SeDuMi教程(2)_线性规划的两种求解器的对比

对于标准的线性规划我们可以写成如下的形式，如果有不等式约束，则写出他的对偶问题：


我们以一个实例来看

解一下线性规划问题，通过linprog求解器和sedumi求解器分别求解。注意，使用sedumi求解前需要写出对偶形式，对对偶进行求解。

%%
clc
clear all
tic 
b=[1 4]';
A=[-1 1 0 1 1;0 0 -1 1 2];
c=[0 2 0 3.5 6];
[x, y, info]=sedumi(A,b,c)
toc
time = toc
%%
tic 
clear all
f = [-1;-4];
b = [0 2 0 3.5 6]';
A = [-1 1 0 1 1; 0 0 -1 1 2]';


options = optimoptions('linprog','Algorithm','dual-simplex','Display','iter','MaxIterations',20)

[x,fval,exitflag,output,lambda]  = linprog(f,A,b,[],[],[],[],options)
toc
time = toc

我们对求解器运行时间也做了计时，进行对比：

SeDuMi 1.3 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 2, order n = 6, dim = 6, blocks = 1
nnz(A) = 7 + 0, nnz(ADA) = 4, nnz(L) = 3
 it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
  0 :            4.58E+01 0.000
  1 :   8.27E+00 1.37E+01 0.000 0.2992 0.9000 0.9000   1.99  1  1  1.3E+00
  2 :   1.15E+01 3.16E+00 0.000 0.2304 0.9000 0.9000   1.81  1  1  2.4E-01
  3 :   1.19E+01 5.77E-01 0.000 0.1826 0.9000 0.9000   1.19  1  1  4.1E-02
  4 :   1.20E+01 2.97E-03 0.000 0.0051 0.9990 0.9990   1.01  1  1  
iter seconds digits       c*x               b*y
  4      0.2   Inf  1.2000000000e+01  1.2000000000e+01
|Ax-b| =   0.0e+00, [Ay-c]_+ =   0.0E+00, |x|=  2.2e+00, |y|=  3.0e+00

Detailed timing (sec)
   Pre          IPM          Post
6.900E-02    1.440E-01    1.700E-02    
Max-norms: ||b||=4, ||c|| = 6,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.80888.
x =
   (1,1)        1
   (5,1)        2
y =
     0
     3
info = 
  struct with fields:

         iter: 4
    feasratio: 1
         pinf: 0
         dinf: 0
       numerr: 0
       timing: [0.0690 0.1440 0.0170]
      wallsec: 0.2300
       cpusec: 0.2500
Elapsed time is 0.250347 seconds.
time =
    0.2504
options = 
  linprog options:

   Options used by current Algorithm ('dual-simplex'):
   (Other available algorithms: 'interior-point', 'interior-point-legacy')

   Set properties:
              Algorithm: 'dual-simplex'
                Display: 'iter'
          MaxIterations: 20
   Default properties:
    ConstraintTolerance: 1.0000e-04
                MaxTime: Inf
    OptimalityTolerance: 1.0000e-07

LP preprocessing removed 1 inequalities, 0 equalities,
0 variables, and 1 non-zero elements.

 Iter      Time            Fval  Primal Infeas    Dual Infeas  
    0     0.002    0.000000e+00   0.000000e+00   2.672696e+00  
    2     0.002   -1.200000e+01   0.000000e+00   0.000000e+00  

Optimal solution found.

x =
     0
     3
fval =
   -12
exitflag =
     1
output = 
  struct with fields:

         iterations: 2
    constrviolation: 0
            message: 'Optimal solution found.'
          algorithm: 'dual-simplex'
      firstorderopt: 0
lambda = 
  struct with fields:

      lower: [2×1 double]
      upper: [2×1 double]
      eqlin: []
    ineqlin: [5×1 double]
Elapsed time is 0.992132 seconds.
time =
    0.9922

linprog之前博客已经解释过，我们现在对sedumi做一次解释，因为对偶问题是最大化问题，这里求解信息中第一个b*y值可以看到在不断增大，在关注一个feasibility值，表示结果置信度，最后为0.999。gap是对偶间隙，这个对偶间隙要求越小越好，也可以看到，最后gap也缩小到e-3了。sedumi中还给出了计算时间，包括预处理时间，处理时间，后处理时间，这个求解过程占用时间都分别给出来了。

我们通过tic toc计时，发现sedumi求解速度比linprog快速很多。