求解下面的帶有等式約束和簡單的邊框約束(box constraints)的優化問題：

minx,y(x?1)2+(y?2)2s.t.0≤x≤3,1≤y≤4,2x+3y=5

% 求解下面的最小化問題： % min_{x,y} (x-1)^2 + (y-2)^2 % s.t. 0 \leq x \leq 3 % 1 \leq y \leq 4 % 2x + 3y = 5% augumented lagrangian function: % L_{rho}(x,y,lambda) = (x-1)^2 + (y-2)^2 + lambda*(2x + 3y -5) + rho/2(2x + 3y -5)^2 % solve x min f(x) = (x-1)^2 + lambda*(2x + 3y -5) + rho/2(2x + 3y -5)^2,s.t. 0<= x <=3 % solve y min f(y) = (y-2)^2 + lambda*(2x + 3y -5) + rho/2(2x + 3y -5)^2, s.t. 1<= y <=4 % sovle lambda :update lambda = lambda + rho(2x + 3y -5) % rho ,we set rho = min(rho_max,beta*rho) beta is a constant ,we set to 1.1,rho0 = 0.5;% x0,y0 都為1是一個可行解。 param.x0 = 1; param.y0 = 1; param.lambda = 1; param.maxIter = 30; param.beta = 1.1; % a constant param.rho = 0.5; param.rhomax = 2000;[Hx,Fx] = getHession_F('f1'); [Hy,Fy] = getHession_F('f2');param.Hx = Hx; param.Fx = Fx; param.Hy = Hy; param.Fy = Fy;% solve problem using admm algrithm [x,y] = solve_admm(param);% disp minimum disp(['[x,y]:' num2str(x) ',' num2str(y)]);function [H,F] = getHession_F(fn) % fn : function name % H :hessian matrix % F :一次項系數 syms x y lambda rho; if strcmp(fn,'f1')f = (x-1)^2 + lambda*(2*x + 3*y -5) + rho/2*(2*x + 3*y -5)^2;H = hessian(f,x);F = (2*lambda + (rho*(12*y - 20))/2 - 2);%%%%%%%%%%%%%%%%%%%%%%%%%%%%fcol = collect(f,{'x'});disp(fcol); elseif strcmp(fn,'f2')f = (y-2)^2 + lambda*(2*x + 3*y -5) + rho/2*(2*x + 3*y -5)^2;H = hessian(f,y);F = (3*lambda + (rho*(12*x - 30))/2 - 4);fcol = collect(f,{'y'});disp(fcol); end % fcol = collect(f,{'x'}); % fcol = collect(f,{'y'}); % disp(fcol); endfunction [x,y] = solve_admm(param)x = param.x0; y = param.y0; lambda = param.lambda; beta = param.beta; rho = param.rho; rhomax = param.rhomax; Hx = param.Hx; Fx = param.Fx; Hy = param.Hy; Fy = param.Fy; %% options = optimoptions('quadprog','Algorithm','interior-point-convex'); xlb = 0; xub = 3; ylb = 1; yub = 4; maxIter = param.maxIter; i = 1; % for plot funval = zeros(maxIter-1,1); iterNum = zeros(maxIter-1,1); while 1if i == maxIterbreak;end% solve xHxx = eval(Hx);Fxx = eval(Fx);x = quadprog(Hxx,Fxx,[],[],[],[],xlb,xub,[],options);% descend% solve yHyy = eval(Hy);Fyy = eval(Fy);y = quadprog(Hyy,Fyy,[],[],[],[],ylb,yub,[],options);%descend% update lambdalambda = lambda + rho*(2*x + 3*y -5); % ascend% rho = min(rhomax,beta*rho);funval(i) = compute_fval(x,y);iterNum(i) = i;i = i + 1; endplot(iterNum,funval,'-r');endfunction fval = compute_fval(x,y) fval = (x-1)^2 + (y-2)^2; end

結果：

[x,y]:0.53846,1.3077

從上圖我們可以看到，算法在第5次迭代后就幾乎收斂。

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