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具有约束约束的二次编程：基于问题的

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This example shows how to formulate and solve a scalable bound-constrained problem with a quadratic objective function. The example shows the solution behavior using several algorithms. The problem can have any number of variables; the number of variables is the scale. For the solver-based version of this example, seeQuadratic Minimization with Bound Constraints。

The objective function, as a function of the number of problem variablesn， 是

2 ∑ i = 1 n x i 2 - 2 ∑ i = 1 n - 1 x i x i + 1 - 2 x 1 - 2 x n 。

Create Problem

Create a problem variable namedxthat has 400 components. Also, create an expression namedobjecfor the objective function. Bound each variable below by 0 and above by 0.9, except allow x n to be unbounded.

n = 400; x = optimvar('x',n,'LowerBound',0,'UpperBound',0.9); x(n).LowerBound = -Inf; x(n).UpperBound = Inf; prevtime = 1:n-1; nexttime = 2:n; objec = 2*sum(x.^2) - 2*sum(x(nexttime).*x(prevtime)) - 2*x(1) - 2*x(end);

Create an optimization problem namedqprob。在问题中包括目标函数。

qprob = optimproblem('Objective'，objec）;

Create options that specify theQuadprog“信任区域反射”algorithm and no display. Create an initial point approximately centered between the bounds.

opts = optimoptions('quadprog','Algorithm',“信任区域反射”,'展示','off'）；x0 = 0.5*一个（n，1）;x00 = struct（'x',x0);

Solve Problem and Examine Solution

解决这个问题。

[SOL，QFVAL，QEXITFLAG，QOUTPUT] = SOLVE（QPROB，X00，'选项',opts);

Plot the solution.

plot(sol.x,'b-'）xlabel（'Index'）ylabel（'x(index)')

Figure contains an axes object. The axes object contains an object of type line.

Report the exit flag, the number of iterations, and the number of conjugate gradient iterations.

fprintf('Exit flag = %d, iterations = %d, cg iterations = %d\n',。。。double(qexitflag),qoutput.iterations,qoutput.cgiterations)
Exit flag = 3, iterations = 19, cg iterations = 1668

There were a lot of conjugate gradient iterations.

调整选项,提高效率

通过设置结合梯度迭代的数量SubproblemAlgorithm选项'factorization'。This option causes the solver to use a more expensive internal solution technique that eliminates conjugate gradient steps, for a net overall savings of time in this case.

opts.SubproblemAlgorithm ='factorization'; [sol2,qfval2,qexitflag2,qoutput2] = solve(qprob,x00,'选项',opts); fprintf('Exit flag = %d, iterations = %d, cg iterations = %d\n',。。。double(qexitflag2),qoutput2.iterations,qoutput2.cgiterations)
Exit flag = 3, iterations = 10, cg iterations = 0

The number of iterations and of conjugate gradient iterations decreased.

Compare Solutions With“内点”Solution

将这些解决方案与使用默认的解金宝搏官方网站决方案进行比较“内点”algorithm. The“内点”算法不使用初始点，因此请勿通过x00tosolve。

opts = optimoptions('quadprog','Algorithm','interior-point-convex','展示','off'）；[sol3，qfval3，qexitflag3，qoutput3] = solve（qprob，'选项',opts); fprintf('Exit flag = %d, iterations = %d, cg iterations = %d\n',。。。double(qexitflag3),qoutput3.iterations,0)
退出标志= 1，迭代= 8，CG迭代= 0
middle = floor(n/2); fprintf('The three solutions are slightly different.\nThe middle component is %f, %f, or %f.\n',。。。sol.x(middle),sol2.x(middle),sol3.x(middle))
The three solutions are slightly different. The middle component is 0.896796, 0.898676, or 0.857389.
fprintf('sol -sol2的相对标准为％f。\ n',norm(sol.x-sol2.x)/norm(sol.x))
The relative norm of sol - sol2 is 0.001445.
fprintf('The relative norm of sol2 - sol3 is %f.\n',norm(sol2.x-sol3.x)/norm(sol2.x))
The relative norm of sol2 - sol3 is 0.035894.
fprintf(['三个目标函数值为％f，％f和％f。\ n'。。。“内点”“算法”的准确性略低。],qfval,qfval2,qfval3)
The three objective function values are -1.985000, -1.985000, and -1.984963. The 'interior-point' algorithm is slightly less accurate.

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