int<\/span> j =<\/span> i;<\/span> j <<\/span> n;<\/span> ++<\/span>j)<\/span> {<\/span>
\n double<\/span> hi =<\/span> eps *<\/span> std::<\/span>max<\/span>(<\/span>1.0<\/span>,<\/span> std::<\/span>abs<\/span>(<\/span>z[<\/span>i]<\/span>)<\/span>)<\/span>;<\/span>
\n double<\/span> hj =<\/span> eps *<\/span> std::<\/span>max<\/span>(<\/span>1.0<\/span>,<\/span> std::<\/span>abs<\/span>(<\/span>z[<\/span>j]<\/span>)<\/span>)<\/span>;<\/span>
\n Vec zpp =<\/span> z,<\/span> zpm =<\/span> z,<\/span> zmp =<\/span> z,<\/span> zmm =<\/span> z;<\/span>
\n zpp[<\/span>i]<\/span> +=<\/span> hi;<\/span> zpp[<\/span>j]<\/span> +=<\/span> hj;<\/span>
\n zpm[<\/span>i]<\/span> +=<\/span> hi;<\/span> zpm[<\/span>j]<\/span> -=<\/span> hj;<\/span>
\n zmp[<\/span>i]<\/span> -=<\/span> hi;<\/span> zmp[<\/span>j]<\/span> +=<\/span> hj;<\/span>
\n zmm[<\/span>i]<\/span> -=<\/span> hi;<\/span> zmm[<\/span>j]<\/span> -=<\/span> hj;<\/span>
\n double<\/span> val =<\/span> (<\/span>phi<\/span>(<\/span>zpp)<\/span> –<\/span> phi<\/span>(<\/span>zpm)<\/span> –<\/span> phi<\/span>(<\/span>zmp)<\/span> +<\/span> phi<\/span>(<\/span>zmm)<\/span>)<\/span>
\n \/<\/span> (<\/span>4.0<\/span> *<\/span> hi *<\/span> hj)<\/span>;<\/span>
\n H[<\/span>i]<\/span>[<\/span>j]<\/span> =<\/span> H[<\/span>j]<\/span>[<\/span>i]<\/span> =<\/span> val;<\/span>
\n }<\/span>
\n }<\/span>
\n \/\/ \u5bf9\u89d2\u6b63\u5219\u5316（\u4fdd\u8bc1\u53ef\u89e3）<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> H[<\/span>i]<\/span>[<\/span>i]<\/span> +=<\/span> 1e-7<\/span>;<\/span>
\n \/\/ \u725b\u987f\u65b9\u5411 H d = -g<\/span>
\n Vec rhs<\/span>(<\/span>n)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> rhs[<\/span>i]<\/span> =<\/span> –<\/span>g[<\/span>i]<\/span>;<\/span>
\n Vec d;<\/span>
\n try<\/span> {<\/span>
\n d =<\/span> solve_lin<\/span>(<\/span>H,<\/span> rhs)<\/span>;<\/span>
\n }<\/span> catch<\/span> (<\/span>.<\/span>.<\/span>.<\/span>)<\/span> {<\/span>
\n \/\/ \u56de\u9000\u5230\u6700\u901f\u4e0b\u964d<\/span>
\n d =<\/span> –<\/span>1.0<\/span> *<\/span> g;<\/span>
\n double<\/span> dn =<\/span> norm2<\/span>(<\/span>d)<\/span>;<\/span>
\n if<\/span> (<\/span>dn ><\/span> 1e-12<\/span>)<\/span> d =<\/span> (<\/span>1.0<\/span> \/<\/span> dn)<\/span> *<\/span> d;<\/span>
\n }<\/span>
\n \/\/ \u9650\u5236\u6b65\u957f，\u9632\u6b62\u6570\u503c\u6ea2\u51fa<\/span>
\n double<\/span> dn =<\/span> norm2<\/span>(<\/span>d)<\/span>;<\/span>
\n if<\/span> (<\/span>dn ><\/span> 5.0<\/span>)<\/span> d =<\/span> (<\/span>5.0<\/span> \/<\/span> dn)<\/span> *<\/span> d;<\/span>
\n \/\/ \u56de\u6eaf\u7ebf\u641c\u7d22（\u4fdd\u8bc1\u53ef\u884c + \u4e0b\u964d）<\/span>
\n double<\/span> alpha =<\/span> 1.0<\/span>;<\/span>
\n bool<\/span> ok =<\/span> false<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> ls =<\/span> 0<\/span>;<\/span> ls <<\/span> 60<\/span>;<\/span> ++<\/span>ls)<\/span> {<\/span>
\n Vec znew =<\/span> z +<\/span> alpha *<\/span> d;<\/span>
\n Vec c =<\/span> eval_ineq<\/span>(<\/span>znew)<\/span>;<\/span>
\n bool<\/span> feas =<\/span> true<\/span>;<\/span>
\n for<\/span> (<\/span>double<\/span> ci :<\/span> c)<\/span> if<\/span> (<\/span>ci >=<\/span> –<\/span>1e-14<\/span>)<\/span> {<\/span> feas =<\/span> false<\/span>;<\/span> break<\/span>;<\/span> }<\/span>
\n if<\/span> (<\/span>feas)<\/span> {<\/span>
\n double<\/span> fn =<\/span> phi<\/span>(<\/span>znew)<\/span>;<\/span>
\n if<\/span> (<\/span>fn <<\/span> f0)<\/span> {<\/span> z =<\/span> znew;<\/span> ok =<\/span> true<\/span>;<\/span> break<\/span>;<\/span> }<\/span>
\n }<\/span>
\n alpha *=<\/span> 0.5<\/span>;<\/span>
\n }<\/span>
\n if<\/span> (<\/span>!<\/span>ok)<\/span> break<\/span>;<\/span>
\n }<\/span>
\n if<\/span> (<\/span>verbose)<\/span>
\n std::<\/span>cout <<<\/span> " [outer "<\/span> <<<\/span> outer <<<\/span> "] mu="<\/span> <<<\/span> mu
\n <<<\/span> ", gamma="<\/span> <<<\/span> z[<\/span>nv]<\/span> <<<\/span> std::<\/span>endl;<\/span>
\n mu *=<\/span> 0.25<\/span>;<\/span>
\n if<\/span> (<\/span>mu <<\/span> mu_floor)<\/span> break<\/span>;<\/span>
\n }<\/span>
\n SolveResult r;<\/span>
\n r.<\/span>x.<\/span>assign<\/span>(<\/span>z.<\/span>begin<\/span>(<\/span>)<\/span>,<\/span> z.<\/span>begin<\/span>(<\/span>)<\/span> +<\/span> nv)<\/span>;<\/span>
\n r.<\/span>fval =<\/span> P.<\/span>F<\/span>(<\/span>r.<\/span>x)<\/span>;<\/span>
\n r.<\/span>gamma =<\/span> z[<\/span>nv]<\/span>;<\/span>
\n r.<\/span>success =<\/span> true<\/span>;<\/span>
\n r.<\/span>msg =<\/span> "OK"<\/span>;<\/span>
\n return<\/span> r;<\/span>
\n}<\/span>
\n\/\/ ======================= \u6d4b\u8bd5\u4e3b\u7a0b\u5e8f =======================<\/span>
\nint<\/span> main<\/span>(<\/span>)<\/span> {<\/span>
\n std::<\/span>cout <<<\/span> std::<\/span>fixed <<<\/span> std::<\/span>setprecision<\/span>(<\/span>6<\/span>)<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> "================================================\\\\n"<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> " fgoalattain \u72ec\u7acb\u5b9e\u73b0（\u96f6\u5916\u90e8\u4f9d\u8d56）\\\\n"<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> "================================================\\\\n\\\\n"<\/span>;<\/span>
\n \/\/ ———- \u793a\u4f8b 1：MATLAB \u6587\u6863\u793a\u4f8b ———-<\/span>
\n {<\/span>
\n std::<\/span>cout <<<\/span> "— Example 1: MATLAB \u6587\u6863\u793a\u4f8b —\\\\n"<\/span>;<\/span>
\n GoalAttainProblem P;<\/span>
\n P.<\/span>n_vars =<\/span> 1<\/span>;<\/span>
\n P.<\/span>n_objectives =<\/span> 2<\/span>;<\/span>
\n P.<\/span>goal =<\/span> {<\/span>3.0<\/span>,<\/span> 6.0<\/span>}<\/span>;<\/span>
\n P.<\/span>weight =<\/span> {<\/span>1.0<\/span>,<\/span> 1.0<\/span>}<\/span>;<\/span>
\n P.<\/span>F =<\/span> [<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span> x)<\/span> -><\/span> Vec {<\/span>
\n double<\/span> t =<\/span> x[<\/span>0<\/span>]<\/span>;<\/span>
\n return<\/span> {<\/span> 2.0<\/span> +<\/span> (<\/span>t –<\/span> 3.0<\/span>)<\/span> *<\/span> (<\/span>t –<\/span> 3.0<\/span>)<\/span>,<\/span> 5.0<\/span> +<\/span> t *<\/span> t \/<\/span> 4.0<\/span> }<\/span>;<\/span>
\n }<\/span>;<\/span>
\n P.<\/span>nonlin_ineq =<\/span> [<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span>)<\/span> {<\/span> return<\/span> Vec{<\/span>}<\/span>;<\/span> }<\/span>;<\/span>
\n P.<\/span>nonlin_eq =<\/span> [<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span>)<\/span> {<\/span> return<\/span> Vec{<\/span>}<\/span>;<\/span> }<\/span>;<\/span>
\n P.<\/span>lb =<\/span> {<\/span>–<\/span>1e10<\/span>}<\/span>;<\/span>
\n P.<\/span>ub =<\/span> {<\/span> 1e10<\/span>}<\/span>;<\/span>
\n Vec x0 =<\/span> {<\/span>1.0<\/span>}<\/span>;<\/span>
\n SolveResult r =<\/span> fgoalattain<\/span>(<\/span>P,<\/span> x0)<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> " x* = "<\/span> <<<\/span> r.<\/span>x[<\/span>0<\/span>]<\/span> <<<\/span> "\\\\n"<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> " F(x*) = ["<\/span> <<<\/span> r.<\/span>fval[<\/span>0<\/span>]<\/span>
\n <<<\/span> ", "<\/span> <<<\/span> r.<\/span>fval[<\/span>1<\/span>]<\/span> <<<\/span> "]\\\\n"<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> " attainfactor = "<\/span> <<<\/span> r.<\/span>gamma <<<\/span> "\\\\n"<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> " success = "<\/span>
\n <<<\/span> (<\/span>r.<\/span>success ?<\/span> "true"<\/span> :<\/span> "false"<\/span>)<\/span> <<<\/span> "\\\\n\\\\n"<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> " \u671f\u671b\u7ed3\u679c: x*=2, F*=[3,6], gamma=0\\\\n\\\\n"<\/span>;<\/span>
\n }<\/span>
\n \/\/ ———- \u793a\u4f8b 2：\u5e26\u975e\u7ebf\u6027\u4e0d\u7b49\u5f0f\u7ea6\u675f ———-<\/span>
\n {<\/span>
\n std::<\/span>cout <<<\/span> "— Example 2: \u5e26\u7ea6\u675f —\\\\n"<\/span>;<\/span>
\n GoalAttainProblem P;<\/span>
\n P.<\/span>n_vars =<\/span> 2<\/span>;<\/span>
\n P.<\/span>n_objectives =<\/span> 2<\/span>;<\/span>
\n P.<\/span>goal =<\/span> {<\/span>0.0<\/span>,<\/span> 0.0<\/span>}<\/span>;<\/span>
\n P.<\/span>weight =<\/span> {<\/span>1.0<\/span>,<\/span> 1.0<\/span>}<\/span>;<\/span>
\n P.<\/span>F =<\/span> [<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span> x)<\/span> -><\/span> Vec {<\/span>
\n return<\/span> {<\/span> x[<\/span>0<\/span>]<\/span> *<\/span> x[<\/span>0<\/span>]<\/span> +<\/span> x[<\/span>1<\/span>]<\/span> *<\/span> x[<\/span>1<\/span>]<\/span>,<\/span>
\n (<\/span>x[<\/span>0<\/span>]<\/span> –<\/span> 2.0<\/span>)<\/span> *<\/span> (<\/span>x[<\/span>0<\/span>]<\/span> –<\/span> 2.0<\/span>)<\/span> +<\/span> x[<\/span>1<\/span>]<\/span> *<\/span> x[<\/span>1<\/span>]<\/span> }<\/span>;<\/span>
\n }<\/span>;<\/span>
\n \/\/ x0 + x1 >= 1 -> 1 – x0 – x1 <= 0<\/span>
\n P.<\/span>nonlin_ineq =<\/span> [<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span> x)<\/span> -><\/span> Vec {<\/span>
\n return<\/span> {<\/span> 1.0<\/span> –<\/span> x[<\/span>0<\/span>]<\/span> –<\/span> x[<\/span>1<\/span>]<\/span> }<\/span>;<\/span>
\n }<\/span>;<\/span>
\n P.<\/span>nonlin_eq =<\/span> [<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span>)<\/span> {<\/span> return<\/span> Vec{<\/span>}<\/span>;<\/span> }<\/span>;<\/span>
\n P.<\/span>lb =<\/span> {<\/span>–<\/span>10.0<\/span>,<\/span> –<\/span>10.0<\/span>}<\/span>;<\/span>
\n P.<\/span>ub =<\/span> {<\/span> 10.0<\/span>,<\/span> 10.0<\/span>}<\/span>;<\/span>
\n Vec x0 =<\/span> {<\/span>1.0<\/span>,<\/span> 1.0<\/span>}<\/span>;<\/span>
\n SolveResult r =<\/span> fgoalattain<\/span>(<\/span>P,<\/span> x0)<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> " x* = ["<\/span> <<<\/span> r.<\/span>x[<\/span>0<\/span>]<\/span>
\n <<<\/span> ", "<\/span> <<<\/span> r.<\/span>x[<\/span>1<\/span>]<\/span> <<<\/span> "]\\\\n"<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> " F(x*) = ["<\/span> <<<\/span> r.<\/span>fval[<\/span>0<\/span>]<\/span>
\n <<<\/span> ", "<\/span> <<<\/span> r.<\/span>fval[<\/span>1<\/span>]<\/span> <<<\/span> "]\\\\n"<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> " attainfactor = "<\/span> <<<\/span> r.<\/span>gamma <<<\/span> "\\\\n"<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> " success = "<\/span>
\n <<<\/span> (<\/span>r.<\/span>success ?<\/span> "true"<\/span> :<\/span> "false"<\/span>)<\/span> <<<\/span> "\\\\n\\\\n"<\/span>;<\/span>
\n }<\/span>
\n \/\/ ———- \u793a\u4f8b 3：\u4e0d\u540c\u6743\u91cd ———-<\/span>
\n {<\/span>
\n std::<\/span>cout <<<\/span> "— Example 3: \u6743\u91cd [1, 2] —\\\\n"<\/span>;<\/span>
\n GoalAttainProblem P;<\/span>
\n P.<\/span>n_vars =<\/span> 1<\/span>;<\/span>
\n P.<\/span>n_objectives =<\/span> 2<\/span>;<\/span>
\n P.<\/span>goal =<\/span> {<\/span>3.0<\/span>,<\/span> 6.0<\/span>}<\/span>;<\/span>
\n P.<\/span>weight =<\/span> {<\/span>1.0<\/span>,<\/span> 2.0<\/span>}<\/span>;<\/span> \/\/ \u5bf9\u7b2c\u4e8c\u4e2a\u76ee\u6807\u8d4b\u4e88\u66f4\u5927\u6743\u91cd<\/span>
\n P.<\/span>F =<\/span> [<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span> x)<\/span> -><\/span> Vec {<\/span>
\n double<\/span> t =<\/span> x[<\/span>0<\/span>]<\/span>;<\/span>
\n return<\/span> {<\/span> 2.0<\/span> +<\/span> (<\/span>t –<\/span> 3.0<\/span>)<\/span> *<\/span> (<\/span>t –<\/span> 3.0<\/span>)<\/span>,<\/span> 5.0<\/span> +<\/span> t *<\/span> t \/<\/span> 4.0<\/span> }<\/span>;<\/span>
\n }<\/span>;<\/span>
\n P.<\/span>nonlin_ineq =<\/span> [<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span>)<\/span> {<\/span> return<\/span> Vec{<\/span>}<\/span>;<\/span> }<\/span>;<\/span>
\n P.<\/span>nonlin_eq =<\/span> [<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span>)<\/span> {<\/span> return<\/span> Vec{<\/span>}<\/span>;<\/span> }<\/span>;<\/span>
\n P.<\/span>lb =<\/span> {<\/span>–<\/span>1e10<\/span>}<\/span>;<\/span>
\n P.<\/span>ub =<\/span> {<\/span> 1e10<\/span>}<\/span>;<\/span>
\n Vec x0 =<\/span> {<\/span>1.0<\/span>}<\/span>;<\/span>
\n SolveResult r =<\/span> fgoalattain<\/span>(<\/span>P,<\/span> x0)<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> " x* = "<\/span> <<<\/span> r.<\/span>x[<\/span>0<\/span>]<\/span> <<<\/span> "\\\\n"<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> " F(x*) = ["<\/span> <<<\/span> r.<\/span>fval[<\/span>0<\/span>]<\/span>
\n <<<\/span> ", "<\/span> <<<\/span> r.<\/span>fval[<\/span>1<\/span>]<\/span> <<<\/span> "]\\\\n"<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> " attainfactor = "<\/span> <<<\/span> r.<\/span>gamma <<<\/span> "\\\\n"<\/span>;<\/span>
\n std::<\/span>cout <<<\/span> " success = "<\/span>
\n <<<\/span> (<\/span>r.<\/span>success ?<\/span> "true"<\/span> :<\/span> "false"<\/span>)<\/span> <<<\/span> "\\\\n\\\\n"<\/span>;<\/span>
\n }<\/span>
\n return<\/span> 0<\/span>;<\/span>
\n}<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

\/\/ fgoalattain.cpp
\n\/\/ \u96f6\u7b2c\u4e09\u65b9\u4f9d\u8d56：\u4ec5\u9700 C17 \u6807\u51c6\u5e93
\n\/\/ \u7f16\u8bd1: g -stdc17 -O2 fgoalattain_standalone.cpp -o fgoalattain
\n#include
\n#include
\n#include
\n#include
\n#include
\n#include
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